metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C23.3D22, C44⋊C4⋊3C2, (C2×C4).27D22, Dic11⋊C4⋊8C2, C22.7(C4○D4), (C2×C44).2C22, C22⋊C4.2D11, C11⋊2(C42⋊2C2), (C4×Dic11)⋊10C2, (C2×C22).20C23, C23.D11.3C2, C2.7(D4⋊2D11), C2.9(D44⋊5C2), (C22×C22).9C22, C22.40(C22×D11), (C2×Dic11).26C22, (C11×C22⋊C4).2C2, SmallGroup(352,74)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C23.D22
G = < a,b,c,d,e | a2=b2=c2=1, d22=b, e2=cb=bc, eae-1=ab=ba, dad-1=ac=ca, bd=db, be=eb, cd=dc, ce=ec, ede-1=d21 >
Subgroups: 274 in 60 conjugacy classes, 29 normal (all characteristic)
C1, C2, C2, C4, C22, C22, C2×C4, C2×C4, C23, C11, C42, C22⋊C4, C22⋊C4, C4⋊C4, C22, C22, C42⋊2C2, Dic11, C44, C2×C22, C2×C22, C2×Dic11, C2×C44, C22×C22, C4×Dic11, Dic11⋊C4, C44⋊C4, C23.D11, C11×C22⋊C4, C23.D22
Quotients: C1, C2, C22, C23, C4○D4, D11, C42⋊2C2, D22, C22×D11, D44⋊5C2, D4⋊2D11, C23.D22
►On 176 points
Generators in S176
(2 124)(4 126)(6 128)(8 130)(10 132)(12 90)(14 92)(16 94)(18 96)(20 98)(22 100)(24 102)(26 104)(28 106)(30 108)(32 110)(34 112)(36 114)(38 116)(40 118)(42 120)(44 122)(45 67)(46 138)(47 69)(48 140)(49 71)(50 142)(51 73)(52 144)(53 75)(54 146)(55 77)(56 148)(57 79)(58 150)(59 81)(60 152)(61 83)(62 154)(63 85)(64 156)(65 87)(66 158)(68 160)(70 162)(72 164)(74 166)(76 168)(78 170)(80 172)(82 174)(84 176)(86 134)(88 136)(133 155)(135 157)(137 159)(139 161)(141 163)(143 165)(145 167)(147 169)(149 171)(151 173)(153 175)
(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)
(1 123)(2 124)(3 125)(4 126)(5 127)(6 128)(7 129)(8 130)(9 131)(10 132)(11 89)(12 90)(13 91)(14 92)(15 93)(16 94)(17 95)(18 96)(19 97)(20 98)(21 99)(22 100)(23 101)(24 102)(25 103)(26 104)(27 105)(28 106)(29 107)(30 108)(31 109)(32 110)(33 111)(34 112)(35 113)(36 114)(37 115)(38 116)(39 117)(40 118)(41 119)(42 120)(43 121)(44 122)(45 159)(46 160)(47 161)(48 162)(49 163)(50 164)(51 165)(52 166)(53 167)(54 168)(55 169)(56 170)(57 171)(58 172)(59 173)(60 174)(61 175)(62 176)(63 133)(64 134)(65 135)(66 136)(67 137)(68 138)(69 139)(70 140)(71 141)(72 142)(73 143)(74 144)(75 145)(76 146)(77 147)(78 148)(79 149)(80 150)(81 151)(82 152)(83 153)(84 154)(85 155)(86 156)(87 157)(88 158)
(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 51 101 143)(2 72 102 164)(3 49 103 141)(4 70 104 162)(5 47 105 139)(6 68 106 160)(7 45 107 137)(8 66 108 158)(9 87 109 135)(10 64 110 156)(11 85 111 133)(12 62 112 154)(13 83 113 175)(14 60 114 152)(15 81 115 173)(16 58 116 150)(17 79 117 171)(18 56 118 148)(19 77 119 169)(20 54 120 146)(21 75 121 167)(22 52 122 144)(23 73 123 165)(24 50 124 142)(25 71 125 163)(26 48 126 140)(27 69 127 161)(28 46 128 138)(29 67 129 159)(30 88 130 136)(31 65 131 157)(32 86 132 134)(33 63 89 155)(34 84 90 176)(35 61 91 153)(36 82 92 174)(37 59 93 151)(38 80 94 172)(39 57 95 149)(40 78 96 170)(41 55 97 147)(42 76 98 168)(43 53 99 145)(44 74 100 166)
