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G = C11×C4⋊C4order 176 = 24·11

Direct product of C11 and C4⋊C4

direct product, metacyclic, nilpotent (class 2), monomial, 2-elementary

Aliases: C11×C4⋊C4, C4⋊C44, C443C4, C22.3Q8, C22.13D4, C2.(Q8×C11), (C2×C4).1C22, (C2×C44).2C2, C2.2(C2×C44), C2.2(D4×C11), C22.11(C2×C4), C22.3(C2×C22), (C2×C22).14C22, SmallGroup(176,21)

Series: Derived Chief Lower central Upper central

C1C2 — C11×C4⋊C4
C1C2C22C2×C22C2×C44 — C11×C4⋊C4
C1C2 — C11×C4⋊C4
C1C2×C22 — C11×C4⋊C4

Generators and relations for C11×C4⋊C4
 G = < a,b,c | a11=b4=c4=1, ab=ba, ac=ca, cbc-1=b-1 >

2C4
2C4
2C44
2C44

Smallest permutation representation of C11×C4⋊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)
(1 103 74 114)(2 104 75 115)(3 105 76 116)(4 106 77 117)(5 107 67 118)(6 108 68 119)(7 109 69 120)(8 110 70 121)(9 100 71 111)(10 101 72 112)(11 102 73 113)(12 145 176 156)(13 146 166 157)(14 147 167 158)(15 148 168 159)(16 149 169 160)(17 150 170 161)(18 151 171 162)(19 152 172 163)(20 153 173 164)(21 154 174 165)(22 144 175 155)(23 130 36 140)(24 131 37 141)(25 132 38 142)(26 122 39 143)(27 123 40 133)(28 124 41 134)(29 125 42 135)(30 126 43 136)(31 127 44 137)(32 128 34 138)(33 129 35 139)(45 78 58 99)(46 79 59 89)(47 80 60 90)(48 81 61 91)(49 82 62 92)(50 83 63 93)(51 84 64 94)(52 85 65 95)(53 86 66 96)(54 87 56 97)(55 88 57 98)
(1 158 46 133)(2 159 47 134)(3 160 48 135)(4 161 49 136)(5 162 50 137)(6 163 51 138)(7 164 52 139)(8 165 53 140)(9 155 54 141)(10 156 55 142)(11 157 45 143)(12 98 25 112)(13 99 26 113)(14 89 27 114)(15 90 28 115)(16 91 29 116)(17 92 30 117)(18 93 31 118)(19 94 32 119)(20 95 33 120)(21 96 23 121)(22 97 24 111)(34 108 172 84)(35 109 173 85)(36 110 174 86)(37 100 175 87)(38 101 176 88)(39 102 166 78)(40 103 167 79)(41 104 168 80)(42 105 169 81)(43 106 170 82)(44 107 171 83)(56 131 71 144)(57 132 72 145)(58 122 73 146)(59 123 74 147)(60 124 75 148)(61 125 76 149)(62 126 77 150)(63 127 67 151)(64 128 68 152)(65 129 69 153)(66 130 70 154)

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

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

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

C11×C4⋊C4 is a maximal subgroup of
C44.Q8  C4.Dic22  C22.D8  C22.Q16  Dic22⋊C4  C44⋊Q8  Dic11.Q8  C44.3Q8  C4⋊C47D11  D44⋊C4  D22.5D4  C42D44  D22⋊Q8  D222Q8  C4⋊C4⋊D11  D4×C44  Q8×C44

110 conjugacy classes

class 1 2A2B2C4A···4F11A···11J22A···22AD44A···44BH
order12224···411···1122···2244···44
size11112···21···11···12···2

110 irreducible representations

dim1111112222
type+++-
imageC1C2C4C11C22C44D4Q8D4×C11Q8×C11
kernelC11×C4⋊C4C2×C44C44C4⋊C4C2×C4C4C22C22C2C2
# reps134103040111010

Matrix representation of C11×C4⋊C4 in GL3(𝔽89) generated by

100
040
004
,
8800
0088
010
,
3400
05135
03538
G:=sub<GL(3,GF(89))| [1,0,0,0,4,0,0,0,4],[88,0,0,0,0,1,0,88,0],[34,0,0,0,51,35,0,35,38] >;

C11×C4⋊C4 in GAP, Magma, Sage, TeX

C_{11}\times C_4\rtimes C_4
% in TeX

G:=Group("C11xC4:C4");
// GroupNames label

G:=SmallGroup(176,21);
// by ID

G=gap.SmallGroup(176,21);
# by ID

G:=PCGroup([5,-2,-2,-11,-2,-2,440,461,226]);
// Polycyclic

G:=Group<a,b,c|a^11=b^4=c^4=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

Export

Subgroup lattice of C11×C4⋊C4 in TeX

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