metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D44⋊3C4, C44.1D4, C4.9D44, C22.7D8, C22.7SD16, C4⋊C4⋊1D11, C44.3(C2×C4), C4.1(C4×D11), (C2×D44).5C2, (C2×C22).30D4, (C2×C4).35D22, C11⋊1(D4⋊C4), C2.5(D22⋊C4), C2.2(D4⋊D11), C2.2(Q8⋊D11), C22.3(C22⋊C4), (C2×C44).10C22, C22.14(C11⋊D4), (C2×C11⋊C8)⋊1C2, (C11×C4⋊C4)⋊1C2, SmallGroup(352,15)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C22.D8
G = < a,b,c | a22=b8=c2=1, bab-1=cac=a-1, cbc=a11b-1 >
44C2
44C2
4C4
22C22
22C22
44C22
44C22
4D11
4D11
11D4
11D4
22C8
22D4
22C23
2D22
2D22
4D22
4D22
4C44
11C2×C8
11C2×D4
2D44
11D4⋊C4
Smallest permutation representation of C22.D8
►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)
(1 132 148 100 160 61 71 37)(2 131 149 99 161 60 72 36)(3 130 150 98 162 59 73 35)(4 129 151 97 163 58 74 34)(5 128 152 96 164 57 75 33)(6 127 153 95 165 56 76 32)(7 126 154 94 166 55 77 31)(8 125 133 93 167 54 78 30)(9 124 134 92 168 53 79 29)(10 123 135 91 169 52 80 28)(11 122 136 90 170 51 81 27)(12 121 137 89 171 50 82 26)(13 120 138 110 172 49 83 25)(14 119 139 109 173 48 84 24)(15 118 140 108 174 47 85 23)(16 117 141 107 175 46 86 44)(17 116 142 106 176 45 87 43)(18 115 143 105 155 66 88 42)(19 114 144 104 156 65 67 41)(20 113 145 103 157 64 68 40)(21 112 146 102 158 63 69 39)(22 111 147 101 159 62 70 38)
(2 22)(3 21)(4 20)(5 19)(6 18)(7 17)(8 16)(9 15)(10 14)(11 13)(23 113)(24 112)(25 111)(26 132)(27 131)(28 130)(29 129)(30 128)(31 127)(32 126)(33 125)(34 124)(35 123)(36 122)(37 121)(38 120)(39 119)(40 118)(41 117)(42 116)(43 115)(44 114)(45 105)(46 104)(47 103)(48 102)(49 101)(50 100)(51 99)(52 98)(53 97)(54 96)(55 95)(56 94)(57 93)(58 92)(59 91)(60 90)(61 89)(62 110)(63 109)(64 108)(65 107)(66 106)(67 152)(68 151)(69 150)(70 149)(71 148)(72 147)(73 146)(74 145)(75 144)(76 143)(77 142)(78 141)(79 140)(80 139)(81 138)(82 137)(83 136)(84 135)(85 134)(86 133)(87 154)(88 153)(155 165)(156 164)(157 163)(158 162)(159 161)(166 176)(167 175)(168 174)(169 173)(170 172)
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), (1,132,148,100,160,61,71,37)(2,131,149,99,161,60,72,36)(3,130,150,98,162,59,73,35)(4,129,151,97,163,58,74,34)(5,128,152,96,164,57,75,33)(6,127,153,95,165,56,76,32)(7,126,154,94,166,55,77,31)(8,125,133,93,167,54,78,30)(9,124,134,92,168,53,79,29)(10,123,135,91,169,52,80,28)(11,122,136,90,170,51,81,27)(12,121,137,89,171,50,82,26)(13,120,138,110,172,49,83,25)(14,119,139,109,173,48,84,24)(15,118,140,108,174,47,85,23)(16,117,141,107,175,46,86,44)(17,116,142,106,176,45,87,43)(18,115,143,105,155,66,88,42)(19,114,144,104,156,65,67,41)(20,113,145,103,157,64,68,40)(21,112,146,102,158,63,69,39)(22,111,147,101,159,62,70,38), (2,22)(3,21)(4,20)(5,19)(6,18)(7,17)(8,16)(9,15)(10,14)(11,13)(23,113)(24,112)(25,111)(26,132)(27,131)(28,130)(29,129)(30,128)(31,127)(32,126)(33,125)(34,124)(35,123)(36,122)(37,121)(38,120)(39,119)(40,118)(41,117)(42,116)(43,115)(44,114)(45,105)(46,104)(47,103)(48,102)(49,101)(50,100)(51,99)(52,98)(53,97)(54,96)(55,95)(56,94)(57,93)(58,92)(59,91)(60,90)(61,89)(62,110)(63,109)(64,108)(65,107)(66,106)(67,152)(68,151)(69,150)(70,149)(71,148)(72,147)(73,146)(74,145)(75,144)(76,143)(77,142)(78,141)(79,140)(80,139)(81,138)(82,137)(83,136)(84,135)(85,134)(86,133)(87,154)(88,153)(155,165)(156,164)(157,163)(158,162)(159,161)(166,176)(167,175)(168,174)(169,173)(170,172)>;
