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G = C23.28D30order 480 = 25·3·5

4th non-split extension by C23 of D30 acting via D30/C30=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C23.28D30, (C22×C20)⋊9S3, (C22×C60)⋊6C2, (C2×C4).68D30, (C22×C4)⋊5D15, (C22×C12)⋊5D5, D303C42C2, C30.377(C2×D4), (C2×C30).144D4, (C2×C20).383D6, C30.4Q83C2, (C2×C12).384D10, C30.38D46C2, C30.178(C4○D4), C6.106(C4○D20), (C2×C60).466C22, (C2×C30).303C23, (C22×C6).120D10, (C22×C10).138D6, C55(C23.28D6), C22.9(C157D4), C10.106(C4○D12), C35(C23.23D10), C1534(C22.D4), C2.18(D6011C2), C22.55(C22×D15), (C22×C30).143C22, (C2×Dic15).14C22, (C22×D15).10C22, C2.6(C2×C157D4), (C2×C157D4).6C2, C6.101(C2×C5⋊D4), (C2×C6).76(C5⋊D4), C10.101(C2×C3⋊D4), (C2×C10).76(C3⋊D4), (C2×C6).299(C22×D5), (C2×C10).298(C22×S3), SmallGroup(480,894)

Series: Derived Chief Lower central Upper central

C1C2×C30 — C23.28D30
C1C5C15C30C2×C30C22×D15C2×C157D4 — C23.28D30
C15C2×C30 — C23.28D30
C1C22C22×C4

Generators and relations for C23.28D30
 G = < a,b,c,d,e | a2=b2=c2=1, d30=c, e2=cb=bc, ab=ba, eae-1=ac=ca, ad=da, bd=db, be=eb, cd=dc, ce=ec, ede-1=bd29 >

Subgroups: 852 in 156 conjugacy classes, 55 normal (25 characteristic)
C1, C2, C2 [×2], C2 [×3], C3, C4 [×5], C22, C22 [×2], C22 [×5], C5, S3, C6, C6 [×2], C6 [×2], C2×C4 [×2], C2×C4 [×5], D4 [×2], C23, C23, D5, C10, C10 [×2], C10 [×2], Dic3 [×3], C12 [×2], D6 [×3], C2×C6, C2×C6 [×2], C2×C6 [×2], C15, C22⋊C4 [×3], C4⋊C4 [×2], C22×C4, C2×D4, Dic5 [×3], C20 [×2], D10 [×3], C2×C10, C2×C10 [×2], C2×C10 [×2], C2×Dic3 [×3], C3⋊D4 [×2], C2×C12 [×2], C2×C12 [×2], C22×S3, C22×C6, D15, C30, C30 [×2], C30 [×2], C22.D4, C2×Dic5 [×3], C5⋊D4 [×2], C2×C20 [×2], C2×C20 [×2], C22×D5, C22×C10, Dic3⋊C4 [×2], D6⋊C4 [×2], C6.D4, C2×C3⋊D4, C22×C12, Dic15 [×3], C60 [×2], D30 [×3], C2×C30, C2×C30 [×2], C2×C30 [×2], C10.D4 [×2], D10⋊C4 [×2], C23.D5, C2×C5⋊D4, C22×C20, C23.28D6, C2×Dic15, C2×Dic15 [×2], C157D4 [×2], C2×C60 [×2], C2×C60 [×2], C22×D15, C22×C30, C23.23D10, C30.4Q8 [×2], D303C4 [×2], C30.38D4, C2×C157D4, C22×C60, C23.28D30
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D5, D6 [×3], C2×D4, C4○D4 [×2], D10 [×3], C3⋊D4 [×2], C22×S3, D15, C22.D4, C5⋊D4 [×2], C22×D5, C4○D12 [×2], C2×C3⋊D4, D30 [×3], C4○D20 [×2], C2×C5⋊D4, C23.28D6, C157D4 [×2], C22×D15, C23.23D10, D6011C2 [×2], C2×C157D4, C23.28D30

