metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C23.11D22, (C2×C4).26D22, Dic11⋊C4⋊7C2, (C4×Dic11)⋊9C2, (C2×Dic11)⋊3C4, C22.5(C22×C4), C22⋊C4.3D11, C22.6(C4×D11), C11⋊2(C42⋊C2), C22.20(C4○D4), (C2×C22).18C23, (C2×C44).50C22, Dic11.8(C2×C4), C23.D11.1C2, C2.1(D4⋊2D11), (C22×C22).7C22, (C22×Dic11).2C2, C22.12(C22×D11), (C2×Dic11).46C22, C2.7(C2×C4×D11), (C2×C22).4(C2×C4), (C11×C22⋊C4).3C2, SmallGroup(352,72)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C23.11D22
G = < a,b,c,d,e | a2=b2=c2=1, d22=b, e2=cb=bc, ab=ba, dad-1=eae-1=ac=ca, bd=db, be=eb, cd=dc, ce=ec, ede-1=d21 >
Subgroups: 322 in 76 conjugacy classes, 41 normal (15 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C2×C4, C2×C4, C23, C11, C42, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C22, C22, C22, C42⋊C2, Dic11, Dic11, C44, C2×C22, C2×C22, C2×C22, C2×Dic11, C2×Dic11, C2×C44, C22×C22, C4×Dic11, Dic11⋊C4, C23.D11, C11×C22⋊C4, C22×Dic11, C23.11D22
Quotients: C1, C2, C4, C22, C2×C4, C23, C22×C4, C4○D4, D11, C42⋊C2, D22, C4×D11, C22×D11, C2×C4×D11, D4⋊2D11, C23.11D22
►On 176 points
Generators in S176
(2 106)(4 108)(6 110)(8 112)(10 114)(12 116)(14 118)(16 120)(18 122)(20 124)(22 126)(24 128)(26 130)(28 132)(30 90)(32 92)(34 94)(36 96)(38 98)(40 100)(42 102)(44 104)(46 167)(48 169)(50 171)(52 173)(54 175)(56 133)(58 135)(60 137)(62 139)(64 141)(66 143)(68 145)(70 147)(72 149)(74 151)(76 153)(78 155)(80 157)(82 159)(84 161)(86 163)(88 165)
(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)
(1 105)(2 106)(3 107)(4 108)(5 109)(6 110)(7 111)(8 112)(9 113)(10 114)(11 115)(12 116)(13 117)(14 118)(15 119)(16 120)(17 121)(18 122)(19 123)(20 124)(21 125)(22 126)(23 127)(24 128)(25 129)(26 130)(27 131)(28 132)(29 89)(30 90)(31 91)(32 92)(33 93)(34 94)(35 95)(36 96)(37 97)(38 98)(39 99)(40 100)(41 101)(42 102)(43 103)(44 104)(45 166)(46 167)(47 168)(48 169)(49 170)(50 171)(51 172)(52 173)(53 174)(54 175)(55 176)(56 133)(57 134)(58 135)(59 136)(60 137)(61 138)(62 139)(63 140)(64 141)(65 142)(66 143)(67 144)(68 145)(69 146)(70 147)(71 148)(72 149)(73 150)(74 151)(75 152)(76 153)(77 154)(78 155)(79 156)(80 157)(81 158)(82 159)(83 160)(84 161)(85 162)(86 163)(87 164)(88 165)
(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)
(1 145 127 46)(2 166 128 67)(3 143 129 88)(4 164 130 65)(5 141 131 86)(6 162 132 63)(7 139 89 84)(8 160 90 61)(9 137 91 82)(10 158 92 59)(11 135 93 80)(12 156 94 57)(13 133 95 78)(14 154 96 55)(15 175 97 76)(16 152 98 53)(17 173 99 74)(18 150 100 51)(19 171 101 72)(20 148 102 49)(21 169 103 70)(22 146 104 47)(23 167 105 68)(24 144 106 45)(25 165 107 66)(26 142 108 87)(27 163 109 64)(28 140 110 85)(29 161 111 62)(30 138 112 83)(31 159 113 60)(32 136 114 81)(33 157 115 58)(34 134 116 79)(35 155 117 56)(36 176 118 77)(37 153 119 54)(38 174 120 75)(39 151 121 52)(40 172 122 73)(41 149 123 50)(42 170 124 71)(43 147 125 48)(44 168 126 69)
