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G = C4×C44order 176 = 24·11

Abelian group of type [4,44]

direct product, abelian, monomial, 2-elementary

Aliases: C4×C44, SmallGroup(176,19)

Series: Derived Chief Lower central Upper central

C1 — C4×C44
C1C2C22C2×C22C2×C44 — C4×C44
C1 — C4×C44
C1 — C4×C44

Generators and relations for C4×C44
 G = < a,b | a4=b44=1, ab=ba >


Smallest permutation representation of C4×C44
Regular action on 176 points
Generators in S176
(1 70 125 136)(2 71 126 137)(3 72 127 138)(4 73 128 139)(5 74 129 140)(6 75 130 141)(7 76 131 142)(8 77 132 143)(9 78 89 144)(10 79 90 145)(11 80 91 146)(12 81 92 147)(13 82 93 148)(14 83 94 149)(15 84 95 150)(16 85 96 151)(17 86 97 152)(18 87 98 153)(19 88 99 154)(20 45 100 155)(21 46 101 156)(22 47 102 157)(23 48 103 158)(24 49 104 159)(25 50 105 160)(26 51 106 161)(27 52 107 162)(28 53 108 163)(29 54 109 164)(30 55 110 165)(31 56 111 166)(32 57 112 167)(33 58 113 168)(34 59 114 169)(35 60 115 170)(36 61 116 171)(37 62 117 172)(38 63 118 173)(39 64 119 174)(40 65 120 175)(41 66 121 176)(42 67 122 133)(43 68 123 134)(44 69 124 135)
(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)

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

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

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

C4×C44 is a maximal subgroup of
C42.D11  C44⋊C8  D441C4  C442Q8  C44.6Q8  C42⋊D11  C4⋊D44  C4.D44  C422D11

176 conjugacy classes

class 1 2A2B2C4A···4L11A···11J22A···22AD44A···44DP
order12224···411···1122···2244···44
size11111···11···11···11···1

176 irreducible representations

dim111111
type++
imageC1C2C4C11C22C44
kernelC4×C44C2×C44C44C42C2×C4C4
# reps13121030120

Matrix representation of C4×C44 in GL2(𝔽89) generated by

550
01
,
720
020
G:=sub<GL(2,GF(89))| [55,0,0,1],[72,0,0,20] >;

C4×C44 in GAP, Magma, Sage, TeX

C_4\times C_{44}
% in TeX

G:=Group("C4xC44");
// GroupNames label

G:=SmallGroup(176,19);
// by ID

G=gap.SmallGroup(176,19);
# by ID

G:=PCGroup([5,-2,-2,-11,-2,-2,220,446]);
// Polycyclic

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

Export

Subgroup lattice of C4×C44 in TeX

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