direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C2×C8×D11, C88⋊10C22, C44.35C23, (C2×C88)⋊8C2, C22⋊1(C2×C8), C11⋊1(C22×C8), C11⋊C8⋊13C22, C44.26(C2×C4), (C4×D11).5C4, D22.9(C2×C4), C4.23(C4×D11), (C2×C4).97D22, C22.12(C22×C4), (C2×Dic11).8C4, (C22×D11).5C4, C4.35(C22×D11), C22.13(C4×D11), (C2×C44).110C22, Dic11.10(C2×C4), (C4×D11).17C22, (C2×C11⋊C8)⋊13C2, C2.2(C2×C4×D11), (C2×C4×D11).12C2, (C2×C22).14(C2×C4), SmallGroup(352,94)
Series: Derived ►Chief ►Lower central ►Upper central
| C11 — C2×C8×D11 |
Generators and relations for C2×C8×D11
G = < a,b,c,d | a2=b8=c11=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
Subgroups: 346 in 76 conjugacy classes, 49 normal (19 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C8, C8, C2×C4, C2×C4, C23, C11, C2×C8, C2×C8, C22×C4, D11, C22, C22, C22×C8, Dic11, C44, D22, C2×C22, C11⋊C8, C88, C4×D11, C2×Dic11, C2×C44, C22×D11, C8×D11, C2×C11⋊C8, C2×C88, C2×C4×D11, C2×C8×D11
Quotients: C1, C2, C4, C22, C8, C2×C4, C23, C2×C8, C22×C4, D11, C22×C8, D22, C4×D11, C22×D11, C8×D11, C2×C4×D11, C2×C8×D11
►On 176 points
Generators in S176
(1 98)(2 99)(3 89)(4 90)(5 91)(6 92)(7 93)(8 94)(9 95)(10 96)(11 97)(12 100)(13 101)(14 102)(15 103)(16 104)(17 105)(18 106)(19 107)(20 108)(21 109)(22 110)(23 111)(24 112)(25 113)(26 114)(27 115)(28 116)(29 117)(30 118)(31 119)(32 120)(33 121)(34 122)(35 123)(36 124)(37 125)(38 126)(39 127)(40 128)(41 129)(42 130)(43 131)(44 132)(45 133)(46 134)(47 135)(48 136)(49 137)(50 138)(51 139)(52 140)(53 141)(54 142)(55 143)(56 144)(57 145)(58 146)(59 147)(60 148)(61 149)(62 150)(63 151)(64 152)(65 153)(66 154)(67 155)(68 156)(69 157)(70 158)(71 159)(72 160)(73 161)(74 162)(75 163)(76 164)(77 165)(78 166)(79 167)(80 168)(81 169)(82 170)(83 171)(84 172)(85 173)(86 174)(87 175)(88 176)
(1 87 43 65 21 76 32 54)(2 88 44 66 22 77 33 55)(3 78 34 56 12 67 23 45)(4 79 35 57 13 68 24 46)(5 80 36 58 14 69 25 47)(6 81 37 59 15 70 26 48)(7 82 38 60 16 71 27 49)(8 83 39 61 17 72 28 50)(9 84 40 62 18 73 29 51)(10 85 41 63 19 74 30 52)(11 86 42 64 20 75 31 53)(89 166 122 144 100 155 111 133)(90 167 123 145 101 156 112 134)(91 168 124 146 102 157 113 135)(92 169 125 147 103 158 114 136)(93 170 126 148 104 159 115 137)(94 171 127 149 105 160 116 138)(95 172 128 150 106 161 117 139)(96 173 129 151 107 162 118 140)(97 174 130 152 108 163 119 141)(98 175 131 153 109 164 120 142)(99 176 132 154 110 165 121 143)
(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 20)(2 19)(3 18)(4 17)(5 16)(6 15)(7 14)(8 13)(9 12)(10 22)(11 21)(23 40)(24 39)(25 38)(26 37)(27 36)(28 35)(29 34)(30 44)(31 43)(32 42)(33 41)(45 62)(46 61)(47 60)(48 59)(49 58)(50 57)(51 56)(52 66)(53 65)(54 64)(55 63)(67 84)(68 83)(69 82)(70 81)(71 80)(72 79)(73 78)(74 88)(75 87)(76 86)(77 85)(89 106)(90 105)(91 104)(92 103)(93 102)(94 101)(95 100)(96 110)(97 109)(98 108)(99 107)(111 128)(112 127)(113 126)(114 125)(115 124)(116 123)(117 122)(118 132)(119 131)(120 130)(121 129)(133 150)(134 149)(135 148)(136 147)(137 146)(138 145)(139 144)(140 154)(141 153)(142 152)(143 151)(155 172)(156 171)(157 170)(158 169)(159 168)(160 167)(161 166)(162 176)(163 175)(164 174)(165 173)
