metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: D22.C8, C176⋊4C2, C16⋊3D11, C8.20D22, Dic11.C8, C11⋊1M5(2), C88.20C22, C11⋊C16⋊4C2, C11⋊C8.2C4, C22.2(C2×C8), C2.3(C8×D11), C44.22(C2×C4), (C4×D11).2C4, (C8×D11).2C2, C4.17(C4×D11), SmallGroup(352,4)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D22.C8
G = < a,b,c | a22=b2=1, c8=a11, bab=a-1, ac=ca, cbc-1=a11b >
Smallest permutation representation of D22.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 22)(2 21)(3 20)(4 19)(5 18)(6 17)(7 16)(8 15)(9 14)(10 13)(11 12)(23 32)(24 31)(25 30)(26 29)(27 28)(33 44)(34 43)(35 42)(36 41)(37 40)(38 39)(45 66)(46 65)(47 64)(48 63)(49 62)(50 61)(51 60)(52 59)(53 58)(54 57)(55 56)(67 78)(68 77)(69 76)(70 75)(71 74)(72 73)(79 88)(80 87)(81 86)(82 85)(83 84)(89 99)(90 98)(91 97)(92 96)(93 95)(100 110)(101 109)(102 108)(103 107)(104 106)(111 123)(112 122)(113 121)(114 120)(115 119)(116 118)(124 132)(125 131)(126 130)(127 129)(133 141)(134 140)(135 139)(136 138)(142 154)(143 153)(144 152)(145 151)(146 150)(147 149)(155 163)(156 162)(157 161)(158 160)(164 176)(165 175)(166 174)(167 173)(168 172)(169 171)
(1 165 84 112 39 143 56 100 12 176 73 123 28 154 45 89)(2 166 85 113 40 144 57 101 13 155 74 124 29 133 46 90)(3 167 86 114 41 145 58 102 14 156 75 125 30 134 47 91)(4 168 87 115 42 146 59 103 15 157 76 126 31 135 48 92)(5 169 88 116 43 147 60 104 16 158 77 127 32 136 49 93)(6 170 67 117 44 148 61 105 17 159 78 128 33 137 50 94)(7 171 68 118 23 149 62 106 18 160 79 129 34 138 51 95)(8 172 69 119 24 150 63 107 19 161 80 130 35 139 52 96)(9 173 70 120 25 151 64 108 20 162 81 131 36 140 53 97)(10 174 71 121 26 152 65 109 21 163 82 132 37 141 54 98)(11 175 72 122 27 153 66 110 22 164 83 111 38 142 55 99)
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), (1,22)(2,21)(3,20)(4,19)(5,18)(6,17)(7,16)(8,15)(9,14)(10,13)(11,12)(23,32)(24,31)(25,30)(26,29)(27,28)(33,44)(34,43)(35,42)(36,41)(37,40)(38,39)(45,66)(46,65)(47,64)(48,63)(49,62)(50,61)(51,60)(52,59)(53,58)(54,57)(55,56)(67,78)(68,77)(69,76)(70,75)(71,74)(72,73)(79,88)(80,87)(81,86)(82,85)(83,84)(89,99)(90,98)(91,97)(92,96)(93,95)(100,110)(101,109)(102,108)(103,107)(104,106)(111,123)(112,122)(113,121)(114,120)(115,119)(116,118)(124,132)(125,131)(126,130)(127,129)(133,141)(134,140)(135,139)(136,138)(142,154)(143,153)(144,152)(145,151)(146,150)(147,149)(155,163)(156,162)(157,161)(158,160)(164,176)(165,175)(166,174)(167,173)(168,172)(169,171), (1,165,84,112,39,143,56,100,12,176,73,123,28,154,45,89)(2,166,85,113,40,144,57,101,13,155,74,124,29,133,46,90)(3,167,86,114,41,145,58,102,14,156,75,125,30,134,47,91)(4,168,87,115,42,146,59,103,15,157,76,126,31,135,48,92)(5,169,88,116,43,147,60,104,16,158,77,127,32,136,49,93)(6,170,67,117,44,148,61,105,17,159,78,128,33,137,50,94)(7,171,68,118,23,149,62,106,18,160,79,129,34,138,51,95)(8,172,69,119,24,150,63,107,19,161,80,130,35,139,52,96)(9,173,70,120,25,151,64,108,20,162,81,131,36,140,53,97)(10,174,71,121,26,152,65,109,21,163,82,132,37,141,54,98)(11,175,72,122,27,153,66,110,22,164,83,111,38,142,55,99)>;
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), (1,22)(2,21)(3,20)(4,19)(5,18)(6,17)(7,16)(8,15)(9,14)(10,13)(11,12)(23,32)(24,31)(25,30)(26,29)(27,28)(33,44)(34,43)(35,42)(36,41)(37,40)(38,39)(45,66)(46,65)(47,64)(48,63)(49,62)(50,61)(51,60)(52,59)(53,58)(54,57)(55,56)(67,78)(68,77)(69,76)(70,75)(71,74)(72,73)(79,88)(80,87)(81,86)(82,85)(83,84)(89,99)(90,98)(91,97)(92,96)(93,95)(100,110)(101,109)(102,108)(103,107)(104,106)(111,123)(112,122)(113,121)(114,120)(115,119)(116,118)(124,132)(125,131)(126,130)(127,129)(133,141)(134,140)(135,139)(136,138)(142,154)(143,153)(144,152)(145,151)(146,150)(147,149)(155,163)(156,