metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C44.10D4, (C2×C44).1C4, (C2×C4).4D22, (C2×C4).Dic11, (Q8×C22).2C2, (C2×Q8).2D11, C44.C4.4C2, C4.15(C11⋊D4), C11⋊2(C4.10D4), (C2×C44).19C22, C22.17(C22⋊C4), C2.7(C23.D11), C22.4(C2×Dic11), (C2×C22).30(C2×C4), SmallGroup(352,42)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C44.10D4
G = < a,b,c | a44=1, b4=a22, c2=a33, bab-1=a-1, cac-1=a21, cbc-1=a33b3 >
Smallest permutation representation of C44.10D4
►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)
(1 140 12 173 23 162 34 151)(2 139 13 172 24 161 35 150)(3 138 14 171 25 160 36 149)(4 137 15 170 26 159 37 148)(5 136 16 169 27 158 38 147)(6 135 17 168 28 157 39 146)(7 134 18 167 29 156 40 145)(8 133 19 166 30 155 41 144)(9 176 20 165 31 154 42 143)(10 175 21 164 32 153 43 142)(11 174 22 163 33 152 44 141)(45 132 56 121 67 110 78 99)(46 131 57 120 68 109 79 98)(47 130 58 119 69 108 80 97)(48 129 59 118 70 107 81 96)(49 128 60 117 71 106 82 95)(50 127 61 116 72 105 83 94)(51 126 62 115 73 104 84 93)(52 125 63 114 74 103 85 92)(53 124 64 113 75 102 86 91)(54 123 65 112 76 101 87 90)(55 122 66 111 77 100 88 89)
(1 107 34 96 23 129 12 118)(2 128 35 117 24 106 13 95)(3 105 36 94 25 127 14 116)(4 126 37 115 26 104 15 93)(5 103 38 92 27 125 16 114)(6 124 39 113 28 102 17 91)(7 101 40 90 29 123 18 112)(8 122 41 111 30 100 19 89)(9 99 42 132 31 121 20 110)(10 120 43 109 32 98 21 131)(11 97 44 130 33 119 22 108)(45 165 78 154 67 143 56 176)(46 142 79 175 68 164 57 153)(47 163 80 152 69 141 58 174)(48 140 81 173 70 162 59 151)(49 161 82 150 71 139 60 172)(50 138 83 171 72 160 61 149)(51 159 84 148 73 137 62 170)(52 136 85 169 74 158 63 147)(53 157 86 146 75 135 64 168)(54 134 87 167 76 156 65 145)(55 155 88 144 77 133 66 166)
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), (1,140,12,173,23,162,34,151)(2,139,13,172,24,161,35,150)(3,138,14,171,25,160,36,149)(4,137,15,170,26,159,37,148)(5,136,16,169,27,158,38,147)(6,135,17,168,28,157,39,146)(7,134,18,167,29,156,40,145)(8,133,19,166,30,155,41,144)(9,176,20,165,31,154,42,143)(10,175,21,164,32,153,43,142)(11,174,22,163,33,152,44,141)(45,132,56,121,67,110,78,99)(46,131,57,120,68,109,79,98)(47,130,58,119,69,108,80,97)(48,129,59,118,70,107,81,96)(49,128,60,117,71,106,82,95)(50,127,61,116,72,105,83,94)(51,126,62,115,73,104,84,93)(52,125,63,114,74,103,85,92)(53,124,64,113,75,102,86,91)(54,123,65,112,76,101,87,90)(55,122,66,111,77,100,88,89), (1,107,34,96,23,129,12,118)(2,128,35,117,24,106,13,95)(3,105,36,94,25,127,14,116)(4,126,37,115,26,104,15,93)(5,103,38,92,27,125,16,114)(6,124,39,113,28,102,17,91)(7,101,40,90,29,123,18,112)(8,122,41,111,30,100,19,89)(9,99,42,132,31,121,20,110)(10,120,43,109,32,98,21,131)(11,97,44,130,33,119,22,108)(45,165,78,154,67,143,56,176)(46,142,79,175,68,164,57,153)(47,163,80,152,69,141,58,174)(48,140,81,173,70,162,59,151)(49,161,82,150,71,139,60,172)(50,138,83,171,72,160,61,149)(51,159,84,148,73,137,62,170)(52,136,85,169,74,158,63,147)(53,157,86,146,75,135,64,168)(54,134,87,167,76,156,65,145)(55,155,88,144,77,133,66,166)>;
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), (1,140,12,173,23,162,34,151)(2,139,13,172,24,161,35,150)(3,138,14,171,25,160,36,149)(4,137,15,170,26,159,37,148)(5,136,16,169,27,158,38,147)(6,135,17,168,28,157,39,146)(7,134,18,167,29,156,40,145)(8,133,19,166,30,155,41,144)(9,176,20,165,31,154,42,143)(10,175,21,164,32,153,43,142)(11,174,22,163,33,152,44,141)(45,132,56,121,67,110,78,99)(46,131,57,120,68,109,79,98)(47,130,58,119,69,108,80,97)(48,129,59,118,70,107,81,96)(49,128,60,117,71,106,82,95)(50,127,61,116,72,105,83,94)(51,126,62,115,73,104,84,93)(52