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G = C23.27D28  order 448 = 26·7

6th non-split extension by C23 of D28 acting via D28/C28=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C23.27D28, C24.64D14, C23.16Dic14, (C23×C4).8D7, C28⋊7(C22⋊C4), (C22×C28)⋊14C4, (C2×C28).475D4, C4⋊2(C23.D7), (C22×C4)⋊8Dic7, C2.4(C28⋊7D4), (C23×C28).11C2, C22⋊2(C4⋊Dic7), C22.60(C2×D28), (C22×C14).26Q8, C14.79(C4⋊D4), C7⋊4(C23.7Q8), (C22×C14).143D4, (C22×C4).433D14, C14.68(C22⋊Q8), C23.31(C2×Dic7), C14.C42⋊24C2, C2.5(C28.48D4), C22.63(C4○D28), (C23×C14).99C22, C22.32(C2×Dic14), C23.303(C22×D7), C14.49(C42⋊C2), (C22×C28).484C22, (C22×C14).363C23, C22.50(C22×Dic7), (C22×Dic7).66C22, C2.12(C23.21D14), (C2×C14)⋊7(C4⋊C4), C14.56(C2×C4⋊C4), (C2×C4⋊Dic7)⋊16C2, C2.16(C2×C4⋊Dic7), (C2×C14).44(C2×Q8), (C2×C28).282(C2×C4), C2.6(C2×C23.D7), (C2×C14).549(C2×D4), C14.70(C2×C22⋊C4), (C2×C4).85(C2×Dic7), C22.87(C2×C7⋊D4), (C2×C14).91(C4○D4), (C2×C4).260(C7⋊D4), (C2×C23.D7).18C2, (C2×C14).193(C22×C4), (C22×C14).136(C2×C4), SmallGroup(448,746)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2×C14 — C23.27D28
Chief series
C1 — C7 — C14 — C2×C14 — C22×C14 — C22×Dic7 — C2×C4⋊Dic7 — C23.27D28
Lower central
C7 — C2×C14 — C23.27D28
Upper central
C1 — C23 — C23×C4

Generators and relations for C23.27D28
 G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e14=d, f2=b, ab=ba, faf-1=ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=de13 >

Subgroups: 772 in 234 conjugacy classes, 103 normal (25 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C7, C2×C4, C2×C4, C23, C23, C23, C14, C14, C14, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C24, Dic7, C28, C28, C2×C14, C2×C14, C2×C14, C2.C42, C2×C22⋊C4, C2×C4⋊C4, C23×C4, C2×Dic7, C2×C28, C2×C28, C22×C14, C22×C14, C22×C14, C23.7Q8, C4⋊Dic7, C23.D7, C22×Dic7, C22×C28, C22×C28, C22×C28, C23×C14, C14.C42, C2×C4⋊Dic7, C2×C23.D7, C23×C28, C23.27D28
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, D7, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×Q8, C4○D4, Dic7, D14, C2×C22⋊C4, C2×C4⋊C4, C42⋊C2, C4⋊D4, C22⋊Q8, Dic14, D28, C2×Dic7, C7⋊D4, C22×D7, C23.7Q8, C4⋊Dic7, C23.D7, C2×Dic14, C2×D28, C4○D28, C22×Dic7, C2×C7⋊D4, C28.48D4, C2×C4⋊Dic7, C23.21D14, C28⋊7D4, C2×C23.D7, C23.27D28

Smallest permutation representation of C23.27D28
►On 224 points
Generators in S224
(85 167)(86 168)(87 141)(88 142)(89 143)(90 144)(91 145)(92 146)(93 147)(94 148)(95 149)(96 150)(97 151)(98 152)(99 153)(100 154)(101 155)(102 156)(103 157)(104 158)(105 159)(106 160)(107 161)(108 162)(109 163)(110 164)(111 165)(112 166)(169 222)(170 223)(171 224)(172 197)(173 198)(174 199)(175 200)(176 201)(177 202)(178 203)(179 204)(180 205)(181 206)(182 207)(183 208)(184 209)(185 210)(186 211)(187 212)(188 213)(189 214)(190 215)(191 216)(192 217)(193 218)(194 219)(195 220)(196 221)