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), (1,132,148,100,160,61,71,37)(2,131,149,99,161,60,72,36)(3,130,150,98,162,59,73,35)(4,129,151,97,163,58,74,34)(5,128,152,96,164,57,75,33)(6,127,153,95,165,56,76,32)(7,126,154,94,166,55,77,31)(8,125,133,93,167,54,78,30)(9,124,134,92,168,53,79,29)(10,123,135,91,169,52,80,28)(11,122,136,90,170,51,81,27)(12,121,137,89,171,50,82,26)(13,120,138,110,172,49,83,25)(14,119,139,109,173,48,84,24)(15,118,140,108,174,47,85,23)(16,117,141,107,175,46,86,44)(17,116,142,106,176,45,87,43)(18,115,143,105,155,66,88,42)(19,114,144,104,156,65,67,41)(20,113,145,103,157,64,68,40)(21,112,146,102,158,63,69,39)(22,111,147,101,159,62,70,38), (2,22)(3,21)(4,20)(5,19)(6,18)(7,17)(8,16)(9,15)(10,14)(11,13)(23,113)(24,112)(25,111)(26,132)(27,131)(28,130)(29,129)(30,128)(31,127)(32,126)(33,125)(34,124)(35,123)(36,122)(37,121)(38,120)(39,119)(40,118)(41,117)(42,116)(43,115)(44,114)(45,105)(46,104)(47,103)(48,102)(49,101)(50,100)(51,99)(52,98)(53,97)(54,96)(55,95)(56,94)(57,93)(58,92)(59,91)(60,90)(61,89)(62,110)(63,109)(64,108)(65,107)(66,106)(67,152)(68,151)(69,150)(70,149)(71,148)(72,147)(73,146)(74,145)(75,144)(76,143)(77,142)(78,141)(79,140)(80,139)(81,138)(82,137)(83,136)(84,135)(85,134)(86,133)(87,154)(88,153)(155,165)(156,164)(157,163)(158,162)(159,161)(166,176)(167,175)(168,174)(169,173)(170,172) );
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)], [(1,132,148,100,160,61,71,37),(2,131,149,99,161,60,72,36),(3,130,150,98,162,59,73,35),(4,129,151,97,163,58,74,34),(5,128,152,96,164,57,75,33),(6,127,153,95,165,56,76,32),(7,126,154,94,166,55,77,31),(8,125,133,93,167,54,78,30),(9,124,134,92,168,53,79,29),(10,123,135,91,169,52,80,28),(11,122,136,90,170,51,81,27),(12,121,137,89,171,50,82,26),(13,120,138,110,172,49,83,25),(14,119,139,109,173,48,84,24),(15,118,140,108,174,47,85,23),(16,117,141,107,175,46,86,44),(17,116,142,106,176,45,87,43),(18,115,143,105,155,66,88,42),(19,114,144,104,156,65,67,41),(20,113,145,103,157,64,68,40),(21,112,146,102,158,63,69,39),(22,111,147,101,159,62,70,38)], [(2,22),(3,21),(4,20),(5,19),(6,18),(7,17),(8,16),(9,15),(10,14),(11,13),(23,113),(24,112),(25,111),(26,132),(27,131),(28,130),(29,129),(30,128),(31,127),(32,126),(33,125),(34,124),(35,123),(36,122),(37,121),(38,120),(39,119),(40,118),(41,117),(42,116),(43,115),(44,114),(45,105),(46,104),(47,103),(48,102),(49,101),(50,100),(51,99),(52,98),(53,97),(54,96),(55,95),(56,94),(57,93),(58,92),(59,91),(60,90),(61,89),(62,110),(63,109),(64,108),(65,107),(66,106),(67,152),(68,151),(69,150),(70,149),(71,148),(72,147),(73,146),(74,145),(75,144),(76,143),(77,142),(78,141),(79,140),(80,139),(81,138),(82,137),(83,136),(84,135),(85,134),(86,133),(87,154),(88,153),(155,165),(156,164),(157,163),(158,162),(159,161),(166,176),(167,175),(168,174),(169,173),(170,172)]])
64 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 8A | 8B | 8C | 8D | 11A | ··· | 11E | 22A | ··· | 22O | 44A | ··· | 44AD |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 11 | ··· | 11 | 22 | ··· | 22 | 44 | ··· | 44 |
| size | 1 | 1 | 1 | 1 | 44 | 44 | 2 | 2 | 4 | 4 | 22 | 22 | 22 | 22 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
64 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C4 | D4 | D4 | D8 | SD16 | D11 | D22 | C4×D11 | D44 | C11⋊D4 | D4⋊D11 | Q8⋊D11 |
| kernel | C22.D8 | C2×C11⋊C8 | C11×C4⋊C4 | C2×D44 | D44 | C44 | C2×C22 | C22 | C22 | C4⋊C4 | C2×C4 | C4 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 4 | 1 | 1 | 2 | 2 | 5 | 5 | 10 | 10 | 10 | 5 | 5 |
Matrix representation of C22.D8 ►in GL4(𝔽89) generated by
| 54 | 53 | 0 | 0 |
| 65 | 77 | 0 | 0 |
| 0 | 0 | 88 | 0 |
| 0 | 0 | 0 | 88 |
| 71 | 11 | 0 | 0 |
| 19 | 18 | 0 | 0 |
| 0 | 0 | 0 | 49 |
| 0 | 0 | 20 | 49 |
| 11 | 18 | 0 | 0 |
| 23 | 78 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 1 | 88 |
G:=sub<GL(4,GF(89))| [54,65,0,0,53,77,0,0,0,0,88,0,0,0,0,88],[71,19,0,0,11,18,0,0,0,0,0,20,0,0,49,49],[11,23,0,0,18,78,0,0,0,0,1,1,0,0,0,88] >;
C22.D8 in GAP, Magma, Sage, TeX
C_{22}.D_8 % in TeX
G:=Group("C22.D8"); // GroupNames label
G:=SmallGroup(352,15);
// by ID
G=gap.SmallGroup(352,15);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-11,121,31,579,297,69,11525]);
// Polycyclic
G:=Group<a,b,c|a^22=b^8=c^2=1,b*a*b^-1=c*a*c=a^-1,c*b*c=a^11*b^-1>;
// generators/relations
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