G:=sub<Sym(176)| (1,98)(2,99)(3,89)(4,90)(5,91)(6,92)(7,93)(8,94)(9,95)(10,96)(11,97)(12,100)(13,101)(14,102)(15,103)(16,104)(17,105)(18,106)(19,107)(20,108)(21,109)(22,110)(23,111)(24,112)(25,113)(26,114)(27,115)(28,116)(29,117)(30,118)(31,119)(32,120)(33,121)(34,122)(35,123)(36,124)(37,125)(38,126)(39,127)(40,128)(41,129)(42,130)(43,131)(44,132)(45,133)(46,134)(47,135)(48,136)(49,137)(50,138)(51,139)(52,140)(53,141)(54,142)(55,143)(56,144)(57,145)(58,146)(59,147)(60,148)(61,149)(62,150)(63,151)(64,152)(65,153)(66,154)(67,155)(68,156)(69,157)(70,158)(71,159)(72,160)(73,161)(74,162)(75,163)(76,164)(77,165)(78,166)(79,167)(80,168)(81,169)(82,170)(83,171)(84,172)(85,173)(86,174)(87,175)(88,176), (1,87,43,65,21,76,32,54)(2,88,44,66,22,77,33,55)(3,78,34,56,12,67,23,45)(4,79,35,57,13,68,24,46)(5,80,36,58,14,69,25,47)(6,81,37,59,15,70,26,48)(7,82,38,60,16,71,27,49)(8,83,39,61,17,72,28,50)(9,84,40,62,18,73,29,51)(10,85,41,63,19,74,30,52)(11,86,42,64,20,75,31,53)(89,166,122,144,100,155,111,133)(90,167,123,145,101,156,112,134)(91,168,124,146,102,157,113,135)(92,169,125,147,103,158,114,136)(93,170,126,148,104,159,115,137)(94,171,127,149,105,160,116,138)(95,172,128,150,106,161,117,139)(96,173,129,151,107,162,118,140)(97,174,130,152,108,163,119,141)(98,175,131,153,109,164,120,142)(99,176,132,154,110,165,121,143), (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), (1,20)(2,19)(3,18)(4,17)(5,16)(6,15)(7,14)(8,13)(9,12)(10,22)(11,21)(23,40)(24,39)(25,38)(26,37)(27,36)(28,35)(29,34)(30,44)(31,43)(32,42)(33,41)(45,62)(46,61)(47,60)(48,59)(49,58)(50,57)(51,56)(52,66)(53,65)(54,64)(55,63)(67,84)(68,83)(69,82)(70,81)(71,80)(72,79)(73,78)(74,88)(75,87)(76,86)(77,85)(89,106)(90,105)(91,104)(92,103)(93,102)(94,101)(95,100)(96,110)(97,109)(98,108)(99,107)(111,128)(112,127)(113,126)(114,125)(115,124)(116,123)(117,122)(118,132)(119,131)(120,130)(121,129)(133,150)(134,149)(135,148)(136,147)(137,146)(138,145)(139,144)(140,154)(141,153)(142,152)(143,151)(155,172)(156,171)(157,170)(158,169)(159,168)(160,167)(161,166)(162,176)(163,175)(164,174)(165,173)>;