(1 121)(2 122)(3 123)(4 124)(5 125)(6 126)(7 127)(8 128)(9 129)(10 130)(11 131)(12 132)(13 133)(14 134)(15 135)(16 136)(17 137)(18 138)(19 139)(20 140)(21 113)(22 114)(23 115)(24 116)(25 117)(26 118)(27 119)(28 120)(29 61)(30 62)(31 63)(32 64)(33 65)(34 66)(35 67)(36 68)(37 69)(38 70)(39 71)(40 72)(41 73)(42 74)(43 75)(44 76)(45 77)(46 78)(47 79)(48 80)(49 81)(50 82)(51 83)(52 84)(53 57)(54 58)(55 59)(56 60)(85 203)(86 204)(87 205)(88 206)(89 207)(90 208)(91 209)(92 210)(93 211)(94 212)(95 213)(96 214)(97 215)(98 216)(99 217)(100 218)(101 219)(102 220)(103 221)(104 222)(105 223)(106 224)(107 197)(108 198)(109 199)(110 200)(111 201)(112 202)(141 180)(142 181)(143 182)(144 183)(145 184)(146 185)(147 186)(148 187)(149 188)(150 189)(151 190)(152 191)(153 192)(154 193)(155 194)(156 195)(157 196)(158 169)(159 170)(160 171)(161 172)(162 173)(163 174)(164 175)(165 176)(166 177)(167 178)(168 179)

(1 44)(2 45)(3 46)(4 47)(5 48)(6 49)(7 50)(8 51)(9 52)(10 53)(11 54)(12 55)(13 56)(14 29)(15 30)(16 31)(17 32)(18 33)(19 34)(20 35)(21 36)(22 37)(23 38)(24 39)(25 40)(26 41)(27 42)(28 43)(57 130)(58 131)(59 132)(60 133)(61 134)(62 135)(63 136)(64 137)(65 138)(66 139)(67 140)(68 113)(69 114)(70 115)(71 116)(72 117)(73 118)(74 119)(75 120)(76 121)(77 122)(78 123)(79 124)(80 125)(81 126)(82 127)(83 128)(84 129)(85 167)(86 168)(87 141)(88 142)(89 143)(90 144)(91 145)(92 146)(93 147)(94 148)(95 149)(96 150)(97 151)(98 152)(99 153)(100 154)(101 155)(102 156)(103 157)(104 158)(105 159)(106 160)(107 161)(108 162)(109 163)(110 164)(111 165)(112 166)(169 222)(170 223)(171 224)(172 197)(173 198)(174 199)(175 200)(176 201)(177 202)(178 203)(179 204)(180 205)(181 206)(182 207)(183 208)(184 209)(185 210)(186 211)(187 212)(188 213)(189 214)(190 215)(191 216)(192 217)(193 218)(194 219)(195 220)(196 221)

(1 15)(2 16)(3 17)(4 18)(5 19)(6 20)(7 21)(8 22)(9 23)(10 24)(11 25)(12 26)(13 27)(14 28)(29 43)(30 44)(31 45)(32 46)(33 47)(34 48)(35 49)(36 50)(37 51)(38 52)(39 53)(40 54)(41 55)(42 56)(57 71)(58 72)(59 73)(60 74)(61 75)(62 76)(63 77)(64 78)(65 79)(66 80)(67 81)(68 82)(69 83)(70 84)(85 99)(86 100)(87 101)(88 102)(89 103)(90 104)(91 105)(92 106)(93 107)(94 108)(95 109)(96 110)(97 111)(98 112)(113 127)(114 128)(115 129)(116 130)(117 131)(118 132)(119 133)(120 134)(121 135)(122 136)(123 137)(124 138)(125 139)(126 140)(141 155)(142 156)(143 157)(144 158)(145 159)(146 160)(147 161)(148 162)(149 163)(150 164)(151 165)(152 166)(153 167)(154 168)(169 183)(170 184)(171 185)(172 186)(173 187)(174 188)(175 189)(176 190)(177 191)(178 192)(179 193)(180 194)(181 195)(182 196)(197 211)(198 212)(199 213)(200 214)(201 215)(202 216)(203 217)(204 218)(205 219)(206 220)(207 221)(208 222)(209 223)(210 224)

(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28)(29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56)(57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84)(85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112)(113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140)(141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168)(169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196)(197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224)

