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## G = C23.D22order 352 = 25·11

### 3rd non-split extension by C23 of D22 acting via D22/C11=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C22 — C23.D22
 Chief series C1 — C11 — C22 — C2×C22 — C2×Dic11 — C4×Dic11 — C23.D22
 Lower central C11 — C2×C22 — C23.D22
 Upper central C1 — C22 — C22⋊C4

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, C422C2, 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, C422C2, D22, C22×D11, D445C2, D42D11, C23.D22

Smallest permutation representation of 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`

׿
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