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## G = C60.88D4order 480 = 25·3·5

### 88th non-split extension by C60 of D4 acting via D4/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C30 — C60.88D4
 Chief series C1 — C5 — C15 — C30 — C2×C30 — D5×C2×C6 — D10⋊Dic3 — C60.88D4
 Lower central C15 — C2×C30 — C60.88D4
 Upper central C1 — C22 — C2×C4

Generators and relations for C60.88D4
G = < a,b,c | a60=b4=c2=1, bab-1=a29, cac=a19, cbc=a30b-1 >

Subgroups: 732 in 152 conjugacy classes, 52 normal (22 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C5, C6, C6, C6, C2×C4, C2×C4, D4, Q8, C23, D5, C10, C10, Dic3, C12, C2×C6, C2×C6, C15, C42, C22⋊C4, C2×D4, C2×Q8, Dic5, C20, C20, D10, C2×C10, Dic6, C2×Dic3, C2×Dic3, C2×C12, C3×D4, C22×C6, C3×D5, C30, C30, C4.4D4, D20, C2×Dic5, C2×C20, C2×C20, C5×Q8, C22×D5, C4×Dic3, C6.D4, C2×Dic6, C6×D4, C5×Dic3, Dic15, C60, C6×D5, C2×C30, C4×Dic5, D10⋊C4, C2×D20, Q8×C10, C23.12D6, C3×D20, C5×Dic6, C10×Dic3, C2×Dic15, C2×C60, D5×C2×C6, C20.23D4, D10⋊Dic3, C4×Dic15, C6×D20, C10×Dic6, C60.88D4
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, C2×D4, C4○D4, D10, C3⋊D4, C22×S3, C4.4D4, C5⋊D4, C22×D5, D42S3, C2×C3⋊D4, S3×D5, Q82D5, C2×C5⋊D4, C23.12D6, C15⋊D4, C2×S3×D5, C20.23D4, D20⋊S3, C2×C15⋊D4, C60.88D4

Smallest permutation representation of C60.88D4
On 240 points
Generators in S240
(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)
(1 111 143 212)(2 80 144 181)(3 109 145 210)(4 78 146 239)(5 107 147 208)(6 76 148 237)(7 105 149 206)(8 74 150 235)(9 103 151 204)(10 72 152 233)(11 101 153 202)(12 70 154 231)(13 99 155 200)(14 68 156 229)(15 97 157 198)(16 66 158 227)(17 95 159 196)(18 64 160 225)(19 93 161 194)(20 62 162 223)(21 91 163 192)(22 120 164 221)(23 89 165 190)(24 118 166 219)(25 87 167 188)(26 116 168 217)(27 85 169 186)(28 114 170 215)(29 83 171 184)(30 112 172 213)(31 81 173 182)(32 110 174 211)(33 79 175 240)(34 108 176 209)(35 77 177 238)(36 106 178 207)(37 75 179 236)(38 104 180 205)(39 73 121 234)(40 102 122 203)(41 71 123 232)(42 100 124 201)(43 69 125 230)(44 98 126 199)(45 67 127 228)(46 96 128 197)(47 65 129 226)(48 94 130 195)(49 63 131 224)(50 92 132 193)(51 61 133 222)(52 90 134 191)(53 119 135 220)(54 88 136 189)(55 117 137 218)(56 86 138 187)(57 115 139 216)(58 84 140 185)(59 113 141 214)(60 82 142 183)
(1 128)(2 147)(3 166)(4 125)(5 144)(6 163)(7 122)(8 141)(9 160)(10 179)(11 138)(12 157)(13 176)(14 135)(15 154)(16 173)(17 132)(18 151)(19 170)(20 129)(21 148)(22 167)(23 126)(24 145)(25 164)(26 123)(27 142)(28 161)(29 180)(30 139)(31 158)(32 177)(33 136)(34 155)(35 174)(36 133)(37 152)(38 171)(39 130)(40 149)(41 168)(42 127)(43 146)(44 165)(45 124)(46 143)(47 162)(48 121)(49 140)(50 159)(51 178)(52 137)(53 156)(54 175)(55 134)(56 153)(57 172)(58 131)(59 150)(60 169)(61 76)(62 95)(63 114)(64 73)(65 92)(66 111)(67 70)(68 89)(69 108)(71 86)(72 105)(74 83)(75 102)(77 80)(78 99)(79 118)(81 96)(82 115)(84 93)(85 112)(87 90)(88 109)(91 106)(94 103)(97 100)(98 119)(101 116)(104 113)(107 110)(117 120)(181 238)(182 197)(183 216)(184 235)(185 194)(186 213)(187 232)(188 191)(189 210)(190 229)(192 207)(193 226)(195 204)(196 223)(198 201)(199 220)(200 239)(202 217)(203 236)(205 214)(206 233)(208 211)(209 230)(212 227)(215 224)(218 221)(219 240)(222 237)(225 234)(228 231)

