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## G = C44.C23order 352 = 25·11

### 15th non-split extension by C44 of C23 acting via C23/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C44 — C44.C23
 Chief series C1 — C11 — C22 — C44 — D44 — D44⋊5C2 — C44.C23
 Lower central C11 — C22 — C44 — C44.C23
 Upper central C1 — C2 — C2×C4 — C2×Q8

Generators and relations for C44.C23
G = < a,b,c,d | a44=b2=1, c2=d2=a22, bab=a-1, ac=ca, dad-1=a23, bc=cb, dbd-1=a33b, dcd-1=a22c >

Subgroups: 314 in 60 conjugacy classes, 29 normal (19 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, D4, Q8, Q8, C11, M4(2), SD16, Q16, C2×Q8, C4○D4, D11, C22, C22, C8.C22, Dic11, C44, C44, D22, C2×C22, C11⋊C8, Dic22, C4×D11, D44, C11⋊D4, C2×C44, C2×C44, Q8×C11, Q8×C11, C44.C4, Q8⋊D11, C11⋊Q16, D445C2, Q8×C22, C44.C23
Quotients: C1, C2, C22, D4, C23, C2×D4, D11, C8.C22, D22, C11⋊D4, C22×D11, C2×C11⋊D4, C44.C23

Smallest permutation representation of C44.C23
On 176 points
Generators in S176
```(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)
(2 44)(3 43)(4 42)(5 41)(6 40)(7 39)(8 38)(9 37)(10 36)(11 35)(12 34)(13 33)(14 32)(15 31)(16 30)(17 29)(18 28)(19 27)(20 26)(21 25)(22 24)(45 76)(46 75)(47 74)(48 73)(49 72)(50 71)(51 70)(52 69)(53 68)(54 67)(55 66)(56 65)(57 64)(58 63)(59 62)(60 61)(77 88)(78 87)(79 86)(80 85)(81 84)(82 83)(89 97)(90 96)(91 95)(92 94)(98 132)(99 131)(100 130)(101 129)(102 128)(103 127)(104 126)(105 125)(106 124)(107 123)(108 122)(109 121)(110 120)(111 119)(112 118)(113 117)(114 116)(133 136)(134 135)(137 176)(138 175)(139 174)(140 173)(141 172)(142 171)(143 170)(144 169)(145 168)(146 167)(147 166)(148 165)(149 164)(150 163)(151 162)(152 161)(153 160)(154 159)(155 158)(156 157)
(1 93 23 115)(2 94 24 116)(3 95 25 117)(4 96 26 118)(5 97 27 119)(6 98 28 120)(7 99 29 121)(8 100 30 122)(9 101 31 123)(10 102 32 124)(11 103 33 125)(12 104 34 126)(13 105 35 127)(14 106 36 128)(15 107 37 129)(16 108 38 130)(17 109 39 131)(18 110 40 132)(19 111 41 89)(20 112 42 90)(21 113 43 91)(22 114 44 92)(45 141 67 163)(46 142 68 164)(47 143 69 165)(48 144 70 166)(49 145 71 167)(50 146 72 168)(51 147 73 169)(52 148 74 170)(53 149 75 171)(54 150 76 172)(55 151 77 173)(56 152 78 174)(57 153 79 175)(58 154 80 176)(59 155 81 133)(60 156 82 134)(61 157 83 135)(62 158 84 136)(63 159 85 137)(64 160 86 138)(65 161 87 139)(66 162 88 140)
(1 66 23 88)(2 45 24 67)(3 68 25 46)(4 47 26 69)(5 70 27 48)(6 49 28 71)(7 72 29 50)(8 51 30 73)(9 74 31 52)(10 53 32 75)(11 76 33 54)(12 55 34 77)(13 78 35 56)(14 57 36 79)(15 80 37 58)(16 59 38 81)(17 82 39 60)(18 61 40 83)(19 84 41 62)(20 63 42 85)(21 86 43 64)(22 65 44 87)(89 136 111 158)(90 159 112 137)(91 138 113 160)(92 161 114 139)(93 140 115 162)(94 163 116 141)(95 142 117 164)(96 165 118 143)(97 144 119 166)(98 167 120 145)(99 146 121 168)(100 169 122 147)(101 148 123 170)(102 171 124 149)(103 150 125 172)(104 173 126 151)(105 152 127 174)(106 175 128 153)(107 154 129 176)(108 133 130 155)(109 156 131 134)(110 135 132 157)```

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

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

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

61 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C 4D 4E 8A 8B 11A ··· 11E 22A ··· 22O 44A ··· 44AD order 1 2 2 2 4 4 4 4 4 8 8 11 ··· 11 22 ··· 22 44 ··· 44 size 1 1 2 44 2 2 4 4 44 44 44 2 ··· 2 2 ··· 2 4 ··· 4

61 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 D4 D4 D11 D22 D22 C11⋊D4 C11⋊D4 C8.C22 C44.C23 kernel C44.C23 C44.C4 Q8⋊D11 C11⋊Q16 D44⋊5C2 Q8×C22 C44 C2×C22 C2×Q8 C2×C4 Q8 C4 C22 C11 C1 # reps 1 1 2 2 1 1 1 1 5 5 10 10 10 1 10

Matrix representation of C44.C23 in GL4(𝔽89) generated by

 0 0 51 48 0 0 58 4 38 41 0 0 31 85 0 0
,
 21 82 0 0 12 68 0 0 0 0 68 7 0 0 77 21
,
 0 0 33 39 0 0 36 56 56 50 0 0 53 33 0 0
,
 19 54 44 35 43 44 46 19 44 35 70 35 46 19 46 45
`G:=sub<GL(4,GF(89))| [0,0,38,31,0,0,41,85,51,58,0,0,48,4,0,0],[21,12,0,0,82,68,0,0,0,0,68,77,0,0,7,21],[0,0,56,53,0,0,50,33,33,36,0,0,39,56,0,0],[19,43,44,46,54,44,35,19,44,46,70,46,35,19,35,45] >;`

C44.C23 in GAP, Magma, Sage, TeX

`C_{44}.C_2^3`
`% in TeX`

`G:=Group("C44.C2^3");`
`// GroupNames label`

`G:=SmallGroup(352,137);`
`// by ID`

`G=gap.SmallGroup(352,137);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-11,103,218,188,86,579,159,69,11525]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^44=b^2=1,c^2=d^2=a^22,b*a*b=a^-1,a*c=c*a,d*a*d^-1=a^23,b*c=c*b,d*b*d^-1=a^33*b,d*c*d^-1=a^22*c>;`
`// generators/relations`

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