metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C44.5D4, C8.6D22, C11⋊3SD32, Q16⋊1D11, D88.2C2, C22.10D8, C88.4C22, C11⋊C16⋊3C2, (C11×Q16)⋊1C2, C2.6(D4⋊D11), C4.3(C11⋊D4), SmallGroup(352,34)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C8.6D22
G = < a,b,c | a8=1, b22=a4, c2=a3, bab-1=a-1, ac=ca, cbc-1=a-1b21 >
Smallest permutation representation of C8.6D22
►On 176 points
Generators in S176
(1 45 166 123 23 67 144 101)(2 102 145 68 24 124 167 46)(3 47 168 125 25 69 146 103)(4 104 147 70 26 126 169 48)(5 49 170 127 27 71 148 105)(6 106 149 72 28 128 171 50)(7 51 172 129 29 73 150 107)(8 108 151 74 30 130 173 52)(9 53 174 131 31 75 152 109)(10 110 153 76 32 132 175 54)(11 55 176 89 33 77 154 111)(12 112 155 78 34 90 133 56)(13 57 134 91 35 79 156 113)(14 114 157 80 36 92 135 58)(15 59 136 93 37 81 158 115)(16 116 159 82 38 94 137 60)(17 61 138 95 39 83 160 117)(18 118 161 84 40 96 139 62)(19 63 140 97 41 85 162 119)(20 120 163 86 42 98 141 64)(21 65 142 99 43 87 164 121)(22 122 165 88 44 100 143 66)
(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 22 123 88 144 143 45 122 23 44 101 66 166 165 67 100)(2 99 68 164 167 65 102 43 24 121 46 142 145 87 124 21)(3 20 125 86 146 141 47 120 25 42 103 64 168 163 69 98)(4 97 70 162 169 63 104 41 26 119 48 140 147 85 126 19)(5 18 127 84 148 139 49 118 27 40 105 62 170 161 71 96)(6 95 72 160 171 61 106 39 28 117 50 138 149 83 128 17)(7 16 129 82 150 137 51 116 29 38 107 60 172 159 73 94)(8 93 74 158 173 59 108 37 30 115 52 136 151 81 130 15)(9 14 131 80 152 135 53 114 31 36 109 58 174 157 75 92)(10 91 76 156 175 57 110 35 32 113 54 134 153 79 132 13)(11 12 89 78 154 133 55 112 33 34 111 56 176 155 77 90)
G:=sub<Sym(176)| (1,45,166,123,23,67,144,101)(2,102,145,68,24,124,167,46)(3,47,168,125,25,69,146,103)(4,104,147,70,26,126,169,48)(5,49,170,127,27,71,148,105)(6,106,149,72,28,128,171,50)(7,51,172,129,29,73,150,107)(8,108,151,74,30,130,173,52)(9,53,174,131,31,75,152,109)(10,110,153,76,32,132,175,54)(11,55,176,89,33,77,154,111)(12,112,155,78,34,90,133,56)(13,57,134,91,35,79,156,113)(14,114,157,80,36,92,135,58)(15,59,136,93,37,81,158,115)(16,116,159,82,38,94,137,60)(17,61,138,95,39,83,160,117)(18,118,161,84,40,96,139,62)(19,63,140,97,41,85,162,119)(20,120,163,86,42,98,141,64)(21,65,142,99,43,87,164,121)(22,122,165,88,44,100,143,66), (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), (1,22,123,88,144,143,45,122,23,44,101,66,166,165,67,100)(2,99,68,164,167,65,102,43,24,121,46,142,145,87,124,21)(3,20,125,86,146,141,47,120,25,42,103,64,168,163,69,98)(4,97,70,162,169,63,104,41,26,119,48,140,147,85,126,19)(5,18,127,84,148,139,49,118,27,40,105,62,170,161,71,96)(6,95,72,160,171,61,106,39,28,117,50,138,149,83,128,17)(7,16,129,82,150,137,51,116,29,38,107,60,172,159,73,94)(8,93,74,158,173,59,108,37,30,115,52,136,151,81,130,15)(9,14,131,80,152,135,53,114,31,36,109,58,174,157,75,92)(10,91,76,156,175,57,110,35,32,113,54,134,153,79,132,13)(11,12,89,78,154,133,55,112,33,34,111,56,176,155,77,90)>;
G:=Group( (1,45,166,123,23,67,144,101)(2,102,145,68,24,124,167,46)(3,47,168,125,25,69,146,103)(4,104,147,70,26,126,169,48)(5,49,170,127,27,71,148,105)(6,106,149,72,28,128,171,50)(7,51,172,129,29,73,150,107)(8,108,151,74,30,130,173,52)(9,53,174,131,31,75,152,109)(10,110,153,76,32,132,175,54)(11,55,176,89,33,77,154,111)(12,112,155,78,34,90,133,56)(13,57,134,91,35,79,156,113)(14,114,157,80,36,92,135,58)(15,59,136,93,37,81,158,115)(16,116,159,82,38,94,137,60)(17,61,138,95,39,83,160,117)(18,118,161,84,40,96,139,62)(19,63,140,97,41,85,162,119)(20,120,163,86,42,98,141,64)(21,65,142,99,43,87,164,121)(22,122