metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C23.21D22, (C2×C44)⋊6C4, C44.37(C2×C4), C44⋊C4⋊17C2, (C2×C4)⋊4Dic11, (C2×C4).102D22, (C22×C4).7D11, C11⋊4(C42⋊C2), (C4×Dic11)⋊15C2, C22.16(C4○D4), (C2×C22).44C23, (C22×C44).10C2, (C2×C44).93C22, C22.24(C22×C4), C4.15(C2×Dic11), C23.D11.5C2, C2.4(D44⋊5C2), C2.5(C22×Dic11), C22.5(C2×Dic11), (C22×C22).36C22, C22.22(C22×D11), (C2×Dic11).35C22, (C2×C22).35(C2×C4), SmallGroup(352,121)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C23.21D22
G = < a,b,c,d,e | a2=b2=c2=1, d22=c, e2=cb=bc, ab=ba, eae-1=ac=ca, ad=da, bd=db, be=eb, cd=dc, ce=ec, ede-1=d21 >
Subgroups: 282 in 76 conjugacy classes, 49 normal (15 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C2×C4, C2×C4, C2×C4, C23, C11, C42, C22⋊C4, C4⋊C4, C22×C4, C22, C22, C22, C42⋊C2, Dic11, C44, C2×C22, C2×C22, C2×C22, C2×Dic11, C2×C44, C2×C44, C22×C22, C4×Dic11, C44⋊C4, C23.D11, C22×C44, C23.21D22
Quotients: C1, C2, C4, C22, C2×C4, C23, C22×C4, C4○D4, D11, C42⋊C2, Dic11, D22, C2×Dic11, C22×D11, D44⋊5C2, C22×Dic11, C23.21D22
►On 176 points
Generators in S176
(89 111)(90 112)(91 113)(92 114)(93 115)(94 116)(95 117)(96 118)(97 119)(98 120)(99 121)(100 122)(101 123)(102 124)(103 125)(104 126)(105 127)(106 128)(107 129)(108 130)(109 131)(110 132)(133 155)(134 156)(135 157)(136 158)(137 159)(138 160)(139 161)(140 162)(141 163)(142 164)(143 165)(144 166)(145 167)(146 168)(147 169)(148 170)(149 171)(150 172)(151 173)(152 174)(153 175)(154 176)
(1 50)(2 51)(3 52)(4 53)(5 54)(6 55)(7 56)(8 57)(9 58)(10 59)(11 60)(12 61)(13 62)(14 63)(15 64)(16 65)(17 66)(18 67)(19 68)(20 69)(21 70)(22 71)(23 72)(24 73)(25 74)(26 75)(27 76)(28 77)(29 78)(30 79)(31 80)(32 81)(33 82)(34 83)(35 84)(36 85)(37 86)(38 87)(39 88)(40 45)(41 46)(42 47)(43 48)(44 49)(89 148)(90 149)(91 150)(92 151)(93 152)(94 153)(95 154)(96 155)(97 156)(98 157)(99 158)(100 159)(101 160)(102 161)(103 162)(104 163)(105 164)(106 165)(107 166)(108 167)(109 168)(110 169)(111 170)(112 171)(113 172)(114 173)(115 174)(116 175)(117 176)(118 133)(119 134)(120 135)(121 136)(122 137)(123 138)(124 139)(125 140)(126 141)(127 142)(128 143)(129 144)(130 145)(131 146)(132 147)
(1 23)(2 24)(3 25)(4 26)(5 27)(6 28)(7 29)(8 30)(9 31)(10 32)(11 33)(12 34)(13 35)(14 36)(15 37)(16 38)(17 39)(18 40)(19 41)(20 42)(21 43)(22 44)(45 67)(46 68)(47 69)(48 70)(49 71)(50 72)(51 73)(52 74)(53 75)(54 76)(55 77)(56 78)(57 79)(58 80)(59 81)(60 82)(61 83)(62 84)(63 85)(64 86)(65 87)(66 88)(89 111)(90 112)(91 113)(92 114)(93 115)(94 116)(95 117)(96 118)(97 119)(98 120)(99 121)(100 122)(101 123)(102 124)(103 125)(104 126)(105 127)(106 128)(107 129)(108 130)(109 131)(110 132)(133 155)(134 156)(135 157)(136 158)(137 159)(138 160)(139 161)(140 162)(141 163)(142 164)(143 165)(144 166)(145 167)(146 168)(147 169)(148 170)(149 171)(150 172)(151 173)(152 174)(153 175)(154 176)
(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)
(1 135 72 98)(2 156 73 119)(3 133 74 96)(4 154 75 117)(5 175 76 94)(6 152 77 115)(7 173 78 92)(8 150 79 113)(9 171 80 90)(10 148 81 111)(11 169 82 132)(12 146 83 109)(13 167 84 130)(14 144 85 107)(15 165 86 128)(16 142 87 105)(17 163 88 126)(18 140 45 103)(19 161 46 124)(20 138 47 101)(21 159 48 122)(22 136 49 99)(23 157 50 120)(24 134 51 97)(25 155 52 118)(26 176 53 95)(27 153 54 116)(28 174 55 93)(29 151 56 114)(30 172 57 91)(31 149 58 112)(32 170 59 89)(33 147 60 110)(34 168 61 131)(35 145 62 108)(36 166 63 129)(37 143 64 106)(38 164 65 127)(39 141 66 104)(40 162 67 125)(41 139 68 102)(42 160 69 123)(43 137 70 100)(44 158 71 121)