G:=Group( (1,98)(2,99)(3,89)(4,90)(5,91)(6,92)(7,93)(8,94)(9,95)(10,96)(11,97)(12,100)(13,101)(14,102)(15,103)(16,104)(17,105)(18,106)(19,107)(20,108)(21,109)(22,110)(23,111)(24,112)(25,113)(26,114)(27,115)(28,116)(29,117)(30,118)(31,119)(32,120)(33,121)(34,122)(35,123)(36,124)(37,125)(38,126)(39,127)(40,128)(41,129)(42,130)(43,131)(44,132)(45,133)(46,134)(47,135)(48,136)(49,137)(50,138)(51,139)(52,140)(53,141)(54,142)(55,143)(56,144)(57,145)(58,146)(59,147)(60,148)(61,149)(62,150)(63,151)(64,152)(65,153)(66,154)(67,155)(68,156)(69,157)(70,158)(71,159)(72,160)(73,161)(74,162)(75,163)(76,164)(77,165)(78,166)(79,167)(80,168)(81,169)(82,170)(83,171)(84,172)(85,173)(86,174)(87,175)(88,176), (1,87,43,65,21,76,32,54)(2,88,44,66,22,77,33,55)(3,78,34,56,12,67,23,45)(4,79,35,57,13,68,24,46)(5,80,36,58,14,69,25,47)(6,81,37,59,15,70,26,48)(7,82,38,60,16,71,27,49)(8,83,39,61,17,72,28,50)(9,84,40,62,18,73,29,51)(10,85,41,63,19,74,30,52)(11,86,42,64,20,75,31,53)(89,166,122,144,100,155,111,133)(90,167,123,145,101,156,112,134)(91,168,124,146,102,157,113,135)(92,169,125,147,103,158,114,136)(93,170,126,148,104,159,115,137)(94,171,127,149,105,160,116,138)(95,172,128,150,106,161,117,139)(96,173,129,151,107,162,118,140)(97,174,130,152,108,163,119,141)(98,175,131,153,109,164,120,142)(99,176,132,154,110,165,121,143), (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), (1,20)(2,19)(3,18)(4,17)(5,16)(6,15)(7,14)(8,13)(9,12)(10,22)(11,21)(23,40)(24,39)(25,38)(26,37)(27,36)(28,35)(29,34)(30,44)(31,43)(32,42)(33,41)(45,62)(46,61)(47,60)(48,59)(49,58)(50,57)(51,56)(52,66)(53,65)(54,64)(55,63)(67,84)(68,83)(69,82)(70,81)(71,80)(72,79)(73,78)(74,88)(75,87)(76,86)(77,85)(89,106)(90,105)(91,104)(92,103)(93,102)(94,101)(95,100)(96,110)(97,109)(98,108)(99,107)(111,128)(112,127)(113,126)(114,125)(115,124)(116,123)(117,122)(118,132)(119,131)(120,130)(121,129)(133,150)(134,149)(135,148)(136,147)(137,146)(138,145)(139,144)(140,154)(141,153)(142,152)(143,151)(155,172)(156,171)(157,170)(158,169)(159,168)(160,167)(161,166)(162,176)(163,175)(164,174)(165,173) );
G=PermutationGroup([[(1,98),(2,99),(3,89),(4,90),(5,91),(6,92),(7,93),(8,94),(9,95),(10,96),(11,97),(12,100),(13,101),(14,102),(15,103),(16,104),(17,105),(18,106),(19,107),(20,108),(21,109),(22,110),(23,111),(24,112),(25,113),(26,114),(27,115),(28,116),(29,117),(30,118),(31,119),(32,120),(33,121),(34,122),(35,123),(36,124),(37,125),(38,126),(39,127),(40,128),(41,129),(42,130),(43,131),(44,132),(45,133),(46,134),(47,135),(48,136),(49,137),(50,138),(51,139),(52,140),(53,141),(54,142),(55,143),(56,144),(57,145),(58,146),(59,147),(60,148),(61,149),(62,150),(63,151),(64,152),(65,153),(66,154),(67,155),(68,156),(69,157),(70,158),(71,159),(72,160),(73,161),(74,162),(75,163),(76,164),(77,165),(78,166),(79,167),(80,168),(81,169),(82,170),(83,171),(84,172),(85,173),(86,174),(87,175),(88,176)], [(1,87,43,65,21,76,32,54),(2,88,44,66,22,77,33,55),(3,78,34,56,12,67,23,45),(4,79,35,57,13,68,24,46),(5,80,36,58,14,69,25,47),(6,81,37,59,15,70,26,48),(7,82,38,60,16,71,27,49),(8,83,39,61,17,72,28,50),(9,84,40,62,18,73,29,51),(10,85,41,63,19,74,30,52),(11,86,42,64,20,75,31,53),(89,166,122,144,100,155,111,133),(90,167,123,145,101,156,112,134),(91,168,124,146