metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C88.4C4, C44.1C8, C8.22D22, C11⋊2M5(2), C8.2Dic11, C88.22C22, C4.(C11⋊C8), C11⋊C16⋊5C2, (C2×C22).3C8, C22.9(C2×C8), (C2×C44).8C4, C22.(C11⋊C8), (C2×C8).7D11, (C2×C88).10C2, C44.39(C2×C4), (C2×C4).5Dic11, C4.11(C2×Dic11), C2.4(C2×C11⋊C8), SmallGroup(352,18)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C44.C8
G = < a,b | a44=1, b8=a22, bab-1=a-1 >
Smallest permutation representation of C44.C8
►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)
(1 161 83 129 34 172 72 96 23 139 61 107 12 150 50 118)(2 160 84 128 35 171 73 95 24 138 62 106 13 149 51 117)(3 159 85 127 36 170 74 94 25 137 63 105 14 148 52 116)(4 158 86 126 37 169 75 93 26 136 64 104 15 147 53 115)(5 157 87 125 38 168 76 92 27 135 65 103 16 146 54 114)(6 156 88 124 39 167 77 91 28 134 66 102 17 145 55 113)(7 155 45 123 40 166 78 90 29 133 67 101 18 144 56 112)(8 154 46 122 41 165 79 89 30 176 68 100 19 143 57 111)(9 153 47 121 42 164 80 132 31 175 69 99 20 142 58 110)(10 152 48 120 43 163 81 131 32 174 70 98 21 141 59 109)(11 151 49 119 44 162 82 130 33 173 71 97 22 140 60 108)
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), (1,161,83,129,34,172,72,96,23,139,61,107,12,150,50,118)(2,160,84,128,35,171,73,95,24,138,62,106,13,149,51,117)(3,159,85,127,36,170,74,94,25,137,63,105,14,148,52,116)(4,158,86,126,37,169,75,93,26,136,64,104,15,147,53,115)(5,157,87,125,38,168,76,92,27,135,65,103,16,146,54,114)(6,156,88,124,39,167,77,91,28,134,66,102,17,145,55,113)(7,155,45,123,40,166,78,90,29,133,67,101,18,144,56,112)(8,154,46,122,41,165,79,89,30,176,68,100,19,143,57,111)(9,153,47,121,42,164,80,132,31,175,69,99,20,142,58,110)(10,152,48,120,43,163,81,131,32,174,70,98,21,141,59,109)(11,151,49,119,44,162,82,130,33,173,71,97,22,140,60,108)>;
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), (1,161,83,129,34,172,72,96,23,139,61,107,12,150,50,118)(2,160,84,128,35,171,73,95,24,138,62,106,13,149,51,117)(3,159,85,127,36,170,74,94,25,137,63,105,14,148,52,116)(4,158,86,126,37,169,75,93,26,136,64,104,15,147,53,115)(5,157,87,125,38,168,76,92,27,135,65,103,16,146,54,114)(6,156,88,124,39,167,77,91,28,134,66,102,17,145,55,113)(7,155,45,123,40,166,78,90,29,133,67,101,18,144,56,112)(8,154,46,122,41,165,79,89,30,176,68,100,19,143,57,111)(9,153,47,121,42,164,80,132,31,175,69,99,20,142,58,110)(10,152,48,120,43,163,81,131,32,174,70,98,21,141,59,109)(11,151,49,119,44,162,82,130,33,173,71,97,22,140,60,108) );
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)], [(1,161,83,129,34,172,72,96,23,139,61,107,12,150,50,118),(2,160,84,128,35,171,73,95,24,138,62,106,13,149,51,117),(3,159,85,127,36,170,74,94,25,137,63,105,14,148,52,116),(4,158,86,126,37,169,75,93,26,136,64,104,15,147,53,115),(5,157,87,125,38,168,76,92,27,135,65,103,16,146,54,114),(6,156,88,124,39,167,77,91,28,134,66,102,17,145,55,113),(7,155,45,123,40,166,78,90,29,133,67,101,18,144,56,112),(8,154,46,122,41,165,79,89,30,176,68,100,19,143,57,111),(9,153,47,121,42,164,80,132,31,175,69,99,20,142,58,110),(10,152,48,120,43,163,81,131,32,174,70,98,21,141,59,109),(11,151,49,119,44,162,82,130,33,173,71,97,22,140,60,108)]])
100 conjugacy classes
| class | 1 | 2A | 2B | 4A | 4B | 4C | 8A | 8B | 8C | 8D | 8E | 8F | 11A | ··· | 11E | 16A | ··· | 16H | 22A | ··· | 22O | 44A | ··· | 44T | 88A | ··· | 88AN |
| order | 1 | 2 | 2 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | 8 | 11 | ··· | 11 | 16 | ··· | 16 | 22 | ··· | 22 | 44 | ··· | 44 | 88 | ··· | 88 |
| size | 1 | 1 | 2 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | ··· | 2 | 22 | ··· | 22 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
100 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | - | ||||||||
| image | C1 | C2 | C2 | C4 | C4 | C8 | C8 | D11 | M5(2) | Dic11 | D22 | Dic11 | C11⋊C8 | C11⋊C8 | C44.C8 |
| kernel | C44.C8 | C11⋊C16 | C2×C88 | C88 | C2×C44 | C44 | C2×C22 | C2×C8 | C11 | C8 | C8 | C2×C4 | C4 | C22 | C1 |
| # reps | 1 | 2 | 1 | 2 | 2 | 4 | 4 | 5 | 4 | 5 | 5 | 5 | 10 | 10 | 40 |
Matrix representation of C44.C8 ►in GL2(𝔽353) generated by
| 319 | 305 |
| 0 | 218 |
| 245 | 186 |
| 214 | 108 |
G:=sub<GL(2,GF(353))| [319,0,305,218],[245,214,186,108] >;
C44.C8 in GAP, Magma, Sage, TeX
C_{44}.C_8 % in TeX
G:=Group("C44.C8"); // GroupNames label
G:=SmallGroup(352,18);
// by ID
G=gap.SmallGroup(352,18);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-11,24,217,50,69,11525]);
// Polycyclic
G:=Group<a,b|a^44=1,b^8=a^22,b*a*b^-1=a^-1>;
// generators/relations
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