,165,88,44,100,143,66), (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), (1,22,123,88,144,143,45,122,23,44,101,66,166,165,67,100)(2,99,68,164,167,65,102,43,24,121,46,142,145,87,124,21)(3,20,125,86,146,141,47,120,25,42,103,64,168,163,69,98)(4,97,70,162,169,63,104,41,26,119,48,140,147,85,126,19)(5,18,127,84,148,139,49,118,27,40,105,62,170,161,71,96)(6,95,72,160,171,61,106,39,28,117,50,138,149,83,128,17)(7,16,129,82,150,137,51,116,29,38,107,60,172,159,73,94)(8,93,74,158,173,59,108,37,30,115,52,136,151,81,130,15)(9,14,131,80,152,135,53,114,31,36,109,58,174,157,75,92)(10,91,76,156,175,57,110,35,32,113,54,134,153,79,132,13)(11,12,89,78,154,133,55,112,33,34,111,56,176,155,77,90) );
G=PermutationGroup([[(1,45,166,123,23,67,144,101),(2,102,145,68,24,124,167,46),(3,47,168,125,25,69,146,103),(4,104,147,70,26,126,169,48),(5,49,170,127,27,71,148,105),(6,106,149,72,28,128,171,50),(7,51,172,129,29,73,150,107),(8,108,151,74,30,130,173,52),(9,53,174,131,31,75,152,109),(10,110,153,76,32,132,175,54),(11,55,176,89,33,77,154,111),(12,112,155,78,34,90,133,56),(13,57,134,91,35,79,156,113),(14,114,157,80,36,92,135,58),(15,59,136,93,37,81,158,115),(16,116,159,82,38,94,137,60),(17,61,138,95,39,83,160,117),(18,118,161,84,40,96,139,62),(19,63,140,97,41,85,162,119),(20,120,163,86,42,98,141,64),(21,65,142,99,43,87,164,121),(22,122,165,88,44,100,143,66)], [(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)], [(1,22,123,88,144,143,45,122,23,44,101,66,166,165,67,100),(2,99,68,164,167,65,102,43,24,121,46,142,145,87,124,21),(3,20,125,86,146,141,47,120,25,42,103,64,168,163,69,98),(4,97,70,162,169,63,104,41,26,119,48,140,147,85,126,19),(5,18,127,84,148,139,49,118,27,40,105,62,170,161,71,96),(6,95,72,160,171,61,106,39,28,117,50,138,149,83,128,17),(7,16,129,82,150,137,51,116,29,38,107,60,172,159,73,94),(8,93,74,158,173,59,108,37,30,115,52,136,151,81,130,15),(9,14,131,80,152,135,53,114,31,36,109,58,174,157,75,92),(10,91,76,156,175,57,110,35,32,113,54,134,153,79,132,13),(11,12,89,78,154,133,55,112,33,34,111,56,176,155,77,90)]])
46 conjugacy classes
| class | 1 | 2A | 2B | 4A | 4B | 8A | 8B | 11A | ··· | 11E | 16A | 16B | 16C | 16D | 22A | ··· | 22E | 44A | ··· | 44E | 44F | ··· | 44O | 88A | ··· | 88J |
| order | 1 | 2 | 2 | 4 | 4 | 8 | 8 | 11 | ··· | 11 | 16 | 16 | 16 | 16 | 22 | ··· | 22 | 44 | ··· | 44 | 44 | ··· | 44 | 88 | ··· | 88 |
| size | 1 | 1 | 88 | 2 | 8 | 2 | 2 | 2 | ··· | 2 | 22 | 22 | 22 | 22 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
46 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | D4 | D8 | D11 | SD32 | D22 | C11⋊D4 | D4⋊D11 | C8.6D22 |
| kernel | C8.6D22 | C11⋊C16 | D88 | C11×Q16 | C44 | C22 | Q16 | C11 | C8 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 2 | 5 | 4 | 5 | 10 | 5 | 10 |
Matrix representation of C8.6D22 ►in GL4(𝔽353) generated by
| 0 | 19 | 0 | 0 |
| 130 | 186 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 54 | 320 | 0 | 0 |
| 67 | 299 | 0 | 0 |
| 0 | 0 | 223 | 223 |
| 0 | 0 | 130 | 149 |
| 214 | 33 | 0 | 0 |
| 40 | 54 | 0 | 0 |
| 0 | 0 | 223 | 223 |
| 0 | 0 | 149 | 130 |
G:=sub<GL(4,GF(353))| [0,130,0,0,19,186,0,0,0,0,1,0,0,0,0,1],[54,67,0,0,320,299,0,0,0,0,223,130,0,0,223,149],[214,40,0,0,33,54,0,0,0,0,223,149,0,0,223,130] >;
C8.6D22 in GAP, Magma, Sage, TeX
C_8._6D_{22} % in TeX
G:=Group("C8.6D22"); // GroupNames label
G:=SmallGroup(352,34);
// by ID
G=gap.SmallGroup(352,34);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-11,73,103,218,116,122,579,297,69,11525]);
// Polycyclic
G:=Group<a,b,c|a^8=1,b^22=a^4,c^2=a^3,b*a*b^-1=a^-1,a*c=c*a,c*b*c^-1=a^-1*b^21>;
// generators/relations
Export