Smallest permutation representation of C23.28D30
On 240 points
Generators in S240
(61 91)(62 92)(63 93)(64 94)(65 95)(66 96)(67 97)(68 98)(69 99)(70 100)(71 101)(72 102)(73 103)(74 104)(75 105)(76 106)(77 107)(78 108)(79 109)(80 110)(81 111)(82 112)(83 113)(84 114)(85 115)(86 116)(87 117)(88 118)(89 119)(90 120)(121 151)(122 152)(123 153)(124 154)(125 155)(126 156)(127 157)(128 158)(129 159)(130 160)(131 161)(132 162)(133 163)(134 164)(135 165)(136 166)(137 167)(138 168)(139 169)(140 170)(141 171)(142 172)(143 173)(144 174)(145 175)(146 176)(147 177)(148 178)(149 179)(150 180)
(1 227)(2 228)(3 229)(4 230)(5 231)(6 232)(7 233)(8 234)(9 235)(10 236)(11 237)(12 238)(13 239)(14 240)(15 181)(16 182)(17 183)(18 184)(19 185)(20 186)(21 187)(22 188)(23 189)(24 190)(25 191)(26 192)(27 193)(28 194)(29 195)(30 196)(31 197)(32 198)(33 199)(34 200)(35 201)(36 202)(37 203)(38 204)(39 205)(40 206)(41 207)(42 208)(43 209)(44 210)(45 211)(46 212)(47 213)(48 214)(49 215)(50 216)(51 217)(52 218)(53 219)(54 220)(55 221)(56 222)(57 223)(58 224)(59 225)(60 226)(61 145)(62 146)(63 147)(64 148)(65 149)(66 150)(67 151)(68 152)(69 153)(70 154)(71 155)(72 156)(73 157)(74 158)(75 159)(76 160)(77 161)(78 162)(79 163)(80 164)(81 165)(82 166)(83 167)(84 168)(85 169)(86 170)(87 171)(88 172)(89 173)(90 174)(91 175)(92 176)(93 177)(94 178)(95 179)(96 180)(97 121)(98 122)(99 123)(100 124)(101 125)(102 126)(103 127)(104 128)(105 129)(106 130)(107 131)(108 132)(109 133)(110 134)(111 135)(112 136)(113 137)(114 138)(115 139)(116 140)(117 141)(118 142)(119 143)(120 144)
(1 31)(2 32)(3 33)(4 34)(5 35)(6 36)(7 37)(8 38)(9 39)(10 40)(11 41)(12 42)(13 43)(14 44)(15 45)(16 46)(17 47)(18 48)(19 49)(20 50)(21 51)(22 52)(23 53)(24 54)(25 55)(26 56)(27 57)(28 58)(29 59)(30 60)(61 91)(62 92)(63 93)(64 94)(65 95)(66 96)(67 97)(68 98)(69 99)(70 100)(71 101)(72 102)(73 103)(74 104)(75 105)(76 106)(77 107)(78 108)(79 109)(80 110)(81 111)(82 112)(83 113)(84 114)(85 115)(86 116)(87 117)(88 118)(89 119)(90 120)(121 151)(122 152)(123 153)(124 154)(125 155)(126 156)(127 157)(128 158)(129 159)(130 160)(131 161)(132 162)(133 163)(134 164)(135 165)(136 166)(137 167)(138 168)(139 169)(140 170)(141 171)(142 172)(143 173)(144 174)(145 175)(146 176)(147 177)(148 178)(149 179)(150 180)(181 211)(182 212)(183 213)(184 214)(185 215)(186 216)(187 217)(188 218)(189 219)(190 220)(191 221)(192 222)(193 223)(194 224)(195 225)(196 226)(197 227)(198 228)(199 229)(200 230)(201 231)(202 232)(203 233)(204 234)(205 235)(206 236)(207 237)(208 238)(209 239)(210 240)
(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)(181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240)
(1 163 197 109)(2 108 198 162)(3 161 199 107)(4 106 200 160)(5 159 201 105)(6 104 202 158)(7 157 203 103)(8 102 204 156)(9 155 205 101)(10 100 206 154)(11 153 207 99)(12 98 208 152)(13 151 209 97)(14 96 210 150)(15 149 211 95)(16 94 212 148)(17 147 213 93)(18 92 214 146)(19 145 215 91)(20 90 216 144)(21 143 217 89)(22 88 218 142)(23 141 219 87)(24 86 220 140)(25 139 221 85)(26 84 222 138)(27 137 223 83)(28 82 224 136)(29 135 225 81)(30 80 226 134)(31 133 227 79)(32 78 228 132)(33 131 229 77)(34 76 230 130)(35 129 231 75)(36 74 232 128)(37 127 233 73)(38 72 234 126)(39 125 235 71)(40 70 236 124)(41 123 237 69)(42 68 238 122)(43 121 239 67)(44 66 240 180)(45 179 181 65)(46 64 182 178)(47 177 183 63)(48 62 184 176)(49 175 185 61)(50 120 186 174)(51 173 187 119)(52 118 188 172)(53 171 189 117)(54 116 190 170)(55 169 191 115)(56 114 192 168)(57 167 193 113)(58 112 194 166)(59 165 195 111)(60 110 196 164)