G:=sub<Sym(176)| (2,124)(4,126)(6,128)(8,130)(10,132)(12,90)(14,92)(16,94)(18,96)(20,98)(22,100)(24,102)(26,104)(28,106)(30,108)(32,110)(34,112)(36,114)(38,116)(40,118)(42,120)(44,122)(45,67)(46,138)(47,69)(48,140)(49,71)(50,142)(51,73)(52,144)(53,75)(54,146)(55,77)(56,148)(57,79)(58,150)(59,81)(60,152)(61,83)(62,154)(63,85)(64,156)(65,87)(66,158)(68,160)(70,162)(72,164)(74,166)(76,168)(78,170)(80,172)(82,174)(84,176)(86,134)(88,136)(133,155)(135,157)(137,159)(139,161)(141,163)(143,165)(145,167)(147,169)(149,171)(151,173)(153,175), (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), (1,123)(2,124)(3,125)(4,126)(5,127)(6,128)(7,129)(8,130)(9,131)(10,132)(11,89)(12,90)(13,91)(14,92)(15,93)(16,94)(17,95)(18,96)(19,97)(20,98)(21,99)(22,100)(23,101)(24,102)(25,103)(26,104)(27,105)(28,106)(29,107)(30,108)(31,109)(32,110)(33,111)(34,112)(35,113)(36,114)(37,115)(38,116)(39,117)(40,118)(41,119)(42,120)(43,121)(44,122)(45,159)(46,160)(47,161)(48,162)(49,163)(50,164)(51,165)(52,166)(53,167)(54,168)(55,169)(56,170)(57,171)(58,172)(59,173)(60,174)(61,175)(62,176)(63,133)(64,134)(65,135)(66,136)(67,137)(68,138)(69,139)(70,140)(71,141)(72,142)(73,143)(74,144)(75,145)(76,146)(77,147)(78,148)(79,149)(80,150)(81,151)(82,152)(83,153)(84,154)(85,155)(86,156)(87,157)(88,158), (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), (1,51,101,143)(2,72,102,164)(3,49,103,141)(4,70,104,162)(5,47,105,139)(6,68,106,160)(7,45,107,137)(8,66,108,158)(9,87,109,135)(10,64,110,156)(11,85,111,133)(12,62,112,154)(13,83,113,175)(14,60,114,152)(15,81,115,173)(16,58,116,150)(17,79,117,171)(18,56,118,148)(19,77,119,169)(20,54,120,146)(21,75,121,167)(22,52,122,144)(23,73,123,165)(24,50,124,142)(25,71,125,163)(26,48,126,140)(27,69,127,161)(28,46,128,138)(29,67,129,159)(30,88,130,136)(31,65,131,157)(32,86,132,134)(33,63,89,155)(34,84,90,176)(35,61,91,153)(36,82,92,174)(37,59,93,151)(38,80,94,172)(39,57,95,149)(40,78,96,170)(41,55,97,147)(42,76,98,168)(43,53,99,145)(44,74,100,166)>;
G:=Group( (2,124)(4,126)(6,128)(8,130)(10,132)(12,90)(14,92)(16,94)(18,96)(20,98)(22,100)(24,102)(26,104)(28,106)(30,108)(32,110)(34,112)(36,114)(38,116)(40,118)(42,120)(44,122)(45,67)(46,138)(47,69)(48,140)(49,71)(50,142)(51,73)(52,144)(53,75)(54,146)(55,77)(56,148)(57,79)(58,150)(59,81)(60,152)(61,83)(62,154)(63,85)(64,156)(65,87)(66,158)(68,160)(70,162)(72,164)(74,166)(76,168)(78,170)(80,172)(82,174)(84,176)(86,134)(88,136)(133,155)(135,157)(137,159)(139,161)(141,163)(143,165)(145,167)(147,169)(149,171)(151,173)(153,175), (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), (1,123)(2,124)(3,125)(4,126)(5,127)(6,128)(7,129)(8,130)(9,131)(10,132)(11,89)(12,90)(13,91)(14,92)(15,93)(16,94)(17,95)(18,96)(19,97)(20,98)(21,99)(22,100)(23,101)(24,102