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## G = C23.18D22order 352 = 25·11

### 8th non-split extension by C23 of D22 acting via D22/D11=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C22 — C23.18D22
 Chief series C1 — C11 — C22 — C2×C22 — C2×Dic11 — C22×Dic11 — C23.18D22
 Lower central C11 — C2×C22 — C23.18D22
 Upper central C1 — C22 — C2×D4

Generators and relations for C23.18D22
G = < a,b,c,d,e | a2=b2=c2=d22=1, e2=cb=bc, ab=ba, dad-1=eae-1=ac=ca, bd=db, be=eb, cd=dc, ce=ec, ede-1=bd-1 >

Subgroups: 346 in 78 conjugacy classes, 33 normal (17 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C2×C4, C2×C4, D4, C23, C11, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C22, C22, C22, C22.D4, Dic11, C44, C2×C22, C2×C22, C2×C22, C2×Dic11, C2×Dic11, C2×C44, D4×C11, C22×C22, Dic11⋊C4, C23.D11, C23.D11, C22×Dic11, D4×C22, C23.18D22
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, D11, C22.D4, D22, C11⋊D4, C22×D11, D42D11, C2×C11⋊D4, C23.18D22

Smallest permutation representation of C23.18D22
On 176 points
Generators in S176
```(1 55)(2 45)(3 57)(4 47)(5 59)(6 49)(7 61)(8 51)(9 63)(10 53)(11 65)(12 48)(13 60)(14 50)(15 62)(16 52)(17 64)(18 54)(19 66)(20 56)(21 46)(22 58)(23 83)(24 73)(25 85)(26 75)(27 87)(28 77)(29 67)(30 79)(31 69)(32 81)(33 71)(34 72)(35 84)(36 74)(37 86)(38 76)(39 88)(40 78)(41 68)(42 80)(43 70)(44 82)(89 141)(90 171)(91 143)(92 173)(93 145)(94 175)(95 147)(96 155)(97 149)(98 157)(99 151)(100 159)(101 153)(102 161)(103 133)(104 163)(105 135)(106 165)(107 137)(108 167)(109 139)(110 169)(111 140)(112 170)(113 142)(114 172)(115 144)(116 174)(117 146)(118 176)(119 148)(120 156)(121 150)(122 158)(123 152)(124 160)(125 154)(126 162)(127 134)(128 164)(129 136)(130 166)(131 138)(132 168)
(1 30)(2 31)(3 32)(4 33)(5 23)(6 24)(7 25)(8 26)(9 27)(10 28)(11 29)(12 34)(13 35)(14 36)(15 37)(16 38)(17 39)(18 40)(19 41)(20 42)(21 43)(22 44)(45 69)(46 70)(47 71)(48 72)(49 73)(50 74)(51 75)(52 76)(53 77)(54 78)(55 79)(56 80)(57 81)(58 82)(59 83)(60 84)(61 85)(62 86)(63 87)(64 88)(65 67)(66 68)(89 100)(90 101)(91 102)(92 103)(93 104)(94 105)(95 106)(96 107)(97 108)(98 109)(99 110)(111 122)(112 123)(113 124)(114 125)(115 126)(116 127)(117 128)(118 129)(119 130)(120 131)(121 132)(133 173)(134 174)(135 175)(136 176)(137 155)(138 156)(139 157)(140 158)(141 159)(142 160)(143 161)(144 162)(145 163)(146 164)(147 165)(148 166)(149 167)(150 168)(151 169)(152 170)(153 171)(154 172)
(1 19)(2 20)(3 21)(4 22)(5 12)(6 13)(7 14)(8 15)(9 16)(10 17)(11 18)(23 34)(24 35)(25 36)(26 37)(27 38)(28 39)(29 40)(30 41)(31 42)(32 43)(33 44)(45 56)(46 57)(47 58)(48 59)(49 60)(50 61)(51 62)(52 63)(53 64)(54 65)(55 66)(67 78)(68 79)(69 80)(70 81)(71 82)(72 83)(73 84)(74 85)(75 86)(76 87)(77 88)(89 112)(90 113)(91 114)(92 115)(93 116)(94 117)(95 118)(96 119)(97 120)(98 121)(99 122)(100 123)(101 124)(102 125)(103 126)(104 127)(105 128)(106 129)(107 130)(108 131)(109 132)(110 111)(133 162)(134 163)(135 164)(136 165)(137 166)(138 167)(139 168)(140 169)(141 170)(142 171)(143 172)(144 173)(145 174)(146 175)(147 176)(148 155)(149 156)(150 157)(151 158)(152 159)(153 160)(154 161)