,102,157,113,135),(92,169,125,147,103,158,114,136),(93,170,126,148,104,159,115,137),(94,171,127,149,105,160,116,138),(95,172,128,150,106,161,117,139),(96,173,129,151,107,162,118,140),(97,174,130,152,108,163,119,141),(98,175,131,153,109,164,120,142),(99,176,132,154,110,165,121,143)], [(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)], [(1,20),(2,19),(3,18),(4,17),(5,16),(6,15),(7,14),(8,13),(9,12),(10,22),(11,21),(23,40),(24,39),(25,38),(26,37),(27,36),(28,35),(29,34),(30,44),(31,43),(32,42),(33,41),(45,62),(46,61),(47,60),(48,59),(49,58),(50,57),(51,56),(52,66),(53,65),(54,64),(55,63),(67,84),(68,83),(69,82),(70,81),(71,80),(72,79),(73,78),(74,88),(75,87),(76,86),(77,85),(89,106),(90,105),(91,104),(92,103),(93,102),(94,101),(95,100),(96,110),(97,109),(98,108),(99,107),(111,128),(112,127),(113,126),(114,125),(115,124),(116,123),(117,122),(118,132),(119,131),(120,130),(121,129),(133,150),(134,149),(135,148),(136,147),(137,146),(138,145),(139,144),(140,154),(141,153),(142,152),(143,151),(155,172),(156,171),(157,170),(158,169),(159,168),(160,167),(161,166),(162,176),(163,175),(164,174),(165,173)]])
112 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 8A | ··· | 8H | 8I | ··· | 8P | 11A | ··· | 11E | 22A | ··· | 22O | 44A | ··· | 44T | 88A | ··· | 88AN |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 8 | ··· | 8 | 11 | ··· | 11 | 22 | ··· | 22 | 44 | ··· | 44 | 88 | ··· | 88 |
| size | 1 | 1 | 1 | 1 | 11 | 11 | 11 | 11 | 1 | 1 | 1 | 1 | 11 | 11 | 11 | 11 | 1 | ··· | 1 | 11 | ··· | 11 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
112 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C8 | D11 | D22 | D22 | C4×D11 | C4×D11 | C8×D11 |
| kernel | C2×C8×D11 | C8×D11 | C2×C11⋊C8 | C2×C88 | C2×C4×D11 | C4×D11 | C2×Dic11 | C22×D11 | D22 | C2×C8 | C8 | C2×C4 | C4 | C22 | C2 |
| # reps | 1 | 4 | 1 | 1 | 1 | 4 | 2 | 2 | 16 | 5 | 10 | 5 | 10 | 10 | 40 |
Matrix representation of C2×C8×D11 ►in GL4(𝔽89) generated by
| 88 | 0 | 0 | 0 |
| 0 | 88 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 37 | 0 | 0 | 0 |
| 0 | 88 | 0 | 0 |
| 0 | 0 | 55 | 0 |
| 0 | 0 | 0 | 55 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 82 | 1 |
| 0 | 0 | 76 | 78 |
| 1 | 0 | 0 | 0 |
| 0 | 88 | 0 | 0 |
| 0 | 0 | 78 | 88 |
| 0 | 0 | 31 | 11 |
G:=sub<GL(4,GF(89))| [88,0,0,0,0,88,0,0,0,0,1,0,0,0,0,1],[37,0,0,0,0,88,0,0,0,0,55,0,0,0,0,55],[1,0,0,0,0,1,0,0,0,0,82,76,0,0,1,78],[1,0,0,0,0,88,0,0,0,0,78,31,0,0,88,11] >;
C2×C8×D11 in GAP, Magma, Sage, TeX
C_2\times C_8\times D_{11} % in TeX
G:=Group("C2xC8xD11"); // GroupNames label
G:=SmallGroup(352,94);
// by ID
G=gap.SmallGroup(352,94);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-11,50,69,11525]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^8=c^11=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations