metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic11⋊C4, C22.5D4, C22.1Q8, C2.1Dic22, C22.4D22, C11⋊1(C4⋊C4), C22.4(C2×C4), (C2×C44).1C2, (C2×C4).1D11, C2.4(C4×D11), C2.1(C11⋊D4), (C2×C22).4C22, (C2×Dic11).1C2, SmallGroup(176,11)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic11⋊C4
G = < a,b,c | a22=c4=1, b2=a11, bab-1=a-1, ac=ca, cbc-1=a11b >
Smallest permutation representation of Dic11⋊C4
►Regular action 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)
(1 122 12 111)(2 121 13 132)(3 120 14 131)(4 119 15 130)(5 118 16 129)(6 117 17 128)(7 116 18 127)(8 115 19 126)(9 114 20 125)(10 113 21 124)(11 112 22 123)(23 103 34 92)(24 102 35 91)(25 101 36 90)(26 100 37 89)(27 99 38 110)(28 98 39 109)(29 97 40 108)(30 96 41 107)(31 95 42 106)(32 94 43 105)(33 93 44 104)(45 170 56 159)(46 169 57 158)(47 168 58 157)(48 167 59 156)(49 166 60 155)(50 165 61 176)(51 164 62 175)(52 163 63 174)(53 162 64 173)(54 161 65 172)(55 160 66 171)(67 137 78 148)(68 136 79 147)(69 135 80 146)(70 134 81 145)(71 133 82 144)(72 154 83 143)(73 153 84 142)(74 152 85 141)(75 151 86 140)(76 150 87 139)(77 149 88 138)
(1 71 26 49)(2 72 27 50)(3 73 28 51)(4 74 29 52)(5 75 30 53)(6 76 31 54)(7 77 32 55)(8 78 33 56)(9 79 34 57)(10 80 35 58)(11 81 36 59)(12 82 37 60)(13 83 38 61)(14 84 39 62)(15 85 40 63)(16 86 41 64)(17 87 42 65)(18 88 43 66)(19 67 44 45)(20 68 23 46)(21 69 24 47)(22 70 25 48)(89 166 111 133)(90 167 112 134)(91 168 113 135)(92 169 114 136)(93 170 115 137)(94 171 116 138)(95 172 117 139)(96 173 118 140)(97 174 119 141)(98 175 120 142)(99 176 121 143)(100 155 122 144)(101 156 123 145)(102 157 124 146)(103 158 125 147)(104 159 126 148)(105 160 127 149)(106 161 128 150)(107 162 129 151)(108 163 130 152)(109 164 131 153)(110 165 132 154)
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), (1,122,12,111)(2,121,13,132)(3,120,14,131)(4,119,15,130)(5,118,16,129)(6,117,17,128)(7,116,18,127)(8,115,19,126)(9,114,20,125)(10,113,21,124)(11,112,22,123)(23,103,34,92)(24,102,35,91)(25,101,36,90)(26,100,37,89)(27,99,38,110)(28,98,39,109)(29,97,40,108)(30,96,41,107)(31,95,42,106)(32,94,43,105)(33,93,44,104)(45,170,56,159)(46,169,57,158)(47,168,58,157)(48,167,59,156)(49,166,60,155)(50,165,61,176)(51,164,62,175)(52,163,63,174)(53,162,64,173)(54,161,65,172)(55,160,66,171)(67,137,78,148)(68,136,79,147)(69,135,80,146)(70,134,81,145)(71,133,82,144)(72,154,83,143)(73,153,84,142)(74,152,85,141)(75,151,86,140)(76,150,87,139)(77,149,88,138), (1,71,26,49)(2,72,27,50)(3,73,28,51)(4,74,29,52)(5,75,30,53)(6,76,31,54)(7,77,32,55)(8,78,33,56)(9,79,34,57)(10,80,35,58)(11,81,36,59)(12,82,37,60)(13,83,38,61)(14,84,39,62)(15,85,40,63)(16,86,41,64)(17,87,42,65)(18,88,43,66)(19,67,44,45)(20,68,23,46)(21,69,24,47)(22,70,25,48)(89,166,111,133)(90,167,112,134)(91,168,113,135)(92,169,114,136)(93,170,115,137)(94,171,116,138)(95,172,117,139)(96,173,118,140)(97,174,119,141)(98,175,120,142)(99,176,121,143)(100,155,122,144)(101,156,123,145)(102,157,124,146)(103,158,125,147)(104,159,126,148)(105,160,127,149)(106,161,128,150)(107,162,129,151)(108,163,130,152)(109,164,131,153)(110,165,132,154)>;
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), (1,122,12,111)(2,121,13,132)(3,120,14,131)(4,119,15,130)(5,118,16,129)(6,117,17,128)(7,116,18,127)(8,115,19,126)(9,114,20,125)(10,113,21,124)(11,112,22,123)(23,103,34,92)(24,102,35,91)(25,101,36,90)(26,100,37,89)(27,99,38,110)(28,98,39,109)(29,97,40,108)(30,96,41,107)(31,95,42,106)(32,94,43,105)(33,93,44,104)(45,170,56,159)(46,169,57,158)(47,168,58,157)(48,167,59,156)(49,166,60,155)(50,165,61,176)(51,164,62,175)(52,163,63,174)(53,162,64,173)(54,161,65,172)(55,160,66,171)(67,137,78,148)(68,136,79,147)(69,135,80,146)(70,134,81,145)(71,133,82,144)(72,154,