metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C44.55D4, C22.6M4(2), C23.2Dic11, (C2×C22)⋊2C8, C22⋊(C11⋊C8), (C2×C44).4C4, C11⋊2(C22⋊C8), C22.10(C2×C8), (C2×C4).94D22, (C22×C22).5C4, (C2×C4).4Dic11, (C22×C4).1D11, C4.30(C11⋊D4), (C22×C44).11C2, C2.3(C44.C4), C22.12(C22⋊C4), (C2×C44).108C22, C2.1(C23.D11), C22.9(C2×Dic11), C2.5(C2×C11⋊C8), (C2×C11⋊C8)⋊10C2, (C2×C22).27(C2×C4), SmallGroup(352,36)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C44.55D4
G = < a,b,c | a44=1, b4=a22, c2=a33, bab-1=cac-1=a21, cbc-1=a11b3 >
Smallest permutation representation of C44.55D4
►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)
(1 120 63 173 23 98 85 151)(2 97 64 150 24 119 86 172)(3 118 65 171 25 96 87 149)(4 95 66 148 26 117 88 170)(5 116 67 169 27 94 45 147)(6 93 68 146 28 115 46 168)(7 114 69 167 29 92 47 145)(8 91 70 144 30 113 48 166)(9 112 71 165 31 90 49 143)(10 89 72 142 32 111 50 164)(11 110 73 163 33 132 51 141)(12 131 74 140 34 109 52 162)(13 108 75 161 35 130 53 139)(14 129 76 138 36 107 54 160)(15 106 77 159 37 128 55 137)(16 127 78 136 38 105 56 158)(17 104 79 157 39 126 57 135)(18 125 80 134 40 103 58 156)(19 102 81 155 41 124 59 133)(20 123 82 176 42 101 60 154)(21 100 83 153 43 122 61 175)(22 121 84 174 44 99 62 152)
(1 140 34 173 23 162 12 151)(2 161 35 150 24 139 13 172)(3 138 36 171 25 160 14 149)(4 159 37 148 26 137 15 170)(5 136 38 169 27 158 16 147)(6 157 39 146 28 135 17 168)(7 134 40 167 29 156 18 145)(8 155 41 144 30 133 19 166)(9 176 42 165 31 154 20 143)(10 153 43 142 32 175 21 164)(11 174 44 163 33 152 22 141)(45 127 78 116 67 105 56 94)(46 104 79 93 68 126 57 115)(47 125 80 114 69 103 58 92)(48 102 81 91 70 124 59 113)(49 123 82 112 71 101 60 90)(50 100 83 89 72 122 61 111)(51 121 84 110 73 99 62 132)(52 98 85 131 74 120 63 109)(53 119 86 108 75 97 64 130)(54 96 87 129 76 118 65 107)(55 117 88 106 77 95 66 128)
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), (1,120,63,173,23,98,85,151)(2,97,64,150,24,119,86,172)(3,118,65,171,25,96,87,149)(4,95,66,148,26,117,88,170)(5,116,67,169,27,94,45,147)(6,93,68,146,28,115,46,168)(7,114,69,167,29,92,47,145)(8,91,70,144,30,113,48,166)(9,112,71,165,31,90,49,143)(10,89,72,142,32,111,50,164)(11,110,73,163,33,132,51,141)(12,131,74,140,34,109,52,162)(13,108,75,161,35,130,53,139)(14,129,76,138,36,107,54,160)(15,106,77,159,37,128,55,137)(16,127,78,136,38,105,56,158)(17,104,79,157,39,126,57,135)(18,125,80,134,40,103,58,156)(19,102,81,155,41,124,59,133)(20,123,82,176,42,101,60,154)(21,100,83,153,43,122,61,175)(22,121,84,174,44,99,62,152), (1,140,34,173,23,162,12,151)(2,161,35,150,24,139,13,172)(3,138,36,171,25,160,14,149)(4,159,37,148,26,137,15,170)(5,136,38,169,27,158,16,147)(6,157,39,146,28,135,17,168)(7,134,40,167,29,156,18,145)(8,155,41,144,30,133,19,166)(9,176,42,165,31,154,20,143)(10,153,43,142,32,175,21,164)(11,174,44,163,33,152,22,141)(45,127,78,116,67,105,56,94)(46,104,79,93,68,126,57,115)(47,125,80,114,69,103,58,92)(48,102,81,91,70,124,59,113)(49,123,82,112,71,101,60,90)(50,100,83,89,72,122,61,111)(51,121,84,110,73,99,62,132)(52,98,85,131,74,120,63,109)(53,119,86,108,75,97,64,130)(54,96,87,129,76,118,65,107)(55,117,88,106,77,95,66,128)>;