(1 173 121 162)(2 172 122 161)(3 171 123 160)(4 170 124 159)(5 169 125 158)(6 196 126 157)(7 195 127 156)(8 194 128 155)(9 193 129 154)(10 192 130 153)(11 191 131 152)(12 190 132 151)(13 189 133 150)(14 188 134 149)(15 187 135 148)(16 186 136 147)(17 185 137 146)(18 184 138 145)(19 183 139 144)(20 182 140 143)(21 181 113 142)(22 180 114 141)(23 179 115 168)(24 178 116 167)(25 177 117 166)(26 176 118 165)(27 175 119 164)(28 174 120 163)(29 213 61 95)(30 212 62 94)(31 211 63 93)(32 210 64 92)(33 209 65 91)(34 208 66 90)(35 207 67 89)(36 206 68 88)(37 205 69 87)(38 204 70 86)(39 203 71 85)(40 202 72 112)(41 201 73 111)(42 200 74 110)(43 199 75 109)(44 198 76 108)(45 197 77 107)(46 224 78 106)(47 223 79 105)(48 222 80 104)(49 221 81 103)(50 220 82 102)(51 219 83 101)(52 218 84 100)(53 217 57 99)(54 216 58 98)(55 215 59 97)(56 214 60 96)
 
G:=sub<Sym(224)| (85,167)(86,168)(87,141)(88,142)(89,143)(90,144)(91,145)(92,146)(93,147)(94,148)(95,149)(96,150)(97,151)(98,152)(99,153)(100,154)(101,155)(102,156)(103,157)(104,158)(105,159)(106,160)(107,161)(108,162)(109,163)(110,164)(111,165)(112,166)(169,222)(170,223)(171,224)(172,197)(173,198)(174,199)(175,200)(176,201)(177,202)(178,203)(179,204)(180,205)(181,206)(182,207)(183,208)(184,209)(185,210)(186,211)(187,212)(188,213)(189,214)(190,215)(191,216)(192,217)(193,218)(194,219)(195,220)(196,221), (1,121)(2,122)(3,123)(4,124)(5,125)(6,126)(7,127)(8,128)(9,129)(10,130)(11,131)(12,132)(13,133)(14,134)(15,135)(16,136)(17,137)(18,138)(19,139)(20,140)(21,113)(22,114)(23,115)(24,116)(25,117)(26,118)(27,119)(28,120)(29,61)(30,62)(31,63)(32,64)(33,65)(34,66)(35,67)(36,68)(37,69)(38,70)(39,71)(40,72)(41,73)(42,74)(43,75)(44,76)(45,77)(46,78)(47,79)(48,80)(49,81)(50,82)(51,83)(52,84)(53,57)(54,58)(55,59)(56,60)(85,203)(86,204)(87,205)(88,206)(89,207)(90,208)(91,209)(92,210)(93,211)(94,212)(95,213)(96,214)(97,215)(98,216)(99,217)(100,218)(101,219)(102,220)(103,221)(104,222)(105,223)(106,224)(107,197)(108,198)(109,199)(110,200)(111,201)(112,202)(141,180)(142,181)(143,182)(144,183)(145,184)(146,185)(147,186)(148,187)(149,188)(150,189)(151,190)(152,191)(153,192)(154,193)(155,194)(156,195)(157,196)(158,169)(159,170)(160,171)(161,172)(162,173)(163,174)(164,175)(165,176)(166,177)(167,178)(168,179), (1,44)(2,45)(3,46)(4,47)(5,48)(6,49)(7,50)(8,51)(9,52)(10,53)(11,54)(12,55)(13,56)(14,29)(15,30)(16,31)(17,32)(18,33)(19,34)(20,35)(21,36)(22,37)(23,38)(24,39)(25,40)(26,41)(27,42)(28,43)(57,130)(58,131)(59,132)(60,133)(61,134)(62,135)(63,136)(64,137)(65,138)(66,139)(67,140)(68,113)(69,114)(70,115)(71,116)(72,117)(73,118)(74,119)(75,120)(76,121)(77,122)(78,123)(79,124)(80,125)(81,126)(82,127)(83,128)(84,129)(85,167)(86,168)(87,141)(88,142)(89,143)(90,144)(91,145)(92,146)(93,147)(94,148)(95,149)(96,150)(97,151)(98,152)(99,153)(100,154)(101,155)(102,156)(103,157)(104,158)(105,159)(106,160)(107,161)(108,162)(109,163)(110,164)(111,165)(112,166)(169,222)(170,223)(171,224)(172,197)(173,198)(174,199)(175,200)(176,201)(177,202)(178,203)(179,204)(180,205)(181,206)(182,207)(183,208)(184,209)(185,210)(186,211)(187,212)(188,213)(189,214)(190,215)(191,216)(192,217)(193,218)(194,219)(195,220)(196,221), (1,15)(2,16)(3,17)(4,18)(5,19)(6,20)(7,21)(8,22)(9,23)(10,24)(11,25)(12,26)(13,27)(14,28)(29,43)(30,44)(31,45)(32,46)(33,47)(34,48)(35,49)(36,50)(37,51)(38,52)(39,53)(40,54)(41,55)(42,56)(57,71)(58,72)(59,73)(60,74)(61,75)(62,76)(63,77)(64,78)(65,79)(66,80)(67,81)(68,82)(69,83)(70,84)(85,99)(86,100)(87,101)(88,102)(89,103)(90,104)(91,105)(92,106)(93,107)(94,108)(95,109)(96,110)(97,111)(98,112)(113,127)(114,128)(115,129)(116,130)(117,131)(118,132)(119,133)(120,134)(121,135)(122,136)(123,137)(124,138)(125,139)(126,140)(141,155)(142,156)(143,157)(144,158)(145,159)(146,160)(147,161)(148,162)(149,163)(150,164)(151,165)(152,166)(153,167)(154,168)(169,183)(170,184)(171,185)(172,186)(173,187)(174,188)(175,189)(176,190)(177,191)(178,192)(179,193)(180,194)(181,195)(182,196)(197,211)(198,212)(199,213)(200,214)(201,215)(202,216)(203,217)(204,218)(205,219)(206,220)(207,221)(208,222)(209,223)(210,224), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28)(29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84)(85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112)(113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140)(141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168)(169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196)(197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224), (1,173,121,162)(2,172,122,161)(3,171,123,160)(4,170,124,159)(5,169,125,158)(6,196,126,157)(7,195,127,156)(8,194,128,155)(9,193,129,154)(10,192,130,153)(11,191,131,152)(12,190,132,151)(13,189,133,150)(14,188,134,149)(15,187,135,148)(16,186,136,147)(17,185,137,146)(18,184,138,145)(19,183,139,144)(20,182,140,143)(21,181,113,142)(22,180,114,141)(23,179,115,168)(24,178,116,167)(25,177,117,166)(26,176,118,165)(27,175,119,164)(28,174,120,163)(29,213,61,95)(30,212,62,94)(31,211,63,93)(32,210,64,92)(33,209,65,91)(34,208,66,90)(35,207,67,89)(36,206,68,88)(37,205,69,87)(38,204,70,86)(39,203,71,85)(40,202,72,112)(41,201,73,111)(42,200,74,110)(43,199,75,109)(44,198,76,108)(45,197,77,107)(46,224,78,106)(47,223,79,105)(48,222,80,104)(49,221,81,103)(50,220,82,102)(51,219,83,101)(52,218,84,100)(53,217,57,99)(54,216,58,98)(55,215,59,97)(56,214,60,96)>;
 
G:=Group( (85,167)(86,168)(87,141)(88,142)(89,143)(90,144)(91,145)(92,146)(93,147)(94,148)(95,149)(96,150)(97,151)(98,152)(99,153)(100,154)(101,155)(102,156)(103,157)(104,158)(105,159)(106,160)(107,161)(108,162)(109,163)(110,164)(111,165)(112,166)(169,222)(170,223)(171,224)(172,197)(173,198)(174,199)(175,200)(176,201)(177,202)(178,203)(179,204)(180,205)(181,206)(182,207)(183,208)(184,209)(185,210)(186,211)(187,212)(188,213)(189,214)(190,215)(191,216)(192,217)(193,218)(194,219)(195,220)(196,221), (1,121)(2,122)(3,123)(4,124)(5,125)(6,126)(7,127)(8,128)(9,129)(10,130)(11,131)(12,132)(13,133)(14,134)(15,135)(16,136)(17,137)(18,138)(19,139)(20,140)(21,113)(22,114)(23,115)(24,116)(25,117)(26,118)(27,119)(28,120)(29,61)(30,62)(31,63)(32,64)(33,65)(34,66)(35,67)(36,68)(37,69)(38,70)(39,71)(40,72)(41,73)(42,74)(43,75)(44,76)(45,77)(46,78)(47,79)(48,80)(49,81)(50,82)(51,83)(52,84)(53,57)(54,58)(55,59)(56,60)(85,203)(86,204)(87,205)(88,206)(89,207)(90,208)(91,209)(92,210)(93,211)(94,212)(95,213)(96,214)(97,215)(98,216)(99,217)(100,218)(101,219)(102,220)(103,221)(104,222)(105,223)(106,224)(107,197)(108,198)(109,199)(110,200)(111,201)(112,202)(141,180)(142,181)(143,182)(144,183)(145,184)(146,185)(147,186)(148,187)(149,188)(150,189)(151,190)(152,191)(153,192)(154,193)(155,194)(156,195)(157,196)(158,169)(159,170)(160,171)(161,172)(162,173)(163,174)(164,175)(165,176)(166,177)(167,178)(168,179), (1,44)(2,45)(3,46)(4,47)(5,48)(6,49)(7,50)(8,51)(9,52)(10,53)(11,54)(12,55)(13,56)(14,29)(15,30)(16,31)(17,32)(18,33)(19,34)(20,35)(21,36)(22,37)(23,38)(24,39)(25,40)(26,41)(27,42)(28,43)(57,130)(58,131)(59,132)(60,133)(61,134)(62,135)(63,136)(64,137)(65,138)(66,139)(67,140)(68,113)(69,114)(70,115)(71,116)(72,117)(73,118)(74,119)(75,120)(76,121)(77,122)(78,123)(79,124)(80,125)(81,126)(82,127)(83,128)(84,129)(85,167)(86,168)(87,141)(88,142)(89,143)(90,144)(91,145)(92,146)(93,147)(94,148)(95,149)(96,150)(97,151)(98,152)(99,153)(100,154)(101,155)(102,156)(103,157)(104,158)(105,159)(106,160)(107,161)(108,162)(109,163)(110,164)(111,165)(112,166)(169,222)(170,223)(171,224)(172,197)(173,198)(174,199)(175,200)(176,201)(177,202)(178,203)(179,204)(180,205)(181,206)(182,207)(183,208)(184,209)(185,210)(186,211)(187,212)(188,213)(189,214)(190,215)(191,216)(192,217)(193,218)(194,219)(195,220)(196,221), (1,15)(2,16)(3,17)(4,18)(5,19)(6,20)(7,21)(8,22)(9,23)(10,24)(11,25)(12,26)(13,27)(14,28)(29,43)(30,44)(31,45)(32,46)(33,47)(34,48)(35,49)(36,50)(37,51)(38,52)(39,53)(40,54)(41,55)(42,56)(57,71)(58,72)(59,73)(60,74)(61,75)(62,76)(63,77)(64,78)(65,79)(66,80)(67,81)(68,82)(69,83)(70,84)(85,99)(86,100)(87,101)(88,102)(89,103)(90,104)(91,105)(92,106)(93,107)(94,108)(95,109)(96,110)(97,111)(98,112)(113,127)(114,128)(115,129)(116,130)(117,131)(118,132)(119,133)(120,134)(121,135)(122,136)(123,137)(124,138)(125,139)(126,140)(141,155)(142,156)(143,157)(144,158)(145,159)(146,160)(147,161)(148,162)(149,163)(150,164)(151,165)(152,166)(153,167)(154,168)(169,183)(170,184)(171,185)(172,186)(173,187)(174,188)(175,189)(176,190)(177,191)(178,192)(179,193)(180,194)(181,195)(182,196)(197,211)(198,212)(199,213)(200,214)(201,215)(202,216)(203,217)(204,218)(205,219)(206,220)(207,221)(208,222)(209,223)(210,224), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28)(29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84)(85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112)(113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140)(141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168)(169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196)(197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224), (1,173,121,162)(2,172,122,161)(3,171,123,160)(4,170,124,159)(5,169,125,158)(6,196,126,157)(7,195,127,156)(8,194,128,155)(9,193,129,154)(10,192,130,153)(11,191,131,152)(12,190,132,151)(13,189,133,150)(14,188,134,149)(15,187,135,148)(16,186,136,147)(17,185,137,146)(18,184,138,145)(19,183,139,144)(20,182,140,143)(21,181,113,142)(22,180,114,141)(23,179,115,168)(24,178,116,167)(25,177,117,166)(26,176,118,165)(27,175,119,164)(28,174,120,163)(29,213,61,95)(30,212,62,94)(31,211,63,93)(32,210,64,92)(33,209,65,91)(34,208,66,90)(35,207,67,89)(36,206,68,88)(37,205,69,87)(38,204,70,86)(39,203,71,85)(40,202,72,112)(41,201,73,111)(42,200,74,110)(43,199,75,109)(44,198,76,108)(45,197,77,107)(46,224,78,106)(47,223,79,105)(48,222,80,104)(49,221,81,103)(50,220,82,102)(51,219,83,101)(52,218,84,100)(53,217,57,99)(54,216,58,98)(55,215,59,97)(56,214,60,96) );
 