G:=sub<Sym(240)| (61,91)(62,92)(63,93)(64,94)(65,95)(66,96)(67,97)(68,98)(69,99)(70,100)(71,101)(72,102)(73,103)(74,104)(75,105)(76,106)(77,107)(78,108)(79,109)(80,110)(81,111)(82,112)(83,113)(84,114)(85,115)(86,116)(87,117)(88,118)(89,119)(90,120)(121,151)(122,152)(123,153)(124,154)(125,155)(126,156)(127,157)(128,158)(129,159)(130,160)(131,161)(132,162)(133,163)(134,164)(135,165)(136,166)(137,167)(138,168)(139,169)(140,170)(141,171)(142,172)(143,173)(144,174)(145,175)(146,176)(147,177)(148,178)(149,179)(150,180), (1,227)(2,228)(3,229)(4,230)(5,231)(6,232)(7,233)(8,234)(9,235)(10,236)(11,237)(12,238)(13,239)(14,240)(15,181)(16,182)(17,183)(18,184)(19,185)(20,186)(21,187)(22,188)(23,189)(24,190)(25,191)(26,192)(27,193)(28,194)(29,195)(30,196)(31,197)(32,198)(33,199)(34,200)(35,201)(36,202)(37,203)(38,204)(39,205)(40,206)(41,207)(42,208)(43,209)(44,210)(45,211)(46,212)(47,213)(48,214)(49,215)(50,216)(51,217)(52,218)(53,219)(54,220)(55,221)(56,222)(57,223)(58,224)(59,225)(60,226)(61,145)(62,146)(63,147)(64,148)(65,149)(66,150)(67,151)(68,152)(69,153)(70,154)(71,155)(72,156)(73,157)(74,158)(75,159)(76,160)(77,161)(78,162)(79,163)(80,164)(81,165)(82,166)(83,167)(84,168)(85,169)(86,170)(87,171)(88,172)(89,173)(90,174)(91,175)(92,176)(93,177)(94,178)(95,179)(96,180)(97,121)(98,122)(99,123)(100,124)(101,125)(102,126)(103,127)(104,128)(105,129)(106,130)(107,131)(108,132)(109,133)(110,134)(111,135)(112,136)(113,137)(114,138)(115,139)(116,140)(117,141)(118,142)(119,143)(120,144), (1,31)(2,32)(3,33)(4,34)(5,35)(6,36)(7,37)(8,38)(9,39)(10,40)(11,41)(12,42)(13,43)(14,44)(15,45)(16,46)(17,47)(18,48)(19,49)(20,50)(21,51)(22,52)(23,53)(24,54)(25,55)(26,56)(27,57)(28,58)(29,59)(30,60)(61,91)(62,92)(63,93)(64,94)(65,95)(66,96)(67,97)(68,98)(69,99)(70,100)(71,101)(72,102)(73,103)(74,104)(75,105)(76,106)(77,107)(78,108)(79,109)(80,110)(81,111)(82,112)(83,113)(84,114)(85,115)(86,116)(87,117)(88,118)(89,119)(90,120)(121,151)(122,152)(123,153)(124,154)(125,155)(126,156)(127,157)(128,158)(129,159)(130,160)(131,161)(132,162)(133,163)(134,164)(135,165)(136,166)(137,167)(138,168)(139,169)(140,170)(141,171)(142,172)(143,173)(144,174)(145,175)(146,176)(147,177)(148,178)(149,179)(150,180)(181,211)(182,212)(183,213)(184,214)(185,215)(186,216)(187,217)(188,218)(189,219)(190,220)(191,221)(192,222)(193,223)(194,224)(195,225)(196,226)(197,227)(198,228)(199,229)(200,230)(201,231)(202,232)(203,233)(204,234)(205,235)(206,236)(207,237)(208,238)(209,239)(210,240), (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)(181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240), (1,163,197,109)(2,108,198,162)(3,161,199,107)(4,106,200,160)(5,159,201,105)(6,104,202,158)(7,157,203,103)(8,102,204,156)(9,155,205,101)(10,100,206,154)(11,153,207,99)(12,98,208,152)(13,151,209,97)(14,96,210,150)(15,149,211,95)(16,94,212,148)(17,147,213,93)(18,92,214,146)(19,145,215,91)(20,90,216,144)(21,143,217,89)(22,88,218,142)(23,141,219,87)(24,86,220,140)(25,139,221,85)(26,84,222,138)(27,137,223,83)(28,82,224,136)(29,135,225,81)(30,80,226,134)(31,133,227,79)(32,78,228,132)(33,131,229,77)(34,76,230,130)(35,129,231,75)(36,74,232,128)(37,127,233,73)(38,72,234,126)(39,125,235,71)(40,70,236,124)(41,123,237,69)(42,68,238,122)(43,121,239,67)(44,66,240,180)(45,179,181,65)(46,64,182,178)(47,177,183,63)(48,62,184,176)(49,175,185,61)(50,120,186,174)(51,173,187,119)(52,118,188,172)(53,171,189,117)(54,116,190,170)(55,169,191,115)(56,114,192,168)(57,167,193,113)(58,112,194,166)(59,165,195,111)(60,110,196,164)>;