83,143)(73,153,84,142)(74,152,85,141)(75,151,86,140)(76,150,87,139)(77,149,88,138), (1,71,26,49)(2,72,27,50)(3,73,28,51)(4,74,29,52)(5,75,30,53)(6,76,31,54)(7,77,32,55)(8,78,33,56)(9,79,34,57)(10,80,35,58)(11,81,36,59)(12,82,37,60)(13,83,38,61)(14,84,39,62)(15,85,40,63)(16,86,41,64)(17,87,42,65)(18,88,43,66)(19,67,44,45)(20,68,23,46)(21,69,24,47)(22,70,25,48)(89,166,111,133)(90,167,112,134)(91,168,113,135)(92,169,114,136)(93,170,115,137)(94,171,116,138)(95,172,117,139)(96,173,118,140)(97,174,119,141)(98,175,120,142)(99,176,121,143)(100,155,122,144)(101,156,123,145)(102,157,124,146)(103,158,125,147)(104,159,126,148)(105,160,127,149)(106,161,128,150)(107,162,129,151)(108,163,130,152)(109,164,131,153)(110,165,132,154) );
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)], [(1,122,12,111),(2,121,13,132),(3,120,14,131),(4,119,15,130),(5,118,16,129),(6,117,17,128),(7,116,18,127),(8,115,19,126),(9,114,20,125),(10,113,21,124),(11,112,22,123),(23,103,34,92),(24,102,35,91),(25,101,36,90),(26,100,37,89),(27,99,38,110),(28,98,39,109),(29,97,40,108),(30,96,41,107),(31,95,42,106),(32,94,43,105),(33,93,44,104),(45,170,56,159),(46,169,57,158),(47,168,58,157),(48,167,59,156),(49,166,60,155),(50,165,61,176),(51,164,62,175),(52,163,63,174),(53,162,64,173),(54,161,65,172),(55,160,66,171),(67,137,78,148),(68,136,79,147),(69,135,80,146),(70,134,81,145),(71,133,82,144),(72,154,83,143),(73,153,84,142),(74,152,85,141),(75,151,86,140),(76,150,87,139),(77,149,88,138)], [(1,71,26,49),(2,72,27,50),(3,73,28,51),(4,74,29,52),(5,75,30,53),(6,76,31,54),(7,77,32,55),(8,78,33,56),(9,79,34,57),(10,80,35,58),(11,81,36,59),(12,82,37,60),(13,83,38,61),(14,84,39,62),(15,85,40,63),(16,86,41,64),(17,87,42,65),(18,88,43,66),(19,67,44,45),(20,68,23,46),(21,69,24,47),(22,70,25,48),(89,166,111,133),(90,167,112,134),(91,168,113,135),(92,169,114,136),(93,170,115,137),(94,171,116,138),(95,172,117,139),(96,173,118,140),(97,174,119,141),(98,175,120,142),(99,176,121,143),(100,155,122,144),(101,156,123,145),(102,157,124,146),(103,158,125,147),(104,159,126,148),(105,160,127,149),(106,161,128,150),(107,162,129,151),(108,163,130,152),(109,164,131,153),(110,165,132,154)]])
Dic11⋊C4 is a maximal subgroup of
C4×Dic22 C44.6Q8 C42⋊D11 C42⋊2D11 C23.11D22 C22⋊Dic22 C23.D22 Dic11⋊4D4 D22.D4 D22⋊D4 Dic22⋊C4 C44⋊Q8 Dic11.Q8 C44.3Q8 C4⋊C4×D11 D22.5D4 D22⋊Q8 C4⋊C4⋊D11 C44.48D4 C4×C11⋊D4 C23.23D22 C23.18D22 Dic11⋊D4 Dic11⋊Q8 D22⋊3Q8
Dic11⋊C4 is a maximal quotient of
C44.Q8 C4.Dic22 Dic11⋊C8 C44.53D4 C22.C42
50 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 11A | ··· | 11E | 22A | ··· | 22O | 44A | ··· | 44T |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 11 | ··· | 11 | 22 | ··· | 22 | 44 | ··· | 44 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 22 | 22 | 22 | 22 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | + | - | |||
| image | C1 | C2 | C2 | C4 | D4 | Q8 | D11 | D22 | Dic22 | C4×D11 | C11⋊D4 |
| kernel | Dic11⋊C4 | C2×Dic11 | C2×C44 | Dic11 | C22 | C22 | C2×C4 | C22 | C2 | C2 | C2 |
| # reps | 1 | 2 | 1 | 4 | 1 | 1 | 5 | 5 | 10 | 10 | 10 |
Matrix representation of Dic11⋊C4 ►in GL3(𝔽89) generated by
| 1 | 0 | 0 |
| 0 | 0 | 88 |
| 0 | 1 | 18 |
| 88 | 0 | 0 |
| 0 | 32 | 43 |
| 0 | 1 | 57 |
| 55 | 0 | 0 |
| 0 | 86 | 59 |
| 0 | 30 | 3 |
G:=sub<GL(3,GF(89))| [1,0,0,0,0,1,0,88,18],[88,0,0,0,32,1,0,43,57],[55,0,0,0,86,30,0,59,3] >;
Dic11⋊C4 in GAP, Magma, Sage, TeX
{\rm Dic}_{11}\rtimes C_4 % in TeX
G:=Group("Dic11:C4"); // GroupNames label
G:=SmallGroup(176,11);
// by ID
G=gap.SmallGroup(176,11);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-11,40,101,26,4004]);
// Polycyclic
G:=Group<a,b,c|a^22=c^4=1,b^2=a^11,b*a*b^-1=a^-1,a*c=c*a,c*b*c^-1=a^11*b>;
// generators/relations
Export