G:=sub<Sym(176)| (89,111)(90,112)(91,113)(92,114)(93,115)(94,116)(95,117)(96,118)(97,119)(98,120)(99,121)(100,122)(101,123)(102,124)(103,125)(104,126)(105,127)(106,128)(107,129)(108,130)(109,131)(110,132)(133,155)(134,156)(135,157)(136,158)(137,159)(138,160)(139,161)(140,162)(141,163)(142,164)(143,165)(144,166)(145,167)(146,168)(147,169)(148,170)(149,171)(150,172)(151,173)(152,174)(153,175)(154,176), (1,50)(2,51)(3,52)(4,53)(5,54)(6,55)(7,56)(8,57)(9,58)(10,59)(11,60)(12,61)(13,62)(14,63)(15,64)(16,65)(17,66)(18,67)(19,68)(20,69)(21,70)(22,71)(23,72)(24,73)(25,74)(26,75)(27,76)(28,77)(29,78)(30,79)(31,80)(32,81)(33,82)(34,83)(35,84)(36,85)(37,86)(38,87)(39,88)(40,45)(41,46)(42,47)(43,48)(44,49)(89,148)(90,149)(91,150)(92,151)(93,152)(94,153)(95,154)(96,155)(97,156)(98,157)(99,158)(100,159)(101,160)(102,161)(103,162)(104,163)(105,164)(106,165)(107,166)(108,167)(109,168)(110,169)(111,170)(112,171)(113,172)(114,173)(115,174)(116,175)(117,176)(118,133)(119,134)(120,135)(121,136)(122,137)(123,138)(124,139)(125,140)(126,141)(127,142)(128,143)(129,144)(130,145)(131,146)(132,147), (1,23)(2,24)(3,25)(4,26)(5,27)(6,28)(7,29)(8,30)(9,31)(10,32)(11,33)(12,34)(13,35)(14,36)(15,37)(16,38)(17,39)(18,40)(19,41)(20,42)(21,43)(22,44)(45,67)(46,68)(47,69)(48,70)(49,71)(50,72)(51,73)(52,74)(53,75)(54,76)(55,77)(56,78)(57,79)(58,80)(59,81)(60,82)(61,83)(62,84)(63,85)(64,86)(65,87)(66,88)(89,111)(90,112)(91,113)(92,114)(93,115)(94,116)(95,117)(96,118)(97,119)(98,120)(99,121)(100,122)(101,123)(102,124)(103,125)(104,126)(105,127)(106,128)(107,129)(108,130)(109,131)(110,132)(133,155)(134,156)(135,157)(136,158)(137,159)(138,160)(139,161)(140,162)(141,163)(142,164)(143,165)(144,166)(145,167)(146,168)(147,169)(148,170)(149,171)(150,172)(151,173)(152,174)(153,175)(154,176), (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), (1,135,72,98)(2,156,73,119)(3,133,74,96)(4,154,75,117)(5,175,76,94)(6,152,77,115)(7,173,78,92)(8,150,79,113)(9,171,80,90)(10,148,81,111)(11,169,82,132)(12,146,83,109)(13,167,84,130)(14,144,85,107)(15,165,86,128)(16,142,87,105)(17,163,88,126)(18,140,45,103)(19,161,46,124)(20,138,47,101)(21,159,48,122)(22,136,49,99)(23,157,50,120)(24,134,51,97)(25,155,52,118)(26,176,53,95)(27,153,54,116)(28,174,55,93)(29,151,56,114)(30,172,57,91)(31,149,58,112)(32,170,59,89)(33,147,60,110)(34,168,61,131)(35,145,62,108)(36,166,63,129)(37,143,64,106)(38,164,65,127)(39,141,66,104)(40,162,67,125)(41,139,68,102)(42,160,69,123)(43,137,70,100)(44,158,71,121)>;