G:=Group( (61,91)(62,92)(63,93)(64,94)(65,95)(66,96)(67,97)(68,98)(69,99)(70,100)(71,101)(72,102)(73,103)(74,104)(75,105)(76,106)(77,107)(78,108)(79,109)(80,110)(81,111)(82,112)(83,113)(84,114)(85,115)(86,116)(87,117)(88,118)(89,119)(90,120)(121,151)(122,152)(123,153)(124,154)(125,155)(126,156)(127,157)(128,158)(129,159)(130,160)(131,161)(132,162)(133,163)(134,164)(135,165)(136,166)(137,167)(138,168)(139,169)(140,170)(141,171)(142,172)(143,173)(144,174)(145,175)(146,176)(147,177)(148,178)(149,179)(150,180), (1,227)(2,228)(3,229)(4,230)(5,231)(6,232)(7,233)(8,234)(9,235)(10,236)(11,237)(12,238)(13,239)(14,240)(15,181)(16,182)(17,183)(18,184)(19,185)(20,186)(21,187)(22,188)(23,189)(24,190)(25,191)(26,192)(27,193)(28,194)(29,195)(30,196)(31,197)(32,198)(33,199)(34,200)(35,201)(36,202)(37,203)(38,204)(39,205)(40,206)(41,207)(42,208)(43,209)(44,210)(45,211)(46,212)(47,213)(48,214)(49,215)(50,216)(51,217)(52,218)(53,219)(54,220)(55,221)(56,222)(57,223)(58,224)(59,225)(60,226)(61,145)(62,146)(63,147)(64,148)(65,149)(66,150)(67,151)(68,152)(69,153)(70,154)(71,155)(72,156)(73,157)(74,158)(75,159)(76,160)(77,161)(78,162)(79,163)(80,164)(81,165)(82,166)(83,167)(84,168)(85,169)(86,170)(87,171)(88,172)(89,173)(90,174)(91,175)(92,176)(93,177)(94,178)(95,179)(96,180)(97,121)(98,122)(99,123)(100,124)(101,125)(102,126)(103,127)(104,128)(105,129)(106,130)(107,131)(108,132)(109,133)(110,134)(111,135)(112,136)(113,137)(114,138)(115,139)(116,140)(117,141)(118,142)(119,143)(120,144), (1,31)(2,32)(3,33)(4,34)(5,35)(6,36)(7,37)(8,38)(9,39)(10,40)(11,41)(12,42)(13,43)(14,44)(15,45)(16,46)(17,47)(18,48)(19,49)(20,50)(21,51)(22,52)(23,53)(24,54)(25,55)(26,56)(27,57)(28,58)(29,59)(30,60)(61,91)(62,92)(63,93)(64,94)(65,95)(66,96)(67,97)(68,98)(69,99)(70,100)(71,101)(72,102)(73,103)(74,104)(75,105)(76,106)(77,107)(78,108)(79,109)(80,110)(81,111)(82,112)(83,113)(84,114)(85,115)(86,116)(87,117)(88,118)(89,119)(90,120)(121,151)(122,152)(123,153)(124,154)(125,155)(126,156)(127,157)(128,158)(129,159)(130,160)(131,161)(132,162)(133,163)(134,164)(135,165)(136,166)(137,167)(138,168)(139,169)(140,170)(141,171)(142,172)(143,173)(144,174)(145,175)(146,176)(147,177)(148,178)(149,179)(150,180)(181,211)(182,212)(183,213)(184,214)(185,215)(186,216)(187,217)(188,218)(189,219)(190,220)(191,221)(192,222)(193,223)(194,224)(195,225)(196,226)(197,227)(198,228)(199,229)(200,230)(201,231)(202,232)(203,233)(204,234)(205,235)(206,236)(207,237)(208,238)(209,239)(210,240), (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)(181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240), (1,163,197,109)(2,108,198,162)(3,161,199,107)(4,106,200,160)(5,159,201,105)(6,104,202,158)(7,157,203,103)(8,102,204,156)(9,155,205,101)(10,100,206,154)(11,153,207,99)(12,98,208,152)(13,151,209,97)(14,96,210,150)(15,149,211,95)(16,94,212,148)(17,147,213,93)(18,92,214,146)(19,145,215,91)(20,90,216,144)(21,143,217,89)(22,88,218,142)(23,141,219,87)(24,86,220,140)(25,139,221,85)(26,84,222,138)(27,137,223,83)(28,82,224,136)(29,135,225,81)(30,80,226,134)(31,133,227,79)(32,78,228,132)(33,131,229,77)(34,76,230,130)(35,129,231,75)(36,74,232,128)(37,127,233,73)(38,72,234,126)(39,125,235,71)(40,70,236,124)(41,123,237,69)(42,68,238,122)(43,121,239,67)(44,66,240,180)(45,179,181,65)(46,64,182,178)(47,177,183,63)(48,62,184,176)(49,175,185,61)(50,120,186,174)(51,173,187,119)(52,118,188,172)(53,171,189,117)(54,116,190,170)(55,169,191,115)(56,114,192,168)(57,167,193,113)(58,112,194,166)(59,165,195,111)(60,110,196,164) );