)(25,103)(26,104)(27,105)(28,106)(29,107)(30,108)(31,109)(32,110)(33,111)(34,112)(35,113)(36,114)(37,115)(38,116)(39,117)(40,118)(41,119)(42,120)(43,121)(44,122)(45,159)(46,160)(47,161)(48,162)(49,163)(50,164)(51,165)(52,166)(53,167)(54,168)(55,169)(56,170)(57,171)(58,172)(59,173)(60,174)(61,175)(62,176)(63,133)(64,134)(65,135)(66,136)(67,137)(68,138)(69,139)(70,140)(71,141)(72,142)(73,143)(74,144)(75,145)(76,146)(77,147)(78,148)(79,149)(80,150)(81,151)(82,152)(83,153)(84,154)(85,155)(86,156)(87,157)(88,158), (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), (1,51,101,143)(2,72,102,164)(3,49,103,141)(4,70,104,162)(5,47,105,139)(6,68,106,160)(7,45,107,137)(8,66,108,158)(9,87,109,135)(10,64,110,156)(11,85,111,133)(12,62,112,154)(13,83,113,175)(14,60,114,152)(15,81,115,173)(16,58,116,150)(17,79,117,171)(18,56,118,148)(19,77,119,169)(20,54,120,146)(21,75,121,167)(22,52,122,144)(23,73,123,165)(24,50,124,142)(25,71,125,163)(26,48,126,140)(27,69,127,161)(28,46,128,138)(29,67,129,159)(30,88,130,136)(31,65,131,157)(32,86,132,134)(33,63,89,155)(34,84,90,176)(35,61,91,153)(36,82,92,174)(37,59,93,151)(38,80,94,172)(39,57,95,149)(40,78,96,170)(41,55,97,147)(42,76,98,168)(43,53,99,145)(44,74,100,166) );
G=PermutationGroup([[(2,124),(4,126),(6,128),(8,130),(10,132),(12,90),(14,92),(16,94),(18,96),(20,98),(22,100),(24,102),(26,104),(28,106),(30,108),(32,110),(34,112),(36,114),(38,116),(40,118),(42,120),(44,122),(45,67),(46,138),(47,69),(48,140),(49,71),(50,142),(51,73),(52,144),(53,75),(54,146),(55,77),(56,148),(57,79),(58,150),(59,81),(60,152),(61,83),(62,154),(63,85),(64,156),(65,87),(66,158),(68,160),(70,162),(72,164),(74,166),(76,168),(78,170),(80,172),(82,174),(84,176),(86,134),(88,136),(133,155),(135,157),(137,159),(139,161),(141,163),(143,165),(145,167),(147,169),(149,171),(151,173),(153,175)], [(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)], [(1,123),(2,124),(3,125),(4,126),(5,127),(6,128),(7,129),(8,130),(9,131),(10,132),(11,89),(12,90),(13,91),(14,92),(15,93),(16,94),(17,95),(18,96),(19,97),(20,98),(21,99),(22,100),(23,101),(24,102),(25,103),(26,104),(27,105),(28,106),(29,107),(30,108),(31,109),(32,110),(33,111),(34,112),(35,113),(36,114),(37,115),(38,116),(39,117),(40,118),(41,119),(42,120),(43,121),(44,122),(45,159),(46,160),(47,161),(48,162),(49,163),(50,164),(51,165),(52,166),(53,167),(54,168),(55,169),(56,170),(57,171),(58,172),(59,173),(60,174),(61,175),(62,176),(63,133),(64,134),(65,135),(66,136),(67,137),(68,138),(69,139),(70,140),(71,141),(72,142),(73,143),(74,144),(75,145),(76,146),(77,147),(78,148),(79,149),(80,150),(81,151),(82,152),(83,153),(84,154),(85,155),(86,156),(87,157),(88,158)], [(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)], [(1,51,101,143),(2,72,102,164),(3,49,103,141),(4,70,104,162),(5,47,105,139),(6,68