G:=Group( (89,111)(90,112)(91,113)(92,114)(93,115)(94,116)(95,117)(96,118)(97,119)(98,120)(99,121)(100,122)(101,123)(102,124)(103,125)(104,126)(105,127)(106,128)(107,129)(108,130)(109,131)(110,132)(133,155)(134,156)(135,157)(136,158)(137,159)(138,160)(139,161)(140,162)(141,163)(142,164)(143,165)(144,166)(145,167)(146,168)(147,169)(148,170)(149,171)(150,172)(151,173)(152,174)(153,175)(154,176), (1,50)(2,51)(3,52)(4,53)(5,54)(6,55)(7,56)(8,57)(9,58)(10,59)(11,60)(12,61)(13,62)(14,63)(15,64)(16,65)(17,66)(18,67)(19,68)(20,69)(21,70)(22,71)(23,72)(24,73)(25,74)(26,75)(27,76)(28,77)(29,78)(30,79)(31,80)(32,81)(33,82)(34,83)(35,84)(36,85)(37,86)(38,87)(39,88)(40,45)(41,46)(42,47)(43,48)(44,49)(89,148)(90,149)(91,150)(92,151)(93,152)(94,153)(95,154)(96,155)(97,156)(98,157)(99,158)(100,159)(101,160)(102,161)(103,162)(104,163)(105,164)(106,165)(107,166)(108,167)(109,168)(110,169)(111,170)(112,171)(113,172)(114,173)(115,174)(116,175)(117,176)(118,133)(119,134)(120,135)(121,136)(122,137)(123,138)(124,139)(125,140)(126,141)(127,142)(128,143)(129,144)(130,145)(131,146)(132,147), (1,23)(2,24)(3,25)(4,26)(5,27)(6,28)(7,29)(8,30)(9,31)(10,32)(11,33)(12,34)(13,35)(14,36)(15,37)(16,38)(17,39)(18,40)(19,41)(20,42)(21,43)(22,44)(45,67)(46,68)(47,69)(48,70)(49,71)(50,72)(51,73)(52,74)(53,75)(54,76)(55,77)(56,78)(57,79)(58,80)(59,81)(60,82)(61,83)(62,84)(63,85)(64,86)(65,87)(66,88)(89,111)(90,112)(91,113)(92,114)(93,115)(94,116)(95,117)(96,118)(97,119)(98,120)(99,121)(100,122)(101,123)(102,124)(103,125)(104,126)(105,127)(106,128)(107,129)(108,130)(109,131)(110,132)(133,155)(134,156)(135,157)(136,158)(137,159)(138,160)(139,161)(140,162)(141,163)(142,164)(143,165)(144,166)(145,167)(146,168)(147,169)(148,170)(149,171)(150,172)(151,173)(152,174)(153,175)(154,176), (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), (1,135,72,98)(2,156,73,119)(3,133,74,96)(4,154,75,117)(5,175,76,94)(6,152,77,115)(7,173,78,92)(8,150,79,113)(9,171,80,90)(10,148,81,111)(11,169,82,132)(12,146,83,109)(13,167,84,130)(14,144,85,107)(15,165,86,128)(16,142,87,105)(17,163,88,126)(18,140,45,103)(19,161,46,124)(20,138,47,101)(21,159,48,122)(22,136,49,99)(23,157,50,120)(24,134,51,97)(25,155,52,118)(26,176,53,95)(27,153,54,116)(28,174,55,93)(29,151,56,114)(30,172,57,91)(31,149,58,112)(32,170,59,89)(33,147,60,110)(34,168,61,131)(35,145,62,108)(36,166,63,129)(37,143,64,106)(38,164,65,127)(39,141,66,104)(40,162,67,125)(41,139,68,102)(42,160,69,123)(43,137,70,100)(44,158,71,121) );
G=PermutationGroup([[(89,111),(90,112),(91,113),(92,114),(93,115),(94,116),(95,117),(96,118),(97,119),(98,120),(99,121),(100,122),(101,123),(102,124),(103,125),(104,126),(105,127),(106,128),(107,129),(108,130),(109,131),(110,132),(133,155),(134,156),(135,157),(136,158),(137,159),(138,160),(139,161),(140,162),(141,163),(142,164),(143,165),(144,166),(145,167),(146,168),(147,169),(148,170),(149,171),(150,172),(151,173),(152,174),(153,175),(154,176)], [(1,50),(2,51),(3,52),(4,53),(5,54),(6,55),(7,56),(8,57),(9,58),(10,59),(11,60),(12,61),(13,62),(14,63),(15,64),(16,65),(17,66),(18,67),(19,68),(20,69),(21,70),(22,71),(23,72),(24,73),(25,74),(26,75),(27,76),(28,77),(29,78),(30,79),(31,80),(32,81),(33,82),(34,83),(35,84),(36,85),(37,86),(38,87),(39,88),(40,45),(41,46),(42,47),(43,48),(44,49),(89,148),(90,149),(91,150),(92,151),(93,152),(94,153),(95,154),(96,155),(97,156),(98,157),(99,158),(100,159),(101,160),(102,161),(103,162),(104,163),(105,164),(106,165),(107,166),(108,167),(109,168),(110,169),(111,170),(112,171),(113,172),(114,173),(115,174),(116,175),(117,176),(118,133),(119,134),(120,135),(121,136),(122,137),(123,138),(124,139),(125,140),(126,141),(127,142),(128,143),(129,144),