G=PermutationGroup([(61,91),(62,92),(63,93),(64,94),(65,95),(66,96),(67,97),(68,98),(69,99),(70,100),(71,101),(72,102),(73,103),(74,104),(75,105),(76,106),(77,107),(78,108),(79,109),(80,110),(81,111),(82,112),(83,113),(84,114),(85,115),(86,116),(87,117),(88,118),(89,119),(90,120),(121,151),(122,152),(123,153),(124,154),(125,155),(126,156),(127,157),(128,158),(129,159),(130,160),(131,161),(132,162),(133,163),(134,164),(135,165),(136,166),(137,167),(138,168),(139,169),(140,170),(141,171),(142,172),(143,173),(144,174),(145,175),(146,176),(147,177),(148,178),(149,179),(150,180)], [(1,227),(2,228),(3,229),(4,230),(5,231),(6,232),(7,233),(8,234),(9,235),(10,236),(11,237),(12,238),(13,239),(14,240),(15,181),(16,182),(17,183),(18,184),(19,185),(20,186),(21,187),(22,188),(23,189),(24,190),(25,191),(26,192),(27,193),(28,194),(29,195),(30,196),(31,197),(32,198),(33,199),(34,200),(35,201),(36,202),(37,203),(38,204),(39,205),(40,206),(41,207),(42,208),(43,209),(44,210),(45,211),(46,212),(47,213),(48,214),(49,215),(50,216),(51,217),(52,218),(53,219),(54,220),(55,221),(56,222),(57,223),(58,224),(59,225),(60,226),(61,145),(62,146),(63,147),(64,148),(65,149),(66,150),(67,151),(68,152),(69,153),(70,154),(71,155),(72,156),(73,157),(74,158),(75,159),(76,160),(77,161),(78,162),(79,163),(80,164),(81,165),(82,166),(83,167),(84,168),(85,169),(86,170),(87,171),(88,172),(89,173),(90,174),(91,175),(92,176),(93,177),(94,178),(95,179),(96,180),(97,121),(98,122),(99,123),(100,124),(101,125),(102,126),(103,127),(104,128),(105,129),(106,130),(107,131),(108,132),(109,133),(110,134),(111,135),(112,136),(113,137),(114,138),(115,139),(116,140),(117,141),(118,142),(119,143),(120,144)], [(1,31),(2,32),(3,33),(4,34),(5,35),(6,36),(7,37),(8,38),(9,39),(10,40),(11,41),(12,42),(13,43),(14,44),(15,45),(16,46),(17,47),(18,48),(19,49),(20,50),(21,51),(22,52),(23,53),(24,54),(25,55),(26,56),(27,57),(28,58),(29,59),(30,60),(61,91),(62,92),(63,93),(64,94),(65,95),(66,96),(67,97),(68,98),(69,99),(70,100),(71,101),(72,102),(73,103),(74,104),(75,105),(76,106),(77,107),(78,108),(79,109),(80,110),(81,111),(82,112),(83,113),(84,114),(85,115),(86,116),(87,117),(88,118),(89,119),(90,120),(121,151),(122,152),(123,153),(124,154),(125,155),(126,156),(127,157),(128,158),(129,159),(130,160),(131,161),(132,162),(133,163),(134,164),(135,165),(136,166),(137,167),(138,168),(139,169),(140,170),(141,171),(142,172),(143,173),(144,174),(145,175),(146,176),(147,177),(148,178),(149,179),(150,180),(181,211),(182,212),(183,213),(184,214),(185,215),(186,216),(187,217),(188,218),(189,219),(190,220),(191,221),(192,222),(193,223),(194,224),(195,225),(196,226),(197,227),(198,228),(199,229),(200,230),(201,231),(202,232),(203,233),(204,234),(205,235),(206,236),(207,237),(208,238),(209,239),(210,240)], [(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),(181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240)], [(1,163,197,109),(2,108,198,162),(3,161,199,107),(4,106,200,160),(5,159,201,105),(6,104,202,158),(7,157,203,103),(8,102,204,156),(9,155,205,101),(10,100,206,154),(11,153,207,99),(12,98,208,152),(13,151,209,97),(14,96,210,150),(15,149,211,95),(16,94,212,148),(17,147,213,93),(18,92,214,146),(19,145,215,91),(20,90,216,144),(21,143,217,89),(22,88,218,142),(23,141,219,87),(24,86,220,140),(25,139,221,85),(26,84,222,138),(27,137,223,83),(28,82,224,136),(29,135,225,81),(30,80,226,134),(31,133,227,79),(32,78,228,132),(33,131,229,77),(34,76,230,130),(35,129,231,75),(36,74,232,128),(37,127,233,73),(38,72,234,126),(39,125,235,71),(40,70,236,124),(41,123,237,69),(42,68,238,122),(43,121,239,67),(44,66,240,180),(45,179,181,65),(46,64,182,178),(47,177,183,63),(48,62,184,176),(49,175,185,61),(50,120,186,174),(51,173,187,119),(52,118,188,172),(53,171,189,117),(54,116,190,170),(55,169,191,115),(56,114,192,168),(57,167,193,113),(58,112,194,166),(59,165,195,111),(60,110,196,164)])