,106,160),(7,45,107,137),(8,66,108,158),(9,87,109,135),(10,64,110,156),(11,85,111,133),(12,62,112,154),(13,83,113,175),(14,60,114,152),(15,81,115,173),(16,58,116,150),(17,79,117,171),(18,56,118,148),(19,77,119,169),(20,54,120,146),(21,75,121,167),(22,52,122,144),(23,73,123,165),(24,50,124,142),(25,71,125,163),(26,48,126,140),(27,69,127,161),(28,46,128,138),(29,67,129,159),(30,88,130,136),(31,65,131,157),(32,86,132,134),(33,63,89,155),(34,84,90,176),(35,61,91,153),(36,82,92,174),(37,59,93,151),(38,80,94,172),(39,57,95,149),(40,78,96,170),(41,55,97,147),(42,76,98,168),(43,53,99,145),(44,74,100,166)]])
64 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 11A | ··· | 11E | 22A | ··· | 22O | 22P | ··· | 22Y | 44A | ··· | 44T |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 11 | ··· | 11 | 22 | ··· | 22 | 22 | ··· | 22 | 44 | ··· | 44 |
| size | 1 | 1 | 1 | 1 | 4 | 2 | 2 | 4 | 22 | 22 | 22 | 22 | 44 | 44 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
64 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4○D4 | D11 | D22 | D22 | D44⋊5C2 | D4⋊2D11 |
| kernel | C23.D22 | C4×Dic11 | Dic11⋊C4 | C44⋊C4 | C23.D11 | C11×C22⋊C4 | C22 | C22⋊C4 | C2×C4 | C23 | C2 | C2 |
| # reps | 1 | 1 | 2 | 1 | 2 | 1 | 6 | 5 | 10 | 5 | 20 | 10 |
Matrix representation of C23.D22 ►in GL6(𝔽89)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 67 | 88 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 46 | 88 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 88 | 0 | 0 | 0 |
| 0 | 0 | 0 | 88 | 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 | 88 | 0 |
| 0 | 0 | 0 | 0 | 0 | 88 |
| 87 | 0 | 0 | 0 | 0 | 0 |
| 39 | 44 | 0 | 0 | 0 | 0 |
| 0 | 0 | 34 | 0 | 0 | 0 |
| 0 | 0 | 0 | 34 | 0 | 0 |
| 0 | 0 | 0 | 0 | 5 | 23 |
| 0 | 0 | 0 | 0 | 57 | 84 |
| 28 | 49 | 0 | 0 | 0 | 0 |
| 4 | 61 | 0 | 0 | 0 | 0 |
| 0 | 0 | 88 | 8 | 0 | 0 |
| 0 | 0 | 22 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 34 | 0 |
| 0 | 0 | 0 | 0 | 0 | 34 |
G:=sub<GL(6,GF(89))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,67,0,0,0,0,0,88,0,0,0,0,0,0,1,46,0,0,0,0,0,88],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,88,0,0,0,0,0,0,88,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,88,0,0,0,0,0,0,88],[87,39,0,0,0,0,0,44,0,0,0,0,0,0,34,0,0,0,0,0,0,34,0,0,0,0,0,0,5,57,0,0,0,0,23,84],[28,4,0,0,0,0,49,61,0,0,0,0,0,0,88,22,0,0,0,0,8,1,0,0,0,0,0,0,34,0,0,0,0,0,0,34] >;
C23.D22 in GAP, Magma, Sage, TeX
C_2^3.D_{22} % in TeX
G:=Group("C2^3.D22"); // GroupNames label
G:=SmallGroup(352,74);
// by ID
G=gap.SmallGroup(352,74);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-11,96,55,506,188,11525]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=1,d^22=b,e^2=c*b=b*c,e*a*e^-1=a*b=b*a,d*a*d^-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=d^21>;
// generators/relations