(130,145),(131,146),(132,147)], [(1,23),(2,24),(3,25),(4,26),(5,27),(6,28),(7,29),(8,30),(9,31),(10,32),(11,33),(12,34),(13,35),(14,36),(15,37),(16,38),(17,39),(18,40),(19,41),(20,42),(21,43),(22,44),(45,67),(46,68),(47,69),(48,70),(49,71),(50,72),(51,73),(52,74),(53,75),(54,76),(55,77),(56,78),(57,79),(58,80),(59,81),(60,82),(61,83),(62,84),(63,85),(64,86),(65,87),(66,88),(89,111),(90,112),(91,113),(92,114),(93,115),(94,116),(95,117),(96,118),(97,119),(98,120),(99,121),(100,122),(101,123),(102,124),(103,125),(104,126),(105,127),(106,128),(107,129),(108,130),(109,131),(110,132),(133,155),(134,156),(135,157),(136,158),(137,159),(138,160),(139,161),(140,162),(141,163),(142,164),(143,165),(144,166),(145,167),(146,168),(147,169),(148,170),(149,171),(150,172),(151,173),(152,174),(153,175),(154,176)], [(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)], [(1,135,72,98),(2,156,73,119),(3,133,74,96),(4,154,75,117),(5,175,76,94),(6,152,77,115),(7,173,78,92),(8,150,79,113),(9,171,80,90),(10,148,81,111),(11,169,82,132),(12,146,83,109),(13,167,84,130),(14,144,85,107),(15,165,86,128),(16,142,87,105),(17,163,88,126),(18,140,45,103),(19,161,46,124),(20,138,47,101),(21,159,48,122),(22,136,49,99),(23,157,50,120),(24,134,51,97),(25,155,52,118),(26,176,53,95),(27,153,54,116),(28,174,55,93),(29,151,56,114),(30,172,57,91),(31,149,58,112),(32,170,59,89),(33,147,60,110),(34,168,61,131),(35,145,62,108),(36,166,63,129),(37,143,64,106),(38,164,65,127),(39,141,66,104),(40,162,67,125),(41,139,68,102),(42,160,69,123),(43,137,70,100),(44,158,71,121)]])
100 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | ··· | 4N | 11A | ··· | 11E | 22A | ··· | 22AI | 44A | ··· | 44AN |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 11 | ··· | 11 | 22 | ··· | 22 | 44 | ··· | 44 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 22 | ··· | 22 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
100 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | - | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4○D4 | D11 | Dic11 | D22 | D22 | D44⋊5C2 |
| kernel | C23.21D22 | C4×Dic11 | C44⋊C4 | C23.D11 | C22×C44 | C2×C44 | C22 | C22×C4 | C2×C4 | C2×C4 | C23 | C2 |
| # reps | 1 | 2 | 2 | 2 | 1 | 8 | 4 | 5 | 20 | 10 | 5 | 40 |
Matrix representation of C23.21D22 ►in GL4(𝔽89) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 51 | 88 |
| 88 | 0 | 0 | 0 |
| 0 | 88 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 88 | 0 |
| 0 | 0 | 0 | 88 |
| 85 | 0 | 0 | 0 |
| 59 | 22 | 0 | 0 |
| 0 | 0 | 84 | 0 |
| 0 | 0 | 81 | 18 |
| 47 | 72 | 0 | 0 |
| 41 | 42 | 0 | 0 |
| 0 | 0 | 28 | 53 |
| 0 | 0 | 49 | 61 |
G:=sub<GL(4,GF(89))| [1,0,0,0,0,1,0,0,0,0,1,51,0,0,0,88],[88,0,0,0,0,88,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,88,0,0,0,0,88],[85,59,0,0,0,22,0,0,0,0,84,81,0,0,0,18],[47,41,0,0,72,42,0,0,0,0,28,49,0,0,53,61] >;
C23.21D22 in GAP, Magma, Sage, TeX
C_2^3._{21}D_{22} % in TeX
G:=Group("C2^3.21D22"); // GroupNames label
G:=SmallGroup(352,121);
// by ID
G=gap.SmallGroup(352,121);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-11,48,103,362,11525]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=1,d^22=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=d^21>;
// generators/relations