,125,63,114,74,103,85,92)(53,124,64,113,75,102,86,91)(54,123,65,112,76,101,87,90)(55,122,66,111,77,100,88,89), (1,107,34,96,23,129,12,118)(2,128,35,117,24,106,13,95)(3,105,36,94,25,127,14,116)(4,126,37,115,26,104,15,93)(5,103,38,92,27,125,16,114)(6,124,39,113,28,102,17,91)(7,101,40,90,29,123,18,112)(8,122,41,111,30,100,19,89)(9,99,42,132,31,121,20,110)(10,120,43,109,32,98,21,131)(11,97,44,130,33,119,22,108)(45,165,78,154,67,143,56,176)(46,142,79,175,68,164,57,153)(47,163,80,152,69,141,58,174)(48,140,81,173,70,162,59,151)(49,161,82,150,71,139,60,172)(50,138,83,171,72,160,61,149)(51,159,84,148,73,137,62,170)(52,136,85,169,74,158,63,147)(53,157,86,146,75,135,64,168)(54,134,87,167,76,156,65,145)(55,155,88,144,77,133,66,166) );
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)], [(1,140,12,173,23,162,34,151),(2,139,13,172,24,161,35,150),(3,138,14,171,25,160,36,149),(4,137,15,170,26,159,37,148),(5,136,16,169,27,158,38,147),(6,135,17,168,28,157,39,146),(7,134,18,167,29,156,40,145),(8,133,19,166,30,155,41,144),(9,176,20,165,31,154,42,143),(10,175,21,164,32,153,43,142),(11,174,22,163,33,152,44,141),(45,132,56,121,67,110,78,99),(46,131,57,120,68,109,79,98),(47,130,58,119,69,108,80,97),(48,129,59,118,70,107,81,96),(49,128,60,117,71,106,82,95),(50,127,61,116,72,105,83,94),(51,126,62,115,73,104,84,93),(52,125,63,114,74,103,85,92),(53,124,64,113,75,102,86,91),(54,123,65,112,76,101,87,90),(55,122,66,111,77,100,88,89)], [(1,107,34,96,23,129,12,118),(2,128,35,117,24,106,13,95),(3,105,36,94,25,127,14,116),(4,126,37,115,26,104,15,93),(5,103,38,92,27,125,16,114),(6,124,39,113,28,102,17,91),(7,101,40,90,29,123,18,112),(8,122,41,111,30,100,19,89),(9,99,42,132,31,121,20,110),(10,120,43,109,32,98,21,131),(11,97,44,130,33,119,22,108),(45,165,78,154,67,143,56,176),(46,142,79,175,68,164,57,153),(47,163,80,152,69,141,58,174),(48,140,81,173,70,162,59,151),(49,161,82,150,71,139,60,172),(50,138,83,171,72,160,61,149),(51,159,84,148,73,137,62,170),(52,136,85,169,74,158,63,147),(53,157,86,146,75,135,64,168),(54,134,87,167,76,156,65,145),(55,155,88,144,77,133,66,166)]])
61 conjugacy classes
| class | 1 | 2A | 2B | 4A | 4B | 4C | 4D | 8A | 8B | 8C | 8D | 11A | ··· | 11E | 22A | ··· | 22O | 44A | ··· | 44AD |
| order | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 11 | ··· | 11 | 22 | ··· | 22 | 44 | ··· | 44 |
| size | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 44 | 44 | 44 | 44 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
61 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | - | + | - | |||
| image | C1 | C2 | C2 | C4 | D4 | D11 | Dic11 | D22 | C11⋊D4 | C4.10D4 | C44.10D4 |
| kernel | C44.10D4 | C44.C4 | Q8×C22 | C2×C44 | C44 | C2×Q8 | C2×C4 | C2×C4 | C4 | C11 | C1 |
| # reps | 1 | 2 | 1 | 4 | 2 | 5 | 10 | 5 | 20 | 1 | 10 |
Matrix representation of C44.10D4 ►in GL4(𝔽89) generated by
| 85 | 86 | 0 | 0 |
| 70 | 4 | 0 | 0 |
| 29 | 28 | 22 | 44 |
| 69 | 36 | 67 | 67 |
| 44 | 21 | 65 | 33 |
| 26 | 65 | 19 | 64 |
| 57 | 86 | 0 | 0 |
| 31 | 64 | 28 | 69 |
| 0 | 0 | 1 | 0 |
| 53 | 88 | 0 | 62 |
| 88 | 66 | 0 | 0 |
| 32 | 0 | 58 | 1 |
G:=sub<GL(4,GF(89))| [85,70,29,69,86,4,28,36,0,0,22,67,0,0,44,67],[44,26,57,31,21,65,86,64,65,19,0,28,33,64,0,69],[0,53,88,32,0,88,66,0,1,0,0,58,0,62,0,1] >;
C44.10D4 in GAP, Magma, Sage, TeX
C_{44}._{10}D_4 % in TeX
G:=Group("C44.10D4"); // GroupNames label
G:=SmallGroup(352,42);
// by ID
G=gap.SmallGroup(352,42);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-11,24,121,103,188,86,579,11525]);
// Polycyclic
G:=Group<a,b,c|a^44=1,b^4=a^22,c^2=a^33,b*a*b^-1=a^-1,c*a*c^-1=a^21,c*b*c^-1=a^33*b^3>;
// generators/relations
Export