162)(157,161)(158,160)(164,176)(165,175)(166,174)(167,173)(168,172)(169,171), (1,165,84,112,39,143,56,100,12,176,73,123,28,154,45,89)(2,166,85,113,40,144,57,101,13,155,74,124,29,133,46,90)(3,167,86,114,41,145,58,102,14,156,75,125,30,134,47,91)(4,168,87,115,42,146,59,103,15,157,76,126,31,135,48,92)(5,169,88,116,43,147,60,104,16,158,77,127,32,136,49,93)(6,170,67,117,44,148,61,105,17,159,78,128,33,137,50,94)(7,171,68,118,23,149,62,106,18,160,79,129,34,138,51,95)(8,172,69,119,24,150,63,107,19,161,80,130,35,139,52,96)(9,173,70,120,25,151,64,108,20,162,81,131,36,140,53,97)(10,174,71,121,26,152,65,109,21,163,82,132,37,141,54,98)(11,175,72,122,27,153,66,110,22,164,83,111,38,142,55,99) );
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)], [(1,22),(2,21),(3,20),(4,19),(5,18),(6,17),(7,16),(8,15),(9,14),(10,13),(11,12),(23,32),(24,31),(25,30),(26,29),(27,28),(33,44),(34,43),(35,42),(36,41),(37,40),(38,39),(45,66),(46,65),(47,64),(48,63),(49,62),(50,61),(51,60),(52,59),(53,58),(54,57),(55,56),(67,78),(68,77),(69,76),(70,75),(71,74),(72,73),(79,88),(80,87),(81,86),(82,85),(83,84),(89,99),(90,98),(91,97),(92,96),(93,95),(100,110),(101,109),(102,108),(103,107),(104,106),(111,123),(112,122),(113,121),(114,120),(115,119),(116,118),(124,132),(125,131),(126,130),(127,129),(133,141),(134,140),(135,139),(136,138),(142,154),(143,153),(144,152),(145,151),(146,150),(147,149),(155,163),(156,162),(157,161),(158,160),(164,176),(165,175),(166,174),(167,173),(168,172),(169,171)], [(1,165,84,112,39,143,56,100,12,176,73,123,28,154,45,89),(2,166,85,113,40,144,57,101,13,155,74,124,29,133,46,90),(3,167,86,114,41,145,58,102,14,156,75,125,30,134,47,91),(4,168,87,115,42,146,59,103,15,157,76,126,31,135,48,92),(5,169,88,116,43,147,60,104,16,158,77,127,32,136,49,93),(6,170,67,117,44,148,61,105,17,159,78,128,33,137,50,94),(7,171,68,118,23,149,62,106,18,160,79,129,34,138,51,95),(8,172,69,119,24,150,63,107,19,161,80,130,35,139,52,96),(9,173,70,120,25,151,64,108,20,162,81,131,36,140,53,97),(10,174,71,121,26,152,65,109,21,163,82,132,37,141,54,98),(11,175,72,122,27,153,66,110,22,164,83,111,38,142,55,99)]])
100 conjugacy classes
| class | 1 | 2A | 2B | 4A | 4B | 4C | 8A | 8B | 8C | 8D | 8E | 8F | 11A | ··· | 11E | 16A | 16B | 16C | 16D | 16E | 16F | 16G | 16H | 22A | ··· | 22E | 44A | ··· | 44J | 88A | ··· | 88T | 176A | ··· | 176AN |
| order | 1 | 2 | 2 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | 8 | 11 | ··· | 11 | 16 | 16 | 16 | 16 | 16 | 16 | 16 | 16 | 22 | ··· | 22 | 44 | ··· | 44 | 88 | ··· | 88 | 176 | ··· | 176 |
| size | 1 | 1 | 22 | 1 | 1 | 22 | 1 | 1 | 1 | 1 | 22 | 22 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 22 | 22 | 22 | 22 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
100 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | ||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C8 | C8 | D11 | M5(2) | D22 | C4×D11 | C8×D11 | D22.C8 |
| kernel | D22.C8 | C11⋊C16 | C176 | C8×D11 | C11⋊C8 | C4×D11 | Dic11 | D22 | C16 | C11 | C8 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 5 | 4 | 5 | 10 | 20 | 40 |
Matrix representation of D22.C8 ►in GL2(𝔽353) generated by
| 204 | 130 |
| 223 | 130 |
| 130 | 204 |
| 130 | 223 |
| 102 | 34 |
| 319 | 251 |
G:=sub<GL(2,GF(353))| [204,223,130,130],[130,130,204,223],[102,319,34,251] >;
D22.C8 in GAP, Magma, Sage, TeX
D_{22}.C_8 % in TeX
G:=Group("D22.C8"); // GroupNames label
G:=SmallGroup(352,4);
// by ID
G=gap.SmallGroup(352,4);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-11,217,31,50,69,11525]);
// Polycyclic
G:=Group<a,b,c|a^22=b^2=1,c^8=a^11,b*a*b=a^-1,a*c=c*a,c*b*c^-1=a^11*b>;
// generators/relations
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