(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)
(1 102 41 114)(2 90 42 124)(3 100 43 112)(4 110 44 122)(5 98 34 132)(6 108 35 120)(7 96 36 130)(8 106 37 118)(9 94 38 128)(10 104 39 116)(11 92 40 126)(12 121 23 109)(13 131 24 97)(14 119 25 107)(15 129 26 95)(16 117 27 105)(17 127 28 93)(18 115 29 103)(19 125 30 91)(20 113 31 101)(21 123 32 89)(22 111 33 99)(45 142 80 153)(46 159 81 170)(47 140 82 151)(48 157 83 168)(49 138 84 149)(50 155 85 166)(51 136 86 147)(52 175 87 164)(53 134 88 145)(54 173 67 162)(55 154 68 143)(56 171 69 160)(57 152 70 141)(58 169 71 158)(59 150 72 139)(60 167 73 156)(61 148 74 137)(62 165 75 176)(63 146 76 135)(64 163 77 174)(65 144 78 133)(66 161 79 172)```

`G:=sub<Sym(176)| (1,55)(2,45)(3,57)(4,47)(5,59)(6,49)(7,61)(8,51)(9,63)(10,53)(11,65)(12,48)(13,60)(14,50)(15,62)(16,52)(17,64)(18,54)(19,66)(20,56)(21,46)(22,58)(23,83)(24,73)(25,85)(26,75)(27,87)(28,77)(29,67)(30,79)(31,69)(32,81)(33,71)(34,72)(35,84)(36,74)(37,86)(38,76)(39,88)(40,78)(41,68)(42,80)(43,70)(44,82)(89,141)(90,171)(91,143)(92,173)(93,145)(94,175)(95,147)(96,155)(97,149)(98,157)(99,151)(100,159)(101,153)(102,161)(103,133)(104,163)(105,135)(106,165)(107,137)(108,167)(109,139)(110,169)(111,140)(112,170)(113,142)(114,172)(115,144)(116,174)(117,146)(118,176)(119,148)(120,156)(121,150)(122,158)(123,152)(124,160)(125,154)(126,162)(127,134)(128,164)(129,136)(130,166)(131,138)(132,168), (1,30)(2,31)(3,32)(4,33)(5,23)(6,24)(7,25)(8,26)(9,27)(10,28)(11,29)(12,34)(13,35)(14,36)(15,37)(16,38)(17,39)(18,40)(19,41)(20,42)(21,43)(22,44)(45,69)(46,70)(47,71)(48,72)(49,73)(50,74)(51,75)(52,76)(53,77)(54,78)(55,79)(56,80)(57,81)(58,82)(59,83)(60,84)(61,85)(62,86)(63,87)(64,88)(65,67)(66,68)(89,100)(90,101)(91,102)(92,103)(93,104)(94,105)(95,106)(96,107)(97,108)(98,109)(99,110)(111,122)(112,123)(113,124)(114,125)(115,126)(116,127)(117,128)(118,129)(119,130)(120,131)(121,132)(133,173)(134,174)(135,175)(136,176)(137,155)(138,156)(139,157)(140,158)(141,159)(142,160)(143,161)(144,162)(145,163)(146,164)(147,165)(148,166)(149,167)(150,168)(151,169)(152,170)(153,171)(154,172), (1,19)(2,20)(3,21)(4,22)(5,12)(6,13)(7,14)(8,15)(9,16)(10,17)(11,18)(23,34)(24,35)(25,36)(26,37)(27,38)(28,39)(29,40)(30,41)(31,42)(32,43)(33,44)(45,56)(46,57)(47,58)(48,59)(49,60)(50,61)(51,62)(52,63)(53,64)(54,65)(55,66)(67,78)(68,79)(69,80)(70,81)(71,82)(72,83)(73,84)(74,85)(75,86)(76,87)(77,88)(89,112)(90,113)(91,114)(92,115)(93,116)(94,117)(95,118)(96,119)(97,120)(98,121)(99,122)(100,123)(101,124)(102,125)(103,126)(104,127)(105,128)(106,129)(107,130)(108,131)(109,132)(110,111)(133,162)(134,163)(135,164)(136,165)(137,166)(138,167)(139,168)(140,169)(141,170)(142,171)(143,172)(144,173)(145,174)(146,175)(147,176)(148,155)(149,156)(150,157)(151,158)(152,159)(153,160)(154,161), (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), (1,102,41,114)(2,90,42,124)(3,100,43,112)(4,110,44,122)(5,98,34,132)(6,108,35,120)(7,96,36,130)(8,106,37,118)(9,94,38,128)(10,104,39,116)(11,92,40,126)(12,121,23,109)(13,131,24,97)(14,119,25,107)(15,129,26,95)(16,117,27,105)(17,127,28,93)(18,115,29,103)(19,125,30,91)(20,113,31,101)(21,123,32,89)(22,111,33,99)(45,142,80,153)(46,159,81,170)(47,140,82,151)(48,157,83,168)(49,138,84,149)(50,155,85,166)(51,136,86,147)(52,175,87,164)(53,134,88,145)(54,173,67,162)(55,154,68,143)(56,171,69,160)(57,152,70,141)(58,169,71,158)(59,150,72,139)(60,167,73,156)(61,148,74,137)(62,165,75,176)(63,146,76,135)(64,163,77,174)(65,144,78,133)(66,161,79,172)>;`

`G:=Group( (1,55)(2,45)(3,57)(4,47)(5,59)(6,49)(7,61)(8,51)(9,63)(10,53)(11,65)(12,48)(13,60)(14,50)(15,62)(16,52)(17,64)(18,54)(19,66)(20,56)(21,46)(22,58)(23,83)(24,73)(25,85)(26,75)(27,87)(28,77)(29,67)(30,79)(31,69)(32,81)(33,71)(34,72)(35,84)(36,74)(37,86)(38,76)(39,88)(40,78)(41,68)(42,80)(43,70)(44,82)(89,141)(90,171)(91,143)(92,173)(93,145)(94,175)(95,147)(96,155)(97,149)(98,157)(99,151)(100,159)(101,153)(102,161)(103,133)(104,163)(105,135)(106,165)(107,137)(108,167)(109,139)(110,169)(111,140)(112,170)(113,142)(114,172)(115,144)(116,174)(117,146)(118,176)(119,148)(120,156)(121,150)(122,158)(123,152)(124,160)(125,154)(126,162)(127,134)(128,164)(129,136)(130,166)(131,138)(132,168), (1,30)(2,31)(3,32)(4,33)(5,23)(6,24)(7,25)(8,26)(9,27)(10,28)(11,29)(12,34)(13,35)(14,36)(15,37)(16,38)(17,39)(18,40)(19,41)(20,42)(21,43)(22,44)(45,69)(46,70)(47,71)(48,72)(49,73)(50,74)(51,75)(52,76)(53,77)(54,78)(55,79)(56,80)(57,81)(58,82)(59,83)(60,84)(61,85)(62,86)(63,87)(64,88)(65,67)(66,68)(89,100)(90,101)(91,102)(92,103)(93,104)(94,105)(95,106)(96,107)(97,108)(98,109)(99,110)(111,122)(112,123)(113,124)(114,125)(115,126)(116,127)(117,128)(118,129)(119,130)(120,131)(121,132)(133,173)(134,174)(135,175)(136,176)(137,155)(138,156)(139,157)(140,158)(141,159)(142,160)(143,161)(144,162)(145,163)(146,164)(147,165)(148,166)(149,167)(150,168)(151,169)(152,170)(153,171)(154,172), (1,19)(2,20)(3,21)(4,22)(5,12)(6,13)(7,14)(8,15)(9,16)(10,17)(11,18)(23,34)(24,35)(25,36)(26,37)(27,38)(28,39)(29,40)(30,41)(31,42)(32,43)(33,44)(45,56)(46,57)(47,58)(48,59)(49,60)(50,61)(51,62)(52,63)(53,64)(54,65)(55,66)(67,78)(68,79)(69,80)(70,81)(71,82)(72,83)(73,84)(74,85)(75,86)(76,87)(77,88)(89,112)(90,113)(91,114)(92,115)(93,116)(94,117)(95,118)(96,119)(97,120)(98,121)(99,122)(100,123)(101,124)(102,125)(103,126)(104,127)(105,128)(106,129)(107,130)(108,131)(109,132)(110,111)(133,162)(134,163)(135,164)(136,165)(137,166)(138,167)(139,168)(140,169)(141,170)(142,171)(143,172)(144,173)(145,174)(146,175)(147,176)(148,155)(149,156)(150,157)(151,158)(152,159)(153,160)(154,161), (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), (1,102,41,114)(2,90,42,124)(3,100,43,112)(4,110,44,122)(5,98,34,132)(6,108,35,120)(7,96,36,130)(8,106,37,118)(9,94,38,128)(10,104,39,116)(11,92,40,126)(12,121,23,109)(13,131,24,97)(14,119,25,107)(15,129,26,95)(16,117,27,105)(17,127,28,93)(18,115,29,103)(19,125,30,91)(20,113,31,101)(21,123,32,89)(22,111,33,99)(45,142,80,153)(46,159,81,170)(47,140,82,151)(48,157,83,168)(49,138,84,149)(50,155,85,166)(51,136,86,147)(52,175,87,164)(53,134,88,145)(54,173,67,162)(55,154,68,143)(56,171,69,160)(57,152,70,141)(58,169,71,158)(59,150,72,139)(60,167,73,156)(61,148,74,137)(62,165,75,176)(63,146,76,135)(64,163,77,174)(65,144,78,133)(66,161,79,172) );`

`G=PermutationGroup([[(1,55),(2,45),(3,57),(4,47),(5,59),(6,49),(7,61),(8,51),(9,63),(10,53),(11,65),(12,48),(13,60),(14,50),(15,62),(16,52),(17,64),(18,54),(19,66),(20,56),(21,46),(22,58),(23,83),(24,73),(25,85),(26,75),(27,87),(28,77),(29,67),(30,79),(31,69),(32,81),(33,71),(34,72),(35,84),(36,74),(37,86),(38,76),(39,88),(40,78),(41,68),(42,80),(43,70),(44,82),(89,141),(90,171),(91,143),(92,173),(93,145),(94,175),(95,147),(96,155),(97,149),(98,157),(99,151),(100,159),(101,153),(102,161),(103,133),(104,163),(105,135),(106,165),(107,137),(108,167),(109,139),(110,169),(111,140),(112,170),(113,142),(114,172),(115,144),(116,174),(117,146),(118,176),(119,148),(120,156),(121,150),(122,158),(123,152),(124,160),(125,154),(126,162),(127,134),(128,164),(129,136),(130,166),(131,138),(132,168)], [(1,30),(2,31),(3,32),(4,33),(5,23),(6,24),(7,25),(8,26),(9,27),(10,28),(11,29),(12,34),(13,35),(14,36),(15,37),(16,38),(17,39),(18,40),(19,41),(20,42),(21,43),(22,44),(45,69),(46,70),(47,71),(48,72),(49,73),(50,74),(51,75),(52,76),(53,77),(54,78),(55,79),(56,80),(57,81),(58,82),(59,83),(60,84),(61,85),(62,86),(63,87),(64,88),(65,67),(66,68),(89,100),(90,101),(91,102),(92,103),(93,104),(94,105),(95,106),(96,107),(97,108),(98,109),(99,110),(111,122),(112,123),(113,124),(114,125),(115,126),(116,127),(117,128),(118,129),(119,130),(120,131),(121,132),(133,173),(134,174),(135,175),(136,176),(137,155),(138,156),(139,157),(140,158),(141,159),(142,160),(143,161),(144,162),(145,163),(146,164),(147,165),(148,166),(149,167),(150,168),(151,169),(152,170),(153,171),(154,172)], [(1,19),(2,20),(3,21),(4,22),(5,12),(6,13),(7,14),(8,15),(9,16),(10,17),(11,18),(23,34),(24,35),(25,36),(26,37),(27,38),(28,39),(29,40),(30,41),(31,42),(32,43),(33,44),(45,56),(46,57),(47,58),(48,59),(49,60),(50,61),(51,62),(52,63),(53,64),(54,65),(55,66),(67,78),(68,79),(69,80),(70,81),(71,82),(72,83),(73,84),(74,85),(75,86),(76,87),(77,88),(89,112),(90,113),(91,114),(92,115),(93,116),(94,117),(95,118),(96,119),(97,120),(98,121),(99,122),(100,123),(101,124),(102,125),(103,126),(104,127),(105,128),(106,129),(107,130),(108,131),(109,132),(110,111),(133,162),(134,163),(135,164),(136,165),(137,166),(138,167),(139,168),(140,169),(141,170),(142,171),(143,172),(144,173),(145,174),(146,175),(147,176),(148,155),(149,156),(150,157),(151,158),(152,159),(153,160),(154,161)], [(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)], [(1,102,41,114),(2,90,42,124),(3,100,43,112),(4,110,44,122),(5,98,34,132),(6,108,35,120),(7,96,36,130),(8,106,37,118),(9,94,38,128),(10,104,39,116),(11,92,40,126),(12,121,23,109),(13,131,24,97),(14,119,25,107),(15,129,26,95),(16,117,27,105),(17,127,28,93),(18,115,29,103),(19,125,30,91),(20,113,31,101),(21,123,32,89),(22,111,33,99),(45,142,80,153),(46,159,81,170),(47,140,82,151),(48,157,83,168),(49,138,84,149),(50,155,85,166),(51,136,86,147),(52,175,87,164),(53,134,88,145),(54,173,67,162),(55,154,68,143),(56,171,69,160),(57,152,70,141),(58,169,71,158),(59,150,72,139),(60,167,73,156),(61,148,74,137),(62,165,75,176),(63,146,76,135),(64,163,77,174),(65,144,78,133),(66,161,79,172)]])`

64 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 4A 4B 4C 4D 4E 4F 4G 11A ··· 11E 22A ··· 22O 22P ··· 22AI 44A ··· 44J order 1 2 2 2 2 2 2 4 4 4 4 4 4 4 11 ··· 11 22 ··· 22 22 ··· 22 44 ··· 44 size 1 1 1 1 2 2 4 4 22 22 22 22 44 44 2 ··· 2 2 ··· 2 4 ··· 4 4 ··· 4

64 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 4 type + + + + + + + + + - image C1 C2 C2 C2 C2 D4 C4○D4 D11 D22 D22 C11⋊D4 D4⋊2D11 kernel C23.18D22 Dic11⋊C4 C23.D11 C22×Dic11 D4×C22 C2×C22 C22 C2×D4 C2×C4 C23 C22 C2 # reps 1 2 3 1 1 2 4 5 5 10 20 10

Matrix representation of C23.18D22 in GL4(𝔽89) generated by

 1 0 0 0 0 1 0 0 0 0 1 84 0 0 0 88
,
 88 0 0 0 0 88 0 0 0 0 88 0 0 0 0 88
,
 1 0 0 0 0 1 0 0 0 0 88 0 0 0 0 88
,
 32 0 0 0 4 25 0 0 0 0 1 0 0 0 36 88
,
 54 39 0 0 53 35 0 0 0 0 34 8 0 0 67 55
`G:=sub<GL(4,GF(89))| [1,0,0,0,0,1,0,0,0,0,1,0,0,0,84,88],[88,0,0,0,0,88,0,0,0,0,88,0,0,0,0,88],[1,0,0,0,0,1,0,0,0,0,88,0,0,0,0,88],[32,4,0,0,0,25,0,0,0,0,1,36,0,0,0,88],[54,53,0,0,39,35,0,0,0,0,34,67,0,0,8,55] >;`

C23.18D22 in GAP, Magma, Sage, TeX

`C_2^3._{18}D_{22}`
`% in TeX`

`G:=Group("C2^3.18D22");`
`// GroupNames label`

`G:=SmallGroup(352,130);`
`// by ID`

`G=gap.SmallGroup(352,130);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-11,96,218,188,11525]);`
`// Polycyclic`

`G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^22=1,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=b*d^-1>;`
`// generators/relations`

׿
×
𝔽