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), (1,120,63,173,23,98,85,151)(2,97,64,150,24,119,86,172)(3,118,65,171,25,96,87,149)(4,95,66,148,26,117,88,170)(5,116,67,169,27,94,45,147)(6,93,68,146,28,115,46,168)(7,114,69,167,29,92,47,145)(8,91,70,144,30,113,48,166)(9,112,71,165,31,90,49,143)(10,89,72,142,32,111,50,164)(11,110,73,163,33,132,51,141)(12,131,74,140,34,109,52,162)(13,108,75,161,35,130,53,139)(14,129,76,138,36,107,54,160)(15,106,77,159,37,128,55,137)(16,127,78,136,38,105,56,158)(17,104,79,157,39,126,57,135)(18,125,80,134,40,103,58,156)(19,102,81,155,41,124,59,133)(20,123,82,176,42,101,60,154)(21,100,83,153,43,122,61,175)(22,121,84,174,44,99,62,152), (1,140,34,173,23,162,12,151)(2,161,35,150,24,139,13,172)(3,138,36,171,25,160,14,149)(4,159,37,148,26,137,15,170)(5,136,38,169,27,158,16,147)(6,157,39,146,28,135,17,168)(7,134,40,167,29,156,18,145)(8,155,41,144,30,133,19,166)(9,176,42,165,31,154,20,143)(10,153,43,142,32,175,21,164)(11,174,44,163,33,152,22,141)(45,127,78,116,67,105,56,94)(46,104,79,93,68,126,57,115)(47,125,80,114,69,103,58,92)(48,102,81,91,70,124,59,113)(49,123,82,112,71,101,60,90)(50,100,83,89,72,122,61,111)(51,121,84,110,73,99,62,132)(52,98,85,131,74,120,63,109)(53,119,86,108,75,97,64,130)(54,96,87,129,76,118,65,107)(55,117,88,106,77,95,66,128) );
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)], [(1,120,63,173,23,98,85,151),(2,97,64,150,24,119,86,172),(3,118,65,171,25,96,87,149),(4,95,66,148,26,117,88,170),(5,116,67,169,27,94,45,147),(6,93,68,146,28,115,46,168),(7,114,69,167,29,92,47,145),(8,91,70,144,30,113,48,166),(9,112,71,165,31,90,49,143),(10,89,72,142,32,111,50,164),(11,110,73,163,33,132,51,141),(12,131,74,140,34,109,52,162),(13,108,75,161,35,130,53,139),(14,129,76,138,36,107,54,160),(15,106,77,159,37,128,55,137),(16,127,78,136,38,105,56,158),(17,104,79,157,39,126,57,135),(18,125,80,134,40,103,58,156),(19,102,81,155,41,124,59,133),(20,123,82,176,42,101,60,154),(21,100,83,153,43,122,61,175),(22,121,84,174,44,99,62,152)], [(1,140,34,173,23,162,12,151),(2,161,35,150,24,139,13,172),(3,138,36,171,25,160,14,149),(4,159,37,148,26,137,15,170),(5,136,38,169,27,158,16,147),(6,157,39,146,28,135,17,168),(7,134,40,167,29,156,18,145),(8,155,41,144,30,133,19,166),(9,176,42,165,31,154,20,143),(10,153,43,142,32,175,21,164),(11,174,44,163,33,152,22,141),(45,127,78,116,67,105,56,94),(46,104,79,93,68,126,57,115),(47,125,80,114,69,103,58,92),(48,102,81,91,70,124,59,113),(49,123,82,112,71,101,60,90),(50,100,83,89,72,122,61,111),(51,121,84,110,73,99,62,132),(52,98,85,131,74,120,63,109),(53,119,86,108,75,97,64,130),(54,96,87,129,76,118,65,107),(55,117,88,106,77,95,66,128)]])
100 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 8A | ··· | 8H | 11A | ··· | 11E | 22A | ··· | 22AI | 44A | ··· | 44AN |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 11 | ··· | 11 | 22 | ··· | 22 | 44 | ··· | 44 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 22 | ··· | 22 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
100 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | + | - | |||||||
| image | C1 | C2 | C2 | C4 | C4 | C8 | D4 | M4(2) | D11 | Dic11 | D22 | Dic11 | C11⋊D4 | C11⋊C8 | C44.C4 |
| kernel | C44.55D4 | C2×C11⋊C8 | C22×C44 | C2×C44 | C22×C22 | C2×C22 | C44 | C22 | C22×C4 | C2×C4 | C2×C4 | C23 | C4 | C22 | C2 |
| # reps | 1 | 2 | 1 | 2 | 2 | 8 | 2 | 2 | 5 | 5 | 5 | 5 | 20 | 20 | 20 |
Matrix representation of C44.55D4 ►in GL3(𝔽89) generated by
| 34 | 0 | 0 |
| 0 | 2 | 39 |
| 0 | 0 | 45 |
| 37 | 0 | 0 |
| 0 | 21 | 9 |
| 0 | 30 | 68 |
| 52 | 0 | 0 |
| 0 | 68 | 74 |
| 0 | 59 | 21 |
G:=sub<GL(3,GF(89))| [34,0,0,0,2,0,0,39,45],[37,0,0,0,21,30,0,9,68],[52,0,0,0,68,59,0,74,21] >;
C44.55D4 in GAP, Magma, Sage, TeX
C_{44}._{55}D_4 % in TeX
G:=Group("C44.55D4"); // GroupNames label
G:=SmallGroup(352,36);
// by ID
G=gap.SmallGroup(352,36);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-11,24,121,86,11525]);
// Polycyclic
G:=Group<a,b,c|a^44=1,b^4=a^22,c^2=a^33,b*a*b^-1=c*a*c^-1=a^21,c*b*c^-1=a^11*b^3>;
// generators/relations
Export