G=PermutationGroup([[(85,167),(86,168),(87,141),(88,142),(89,143),(90,144),(91,145),(92,146),(93,147),(94,148),(95,149),(96,150),(97,151),(98,152),(99,153),(100,154),(101,155),(102,156),(103,157),(104,158),(105,159),(106,160),(107,161),(108,162),(109,163),(110,164),(111,165),(112,166),(169,222),(170,223),(171,224),(172,197),(173,198),(174,199),(175,200),(176,201),(177,202),(178,203),(179,204),(180,205),(181,206),(182,207),(183,208),(184,209),(185,210),(186,211),(187,212),(188,213),(189,214),(190,215),(191,216),(192,217),(193,218),(194,219),(195,220),(196,221)], [(1,121),(2,122),(3,123),(4,124),(5,125),(6,126),(7,127),(8,128),(9,129),(10,130),(11,131),(12,132),(13,133),(14,134),(15,135),(16,136),(17,137),(18,138),(19,139),(20,140),(21,113),(22,114),(23,115),(24,116),(25,117),(26,118),(27,119),(28,120),(29,61),(30,62),(31,63),(32,64),(33,65),(34,66),(35,67),(36,68),(37,69),(38,70),(39,71),(40,72),(41,73),(42,74),(43,75),(44,76),(45,77),(46,78),(47,79),(48,80),(49,81),(50,82),(51,83),(52,84),(53,57),(54,58),(55,59),(56,60),(85,203),(86,204),(87,205),(88,206),(89,207),(90,208),(91,209),(92,210),(93,211),(94,212),(95,213),(96,214),(97,215),(98,216),(99,217),(100,218),(101,219),(102,220),(103,221),(104,222),(105,223),(106,224),(107,197),(108,198),(109,199),(110,200),(111,201),(112,202),(141,180),(142,181),(143,182),(144,183),(145,184),(146,185),(147,186),(148,187),(149,188),(150,189),(151,190),(152,191),(153,192),(154,193),(155,194),(156,195),(157,196),(158,169),(159,170),(160,171),(161,172),(162,173),(163,174),(164,175),(165,176),(166,177),(167,178),(168,179)], [(1,44),(2,45),(3,46),(4,47),(5,48),(6,49),(7,50),(8,51),(9,52),(10,53),(11,54),(12,55),(13,56),(14,29),(15,30),(16,31),(17,32),(18,33),(19,34),(20,35),(21,36),(22,37),(23,38),(24,39),(25,40),(26,41),(27,42),(28,43),(57,130),(58,131),(59,132),(60,133),(61,134),(62,135),(63,136),(64,137),(65,138),(66,139),(67,140),(68,113),(69,114),(70,115),(71,116),(72,117),(73,118),(74,119),(75,120),(76,121),(77,122),(78,123),(79,124),(80,125),(81,126),(82,127),(83,128),(84,129),(85,167),(86,168),(87,141),(88,142),(89,143),(90,144),(91,145),(92,146),(93,147),(94,148),(95,149),(96,150),(97,151),(98,152),(99,153),(100,154),(101,155),(102,156),(103,157),(104,158),(105,159),(106,160),(107,161),(108,162),(109,163),(110,164),(111,165),(112,166),(169,222),(170,223),(171,224),(172,197),(173,198),(174,199),(175,200),(176,201),(177,202),(178,203),(179,204),(180,205),(181,206),(182,207),(183,208),(184,209),(185,210),(186,211),(187,212),(188,213),(189,214),(190,215),(191,216),(192,217),(193,218),(194,219),(195,220),(196,221)], [(1,15),(2,16),(3,17),(4,18),(5,19),(6,20),(7,21),(8,22),(9,23),(10,24),(11,25),(12,26),(13,27),(14,28),(29,43),(30,44),(31,45),(32,46),(33,47),(34,48),(35,49),(36,50),(37,51),(38,52),(39,53),(40,54),(41,55),(42,56),(57,71),(58,72),(59,73),(60,74),(61,75),(62,76),(63,77),(64,78),(65,79),(66,80),(67,81),(68,82),(69,83),(70,84),(85,99),(86,100),(87,101),(88,102),(89,103),(90,104),(91,105),(92,106),(93,107),(94,108),(95,109),(96,110),(97,111),(98,112),(113,127),(114,128),(115,129),(116,130),(117,131),(118,132),(119,133),(120,134),(121,135),(122,136),(123,137),(124,138),(125,139),(126,140),(141,155),(142,156),(143,157),(144,158),(145,159),(146,160),(147,161),(148,162),(149,163),(150,164),(151,165),(152,166),(153,167),(154,168),(169,183),(170,184),(171,185),(172,186),(173,187),(174,188),(175,189),(176,190),(177,191),(178,192),(179,193),(180,194),(181,195),(182,196),(197,211),(198,212),(199,213),(200,214),(201,215),(202,216),(203,217),(204,218),(205,219),(206,220),(207,221),(208,222),(209,223),(210,224)], [(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28),(29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56),(57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84),(85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112),(113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140),(141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168),(169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196),(197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224)], [(1,173,121,162),(2,172,122,161),(3,171,123,160),(4,170,124,159),(5,169,125,158),(6,196,126,157),(7,195,127,156),(8,194,128,155),(9,193,129,154),(10,192,130,153),(11,191,131,152),(12,190,132,151),(13,189,133,150),(14,188,134,149),(15,187,135,148),(16,186,136,147),(17,185,137,146),(18,184,138,145),(19,183,139,144),(20,182,140,143),(21,181,113,142),(22,180,114,141),(23,179,115,168),(24,178,116,167),(25,177,117,166),(26,176,118,165),(27,175,119,164),(28,174,120,163),(29,213,61,95),(30,212,62,94),(31,211,63,93),(32,210,64,92),(33,209,65,91),(34,208,66,90),(35,207,67,89),(36,206,68,88),(37,205,69,87),(38,204,70,86),(39,203,71,85),(40,202,72,112),(41,201,73,111),(42,200,74,110),(43,199,75,109),(44,198,76,108),(45,197,77,107),(46,224,78,106),(47,223,79,105),(48,222,80,104),(49,221,81,103),(50,220,82,102),(51,219,83,101),(52,218,84,100),(53,217,57,99),(54,216,58,98),(55,215,59,97),(56,214,60,96)]])
 

