direct product, metacyclic, nilpotent (class 2), monomial, 2-elementary
Aliases: C11×C4⋊C4, C4⋊C44, C44⋊3C4, 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
Generators and relations for C11×C4⋊C4
G = < a,b,c | a11=b4=c4=1, ab=ba, ac=ca, cbc-1=b-1 >
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⋊C4⋊7D11 D44⋊C4 D22.5D4 C4⋊2D44 D22⋊Q8 D22⋊2Q8 C4⋊C4⋊D11 D4×C44 Q8×C44
110 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | ··· | 4F | 11A | ··· | 11J | 22A | ··· | 22AD | 44A | ··· | 44BH |
| order | 1 | 2 | 2 | 2 | 4 | ··· | 4 | 11 | ··· | 11 | 22 | ··· | 22 | 44 | ··· | 44 |
| size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 |
110 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | ||||||
| image | C1 | C2 | C4 | C11 | C22 | C44 | D4 | Q8 | D4×C11 | Q8×C11 |
| kernel | C11×C4⋊C4 | C2×C44 | C44 | C4⋊C4 | C2×C4 | C4 | C22 | C22 | C2 | C2 |
| # reps | 1 | 3 | 4 | 10 | 30 | 40 | 1 | 1 | 10 | 10 |
Matrix representation of C11×C4⋊C4 ►in GL3(𝔽89) generated by
| 1 | 0 | 0 |
| 0 | 4 | 0 |
| 0 | 0 | 4 |
| 88 | 0 | 0 |
| 0 | 0 | 88 |
| 0 | 1 | 0 |
| 34 | 0 | 0 |
| 0 | 51 | 35 |
| 0 | 35 | 38 |
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
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