Copied to
clipboard

G = C23.22D30order 480 = 25·3·5

8th non-split extension by C23 of D30 acting via D30/D15=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C23.22D30, (C2×C30).7D4, (C2×D4).5D15, (C6×D4).13D5, (C2×C4).17D30, (D4×C10).13S3, (D4×C30).24C2, C30.381(C2×D4), (C2×C20).250D6, (C2×C12).249D10, C30.38D48C2, C30.4Q836C2, (C22×C6).63D10, (C22×C10).78D6, C30.223(C4○D4), (C2×C60).433C22, (C2×C30).306C23, (C22×Dic15)⋊5C2, C55(C23.23D6), C22.4(C157D4), C2.15(D42D15), C6.102(D42D5), C1537(C22.D4), C35(C23.18D10), (C22×C30).19C22, C10.102(D42S3), C22.57(C22×D15), (C2×Dic15).16C22, C6.106(C2×C5⋊D4), C2.11(C2×C157D4), (C2×C6).19(C5⋊D4), C10.106(C2×C3⋊D4), (C2×C10).18(C3⋊D4), (C2×C6).302(C22×D5), (C2×C10).301(C22×S3), SmallGroup(480,900)

Series: Derived Chief Lower central Upper central

C1C2×C30 — C23.22D30
C1C5C15C30C2×C30C2×Dic15C22×Dic15 — C23.22D30
C15C2×C30 — C23.22D30
C1C22C2×D4

Generators and relations for C23.22D30
 G = < a,b,c,d,e | a2=b2=c2=d30=1, e2=cb=bc, ab=ba, dad-1=eae-1=ac=ca, bd=db, be=eb, cd=dc, ce=ec, ede-1=bd-1 >

Subgroups: 692 in 156 conjugacy classes, 55 normal (31 characteristic)
C1, C2, C2 [×2], C2 [×3], C3, C4 [×5], C22, C22 [×2], C22 [×5], C5, C6, C6 [×2], C6 [×3], C2×C4, C2×C4 [×6], D4 [×2], C23 [×2], C10, C10 [×2], C10 [×3], Dic3 [×4], C12, C2×C6, C2×C6 [×2], C2×C6 [×5], C15, C22⋊C4 [×3], C4⋊C4 [×2], C22×C4, C2×D4, Dic5 [×4], C20, C2×C10, C2×C10 [×2], C2×C10 [×5], C2×Dic3 [×6], C2×C12, C3×D4 [×2], C22×C6 [×2], C30, C30 [×2], C30 [×3], C22.D4, C2×Dic5 [×6], C2×C20, C5×D4 [×2], C22×C10 [×2], Dic3⋊C4 [×2], C6.D4 [×3], C22×Dic3, C6×D4, Dic15 [×4], C60, C2×C30, C2×C30 [×2], C2×C30 [×5], C10.D4 [×2], C23.D5 [×3], C22×Dic5, D4×C10, C23.23D6, C2×Dic15 [×4], C2×Dic15 [×2], C2×C60, D4×C15 [×2], C22×C30 [×2], C23.18D10, C30.4Q8 [×2], C30.38D4, C30.38D4 [×2], C22×Dic15, D4×C30, C23.22D30
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, D42S3 [×2], C2×C3⋊D4, D30 [×3], D42D5 [×2], C2×C5⋊D4, C23.23D6, C157D4 [×2], C22×D15, C23.18D10, D42D15 [×2], C2×C157D4, C23.22D30

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

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

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

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

84 conjugacy classes

class 1 2A2B2C2D2E2F 3 4A4B4C4D4E4F4G5A5B6A6B6C6D6E6F6G10A···10F10G···10N12A12B15A15B15C15D20A20B20C20D30A···30L30M···30AB60A···60H
order12222223444444455666666610···1010···101212151515152020202030···3030···3060···60
size1111224243030303060602222244442···24···444222244442···24···44···4

84 irreducible representations

dim1111122222222222222444
type+++++++++++++++---
imageC1C2C2C2C2S3D4D5D6D6C4○D4D10D10C3⋊D4D15C5⋊D4D30D30C157D4D42S3D42D5D42D15
kernelC23.22D30C30.4Q8C30.38D4C22×Dic15D4×C30D4×C10C2×C30C6×D4C2×C20C22×C10C30C2×C12C22×C6C2×C10C2×D4C2×C6C2×C4C23C22C10C6C2
# reps12311122124244484816248

Matrix representation of C23.22D30 in GL6(𝔽61)

100000
010000
001000
000100
000010
0000060
,
6000000
0600000
001000
000100
0000600
0000060
,
100000
010000
001000
000100
0000600
0000060
,
010000
100000
0047900
00222500
000001
000010
,
18520000
9430000
00495900
00411200
0000050
0000110

G:=sub<GL(6,GF(61))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,60],[60,0,0,0,0,0,0,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,47,22,0,0,0,0,9,25,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[18,9,0,0,0,0,52,43,0,0,0,0,0,0,49,41,0,0,0,0,59,12,0,0,0,0,0,0,0,11,0,0,0,0,50,0] >;

C23.22D30 in GAP, Magma, Sage, TeX

C_2^3._{22}D_{30}
% in TeX

G:=Group("C2^3.22D30");
// GroupNames label

G:=SmallGroup(480,900);
// by ID

G=gap.SmallGroup(480,900);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,254,219,2693,18822]);
// Polycyclic

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

׿
×
𝔽