124 conjugacy classes

class 1 2A···2G2H2I2J2K4A···4H4I···4P7A7B7C14A···14AS28A···28AV
order12···222224···44···477714···1428···28
size11···122222···228···282222···22···2

124 irreducible representations

dim111111222222222222
type+++++++-+-++-+
imageC1C2C2C2C2C4D4D4Q8D7C4○D4Dic7D14D14C7⋊D4Dic14D28C4○D28
kernelC23.27D28C14.C42C2×C4⋊Dic7C2×C23.D7C23×C28C22×C28C2×C28C22×C14C22×C14C23×C4C2×C14C22×C4C22×C4C24C2×C4C23C23C22
# reps12221842234126324121224

Matrix representation of C23.27D28 ►in GL6(𝔽29)

100000
0280000
001000
0002800
000010
000001
,
2800000
0280000
001000
000100
000010
000001
,
2800000
0280000
0028000
0002800
000010
000001
,
100000
010000
001000
000100
0000280
0000028
,
700000
0250000
006000
000500
0000140
0000027
,
0250000
2200000
000500
006000
000002
0000150

G:=sub<GL(6,GF(29))| [1,0,0,0,0,0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[7,0,0,0,0,0,0,25,0,0,0,0,0,0,6,0,0,0,0,0,0,5,0,0,0,0,0,0,14,0,0,0,0,0,0,27],[0,22,0,0,0,0,25,0,0,0,0,0,0,0,0,6,0,0,0,0,5,0,0,0,0,0,0,0,0,15,0,0,0,0,2,0] >;
 

C23.27D28 in GAP, Magma, Sage, TeX 

C_2^3._{27}D_{28}
 
% in TeX
 
G:=Group("C2^3.27D28");
 
// GroupNames label
 
G:=SmallGroup(448,746);
 
// by ID
 
G=gap.SmallGroup(448,746);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,56,477,232,422,18822]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^14=d,f^2=b,a*b=b*a,f*a*f^-1=a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=d*e^13>;
 
// generators/relations
 

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