126 conjugacy classes

class 1 2A2B2C2D2E2F 3 4A4B4C4D4E4F4G5A5B6A···6G10A···10N12A···12H15A15B15C15D20A···20P30A···30AB60A···60AF
order122222234444444556···610···1012···121515151520···2030···3060···60
size1111226022222606060222···22···22···222222···22···22···2

126 irreducible representations

dim11111122222222222222222
type++++++++++++++++
imageC1C2C2C2C2C2S3D4D5D6D6C4○D4D10D10C3⋊D4D15C5⋊D4C4○D12D30D30C4○D20C157D4D6011C2
kernelC23.28D30C30.4Q8D303C4C30.38D4C2×C157D4C22×C60C22×C20C2×C30C22×C12C2×C20C22×C10C30C2×C12C22×C6C2×C10C22×C4C2×C6C10C2×C4C23C6C22C2
# reps12211112221442448884161632

Matrix representation of C23.28D30 in GL4(𝔽61) generated by

1000
06000
0010
0001
,
60000
06000
00600
00060
,
60000
06000
0010
0001
,
7000
03500
00532
002939
,
02600
54000
005046
00011
G:=sub<GL(4,GF(61))| [1,0,0,0,0,60,0,0,0,0,1,0,0,0,0,1],[60,0,0,0,0,60,0,0,0,0,60,0,0,0,0,60],[60,0,0,0,0,60,0,0,0,0,1,0,0,0,0,1],[7,0,0,0,0,35,0,0,0,0,5,29,0,0,32,39],[0,54,0,0,26,0,0,0,0,0,50,0,0,0,46,11] >;

C23.28D30 in GAP, Magma, Sage, TeX

C_2^3._{28}D_{30}
% in TeX

G:=Group("C2^3.28D30");
// GroupNames label

G:=SmallGroup(480,894);
// by ID

G=gap.SmallGroup(480,894);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,253,254,100,2693,18822]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^2=1,d^30=c,e^2=c*b=b*c,a*b=b*a,e*a*e^-1=a*c=c*a,a*d=d*a,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=b*d^29>;
// generators/relations

׿
×
𝔽