G:=sub<Sym(176)| (2,106)(4,108)(6,110)(8,112)(10,114)(12,116)(14,118)(16,120)(18,122)(20,124)(22,126)(24,128)(26,130)(28,132)(30,90)(32,92)(34,94)(36,96)(38,98)(40,100)(42,102)(44,104)(46,167)(48,169)(50,171)(52,173)(54,175)(56,133)(58,135)(60,137)(62,139)(64,141)(66,143)(68,145)(70,147)(72,149)(74,151)(76,153)(78,155)(80,157)(82,159)(84,161)(86,163)(88,165), (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), (1,105)(2,106)(3,107)(4,108)(5,109)(6,110)(7,111)(8,112)(9,113)(10,114)(11,115)(12,116)(13,117)(14,118)(15,119)(16,120)(17,121)(18,122)(19,123)(20,124)(21,125)(22,126)(23,127)(24,128)(25,129)(26,130)(27,131)(28,132)(29,89)(30,90)(31,91)(32,92)(33,93)(34,94)(35,95)(36,96)(37,97)(38,98)(39,99)(40,100)(41,101)(42,102)(43,103)(44,104)(45,166)(46,167)(47,168)(48,169)(49,170)(50,171)(51,172)(52,173)(53,174)(54,175)(55,176)(56,133)(57,134)(58,135)(59,136)(60,137)(61,138)(62,139)(63,140)(64,141)(65,142)(66,143)(67,144)(68,145)(69,146)(70,147)(71,148)(72,149)(73,150)(74,151)(75,152)(76,153)(77,154)(78,155)(79,156)(80,157)(81,158)(82,159)(83,160)(84,161)(85,162)(86,163)(87,164)(88,165), (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), (1,145,127,46)(2,166,128,67)(3,143,129,88)(4,164,130,65)(5,141,131,86)(6,162,132,63)(7,139,89,84)(8,160,90,61)(9,137,91,82)(10,158,92,59)(11,135,93,80)(12,156,94,57)(13,133,95,78)(14,154,96,55)(15,175,97,76)(16,152,98,53)(17,173,99,74)(18,150,100,51)(19,171,101,72)(20,148,102,49)(21,169,103,70)(22,146,104,47)(23,167,105,68)(24,144,106,45)(25,165,107,66)(26,142,108,87)(27,163,109,64)(28,140,110,85)(29,161,111,62)(30,138,112,83)(31,159,113,60)(32,136,114,81)(33,157,115,58)(34,134,116,79)(35,155,117,56)(36,176,118,77)(37,153,119,54)(38,174,120,75)(39,151,121,52)(40,172,122,73)(41,149,123,50)(42,170,124,71)(43,147,125,48)(44,168,126,69)>;
G:=Group( (2,106)(4,108)(6,110)(8,112)(10,114)(12,116)(14,118)(16,120)(18,122)(20,124)(22,126)(24,128)(26,130)(28,132)(30,90)(32,92)(34,94)(36,96)(38,98)(40,100)(42,102)(44,104)(46,167)(48,169)(50,171)(52,173)(54,175)(56,133)(58,135)(60,137)(62,139)(64,141)(66,143)(68,145)(70,147)(72,149)(74,151)(76,153)(78,155)(80,157)(82,159)(84,161)(86,163)(88,165), (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), (1,105)(2,106)(3,107)(4,108)(5,109)(6,110)(7,111)(8,112)(9,113)(10,114)(11,115)(12,116)(13,117)(14,118)(15,119)(16,120)(17,121)(18,122)(19,123)(20,124)(21,125)(22,126)(23,127)(24,128)(25,129)(26,130)(27,131)(28,132)(29,89)(30,90)(31,91)(32,92)(33,93)(34,94)(35,95)(36,96)(37,97)(38,98)(39,99)(40,100)(41,101)(42,102)(43,103)(44,104)(45,166)(46,167)(47,168)(48,169)(49,170)(50,171)(51,172)(52,173)(53,174)(54,175)(55,176)(56,133)(57,134)(58,135)(59,136)(60,137)(61,138)(62,139)(63,140)(64,141)(65,142)(66,143)(67,144)(68,145)(69,146)(70,147)(71,148)(72,149)(73,150)(74,151)(75,152)(76,153)(77,154)(78,155)(79,156)(80,157)(81,158)(82,159)(83,160)(84,161)(85,162)(86,163)(87,164)(88,165), (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), (1,145,127,46)(2,166,128,67)(3,143,129,88)(4,164,130,65)(5,141,131,86)(6,162,132,63)(7,139,89,84)(8,160,90,61)(9,137,91,82)(10,158,92,59)(11,135,93,80)(12,156,94,57)(13,133,95,78)(14,154,96,55)(15,175,97,76)(16,152,98,53)(17,173,99,74)(18,150,100,51)(19,171,101,72)(20,148,102,49)(21,169,103,70)(22,146,104,47)(23,167,105,68)(24,144,106,45)(25,165,107,66)(26,142,108,87)(27,163,109,64)(28,140,110,85)(29,161,111,62)(30,138,112,83)(31,159,113,60)(32,136,114,81)(33,157,115,58)(34,134,116,79)(35,155,117,56)(36,176,118,77)(37,153,119,54)(38,174,120,75)(39,151,121,52)(40,172,122,73)(41,149,123,50)(42,170,124,71)(43,147,125,48)(44,168,126,69) );