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

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

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

60 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3 4A 4B 4C 4D 4E 4F 4G 4H 5A 5B 6A 6B 6C 6D 6E 6F 6G 10A ··· 10F 12A 12B 15A 15B 20A 20B 20C 20D 20E ··· 20L 30A ··· 30F 60A ··· 60H order 1 2 2 2 2 2 3 4 4 4 4 4 4 4 4 5 5 6 6 6 6 6 6 6 10 ··· 10 12 12 15 15 20 20 20 20 20 ··· 20 30 ··· 30 60 ··· 60 size 1 1 1 1 20 20 2 2 2 12 12 30 30 30 30 2 2 2 2 2 20 20 20 20 2 ··· 2 4 4 4 4 4 4 4 4 12 ··· 12 4 ··· 4 4 ··· 4

60 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 type + + + + + + + + + + + + - + + - + image C1 C2 C2 C2 C2 S3 D4 D5 D6 D6 C4○D4 D10 D10 C3⋊D4 C5⋊D4 D4⋊2S3 S3×D5 Q8⋊2D5 C15⋊D4 C2×S3×D5 D20⋊S3 kernel C60.88D4 D10⋊Dic3 C4×Dic15 C6×D20 C10×Dic6 C2×D20 C60 C2×Dic6 C2×C20 C22×D5 C30 C2×Dic3 C2×C12 C20 C12 C10 C2×C4 C6 C4 C22 C2 # reps 1 4 1 1 1 1 2 2 1 2 4 4 2 4 8 2 2 4 4 2 8

Matrix representation of C60.88D4 in GL6(𝔽61)

 0 1 0 0 0 0 60 0 0 0 0 0 0 0 44 44 0 0 0 0 17 60 0 0 0 0 0 0 13 0 0 0 0 0 45 47
,
 50 0 0 0 0 0 0 50 0 0 0 0 0 0 14 45 0 0 0 0 39 47 0 0 0 0 0 0 17 2 0 0 0 0 38 44
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 60 0 0 0 0 0 44 1 0 0 0 0 0 0 60 0 0 0 0 0 17 1

G:=sub<GL(6,GF(61))| [0,60,0,0,0,0,1,0,0,0,0,0,0,0,44,17,0,0,0,0,44,60,0,0,0,0,0,0,13,45,0,0,0,0,0,47],[50,0,0,0,0,0,0,50,0,0,0,0,0,0,14,39,0,0,0,0,45,47,0,0,0,0,0,0,17,38,0,0,0,0,2,44],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,60,44,0,0,0,0,0,1,0,0,0,0,0,0,60,17,0,0,0,0,0,1] >;

C60.88D4 in GAP, Magma, Sage, TeX

C_{60}._{88}D_4
% in TeX

G:=Group("C60.88D4");
// GroupNames label

G:=SmallGroup(480,444);
// by ID

G=gap.SmallGroup(480,444);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,141,120,422,219,100,1356,18822]);
// Polycyclic

G:=Group<a,b,c|a^60=b^4=c^2=1,b*a*b^-1=a^29,c*a*c=a^19,c*b*c=a^30*b^-1>;
// generators/relations

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