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## G = C44⋊C4order 176 = 24·11

### 1st semidirect product of C44 and C4 acting via C4/C2=C2

Aliases: C441C4, C4⋊Dic11, C2.1D44, C22.4D4, C22.2Q8, C2.2Dic22, C22.5D22, C112(C4⋊C4), C22.8(C2×C4), (C2×C44).3C2, (C2×C4).3D11, (C2×C22).5C22, C2.4(C2×Dic11), (C2×Dic11).2C2, SmallGroup(176,12)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C44⋊C4
 Chief series C1 — C11 — C22 — C2×C22 — C2×Dic11 — C44⋊C4
 Lower central C11 — C22 — C44⋊C4
 Upper central C1 — C22 — C2×C4

Generators and relations for C44⋊C4
G = < a,b | a44=b4=1, bab-1=a-1 >

Smallest permutation representation of C44⋊C4
Regular action 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)
(1 130 170 48)(2 129 171 47)(3 128 172 46)(4 127 173 45)(5 126 174 88)(6 125 175 87)(7 124 176 86)(8 123 133 85)(9 122 134 84)(10 121 135 83)(11 120 136 82)(12 119 137 81)(13 118 138 80)(14 117 139 79)(15 116 140 78)(16 115 141 77)(17 114 142 76)(18 113 143 75)(19 112 144 74)(20 111 145 73)(21 110 146 72)(22 109 147 71)(23 108 148 70)(24 107 149 69)(25 106 150 68)(26 105 151 67)(27 104 152 66)(28 103 153 65)(29 102 154 64)(30 101 155 63)(31 100 156 62)(32 99 157 61)(33 98 158 60)(34 97 159 59)(35 96 160 58)(36 95 161 57)(37 94 162 56)(38 93 163 55)(39 92 164 54)(40 91 165 53)(41 90 166 52)(42 89 167 51)(43 132 168 50)(44 131 169 49)```

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

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

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

50 conjugacy classes

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

50 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 2 type + + + + - + - + - + image C1 C2 C2 C4 D4 Q8 D11 Dic11 D22 Dic22 D44 kernel C44⋊C4 C2×Dic11 C2×C44 C44 C22 C22 C2×C4 C4 C22 C2 C2 # reps 1 2 1 4 1 1 5 10 5 10 10

Matrix representation of C44⋊C4 in GL4(𝔽89) generated by

 34 87 0 0 2 13 0 0 0 0 14 73 0 0 2 55
,
 23 63 0 0 82 66 0 0 0 0 8 41 0 0 18 81
`G:=sub<GL(4,GF(89))| [34,2,0,0,87,13,0,0,0,0,14,2,0,0,73,55],[23,82,0,0,63,66,0,0,0,0,8,18,0,0,41,81] >;`

C44⋊C4 in GAP, Magma, Sage, TeX

`C_{44}\rtimes C_4`
`% in TeX`

`G:=Group("C44:C4");`
`// GroupNames label`

`G:=SmallGroup(176,12);`
`// by ID`

`G=gap.SmallGroup(176,12);`
`# by ID`

`G:=PCGroup([5,-2,-2,-2,-2,-11,20,101,46,4004]);`
`// Polycyclic`

`G:=Group<a,b|a^44=b^4=1,b*a*b^-1=a^-1>;`
`// generators/relations`

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