G=PermutationGroup([[(2,106),(4,108),(6,110),(8,112),(10,114),(12,116),(14,118),(16,120),(18,122),(20,124),(22,126),(24,128),(26,130),(28,132),(30,90),(32,92),(34,94),(36,96),(38,98),(40,100),(42,102),(44,104),(46,167),(48,169),(50,171),(52,173),(54,175),(56,133),(58,135),(60,137),(62,139),(64,141),(66,143),(68,145),(70,147),(72,149),(74,151),(76,153),(78,155),(80,157),(82,159),(84,161),(86,163),(88,165)], [(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)], [(1,105),(2,106),(3,107),(4,108),(5,109),(6,110),(7,111),(8,112),(9,113),(10,114),(11,115),(12,116),(13,117),(14,118),(15,119),(16,120),(17,121),(18,122),(19,123),(20,124),(21,125),(22,126),(23,127),(24,128),(25,129),(26,130),(27,131),(28,132),(29,89),(30,90),(31,91),(32,92),(33,93),(34,94),(35,95),(36,96),(37,97),(38,98),(39,99),(40,100),(41,101),(42,102),(43,103),(44,104),(45,166),(46,167),(47,168),(48,169),(49,170),(50,171),(51,172),(52,173),(53,174),(54,175),(55,176),(56,133),(57,134),(58,135),(59,136),(60,137),(61,138),(62,139),(63,140),(64,141),(65,142),(66,143),(67,144),(68,145),(69,146),(70,147),(71,148),(72,149),(73,150),(74,151),(75,152),(76,153),(77,154),(78,155),(79,156),(80,157),(81,158),(82,159),(83,160),(84,161),(85,162),(86,163),(87,164),(88,165)], [(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)], [(1,145,127,46),(2,166,128,67),(3,143,129,88),(4,164,130,65),(5,141,131,86),(6,162,132,63),(7,139,89,84),(8,160,90,61),(9,137,91,82),(10,158,92,59),(11,135,93,80),(12,156,94,57),(13,133,95,78),(14,154,96,55),(15,175,97,76),(16,152,98,53),(17,173,99,74),(18,150,100,51),(19,171,101,72),(20,148,102,49),(21,169,103,70),(22,146,104,47),(23,167,105,68),(24,144,106,45),(25,165,107,66),(26,142,108,87),(27,163,109,64),(28,140,110,85),(29,161,111,62),(30,138,112,83),(31,159,113,60),(32,136,114,81),(33,157,115,58),(34,134,116,79),(35,155,117,56),(36,176,118,77),(37,153,119,54),(38,174,120,75),(39,151,121,52),(40,172,122,73),(41,149,123,50),(42,170,124,71),(43,147,125,48),(44,168,126,69)]])
70 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | ··· | 4N | 11A | ··· | 11E | 22A | ··· | 22O | 22P | ··· | 22Y | 44A | ··· | 44T |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 11 | ··· | 11 | 22 | ··· | 22 | 22 | ··· | 22 | 44 | ··· | 44 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 11 | 11 | 11 | 11 | 22 | ··· | 22 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
70 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4○D4 | D11 | D22 | D22 | C4×D11 | D4⋊2D11 |
| kernel | C23.11D22 | C4×Dic11 | Dic11⋊C4 | C23.D11 | C11×C22⋊C4 | C22×Dic11 | C2×Dic11 | C22 | C22⋊C4 | C2×C4 | C23 | C22 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 8 | 4 | 5 | 10 | 5 | 20 | 10 |
Matrix representation of C23.11D22 ►in GL4(𝔽89) generated by
| 1 | 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 |
| 88 | 0 | 0 | 0 |
| 0 | 88 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 4 | 4 |
| 0 | 0 | 67 | 0 |
| 0 | 55 | 0 | 0 |
| 55 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 |
| 0 | 0 | 22 | 0 |
G:=sub<GL(4,GF(89))| [1,0,0,0,0,88,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,88,0,0,0,0,88],[88,0,0,0,0,88,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,4,67,0,0,4,0],[0,55,0,0,55,0,0,0,0,0,0,22,0,0,4,0] >;
C23.11D22 in GAP, Magma, Sage, TeX
C_2^3._{11}D_{22} % in TeX
G:=Group("C2^3.11D22"); // GroupNames label
G:=SmallGroup(352,72);
// by ID
G=gap.SmallGroup(352,72);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-11,96,188,50,11525]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=1,d^22=b,e^2=c*b=b*c,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=d^21>;
// generators/relations