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## G = C23.11D22order 352 = 25·11

### 1st non-split extension by C23 of D22 acting via D22/D11=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C23.11D22
 Chief series C1 — C11 — C22 — C2×C22 — C2×Dic11 — C22×Dic11 — C23.11D22
 Lower central C11 — C22 — C23.11D22
 Upper central C1 — C22 — C22⋊C4

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, D42D11, C23.11D22

Smallest permutation representation of 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`

׿
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