direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C4×C11⋊D4, C44⋊8D4, C23.22D22, C11⋊4(C4×D4), D22⋊4(C2×C4), D22⋊C4⋊18C2, (C22×C44)⋊9C2, C22.41(C2×D4), (C22×C4)⋊2D11, C22⋊2(C4×D11), Dic11⋊2(C2×C4), (C2×C4).103D22, Dic11⋊C4⋊18C2, (C4×Dic11)⋊16C2, C22.17(C4○D4), (C2×C44).77C22, C22.19(C22×C4), (C2×C22).46C23, C23.D11⋊14C2, C2.5(D44⋊5C2), (C22×C22).38C22, C22.24(C22×D11), (C2×Dic11).36C22, (C22×D11).24C22, (C2×C22)⋊5(C2×C4), (C2×C4×D11)⋊14C2, C2.20(C2×C4×D11), C2.3(C2×C11⋊D4), (C2×C11⋊D4).7C2, SmallGroup(352,123)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4×C11⋊D4
G = < a,b,c,d | a4=b11=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c-1 >
Subgroups: 474 in 94 conjugacy classes, 45 normal (29 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C2×C4, C2×C4, D4, C23, C23, C11, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, D11, C22, C22, C4×D4, Dic11, Dic11, C44, C44, D22, D22, C2×C22, C2×C22, C2×C22, C4×D11, C2×Dic11, C11⋊D4, C2×C44, C2×C44, C22×D11, C22×C22, C4×Dic11, Dic11⋊C4, D22⋊C4, C23.D11, C2×C4×D11, C2×C11⋊D4, C22×C44, C4×C11⋊D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22×C4, C2×D4, C4○D4, D11, C4×D4, D22, C4×D11, C11⋊D4, C22×D11, C2×C4×D11, D44⋊5C2, C2×C11⋊D4, C4×C11⋊D4
►On 176 points
Generators in S176
(1 67 23 45)(2 68 24 46)(3 69 25 47)(4 70 26 48)(5 71 27 49)(6 72 28 50)(7 73 29 51)(8 74 30 52)(9 75 31 53)(10 76 32 54)(11 77 33 55)(12 78 34 56)(13 79 35 57)(14 80 36 58)(15 81 37 59)(16 82 38 60)(17 83 39 61)(18 84 40 62)(19 85 41 63)(20 86 42 64)(21 87 43 65)(22 88 44 66)(89 155 111 133)(90 156 112 134)(91 157 113 135)(92 158 114 136)(93 159 115 137)(94 160 116 138)(95 161 117 139)(96 162 118 140)(97 163 119 141)(98 164 120 142)(99 165 121 143)(100 166 122 144)(101 167 123 145)(102 168 124 146)(103 169 125 147)(104 170 126 148)(105 171 127 149)(106 172 128 150)(107 173 129 151)(108 174 130 152)(109 175 131 153)(110 176 132 154)
(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 89 12 100)(2 99 13 110)(3 98 14 109)(4 97 15 108)(5 96 16 107)(6 95 17 106)(7 94 18 105)(8 93 19 104)(9 92 20 103)(10 91 21 102)(11 90 22 101)(23 111 34 122)(24 121 35 132)(25 120 36 131)(26 119 37 130)(27 118 38 129)(28 117 39 128)(29 116 40 127)(30 115 41 126)(31 114 42 125)(32 113 43 124)(33 112 44 123)(45 133 56 144)(46 143 57 154)(47 142 58 153)(48 141 59 152)(49 140 60 151)(50 139 61 150)(51 138 62 149)(52 137 63 148)(53 136 64 147)(54 135 65 146)(55 134 66 145)(67 155 78 166)(68 165 79 176)(69 164 80 175)(70 163 81 174)(71 162 82 173)(72 161 83 172)(73 160 84 171)(74 159 85 170)(75 158 86 169)(76 157 87 168)(77 156 88 167)
(1 23)(2 33)(3 32)(4 31)(5 30)(6 29)(7 28)(8 27)(9 26)(10 25)(11 24)(12 34)(13 44)(14 43)(15 42)(16 41)(17 40)(18 39)(19 38)(20 37)(21 36)(22 35)(45 67)(46 77)(47 76)(48 75)(49 74)(50 73)(51 72)(52 71)(53 70)(54 69)(55 68)(56 78)(57 88)(58 87)(59 86)(60 85)(61 84)(62 83)(63 82)(64 81)(65 80)(66 79)(89 122)(90 132)(91 131)(92 130)(93 129)(94 128)(95 127)(96 126)(97 125)(98 124)(99 123)(100 111)(101 121)(102 120)(103 119)(104 118)(105 117)(106 116)(107 115)(108 114)(109 113)(110 112)(133 166)(134 176)(135 175)(136 174)(137 173)(138 172)(139 171)(140 170)(141 169)(142 168)(143 167)(144 155)(145 165)(146 164)(147 163)(148 162)(149 161)(150 160)(151 159)(152 158)(153 157)(154 156)
G:=sub<Sym(176)| (1,67,23,45)(2,68,24,46)(3,69,25,47)(4,70,26,48)(5,71,27,49)(6,72,28,50)(7,73,29,51)(8,74,30,52)(9,75,31,53)(10,76,32,54)(11,77,33,55)(12,78,34,56)(13,79,35,57)(14,80,36,58)(15,81,37,59)(16,82,38,60)(17,83,39,61)(18,84,40,62)(19,85,41,63)(20,86,42,64)(21,87,43,65)(22,88,44,66)(89,155,111,133)(90,156,112,134)(91,157,113,135)(92,158,114,136)(93,159,115,137)(94,160,116,138)(95,161,117,139)(96,162,118,140)(97,163,119,141)(98,164,120,142)(99,165,121,143)(100,166,122,144)(101,167,123,145)(102,168,124,146)(103,169,125,147)(104,170,126,148)(105,171,127,149)(106,172,128,150)(107,173,129,151)(108,174,130,152)(109,175,131,153)(110,176,132,154), (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), (1,89,12,100)(2,99,13,110)(3,98,14,109)(4,97,15,108)(5,96,16,107)(6,95,17,106)(7,94,18,105)(8,93,19,104)(9,92,20,103)(10,91,21,102)(11,90,22,101)(23,111,34,122)(24,121,35,132)(25,120,36,131)(26,119,37,130)(27,118,38,129)(28,117,39,128)(29,116,40,127)(30,115,41,126)(31,114,42,125)(32,113,43,124)(33,112,44,123)(45,133,56,144)(46,143,57,154)(47,142,58,153)(48,141,59,152)(49,140,60,151)(50,139,61,150)(51,138,62,149)(52,137,63,148)(53,136,64,147)(54,135,65,146)(55,134,66,145)(67,155,78,166)(68,165,79,176)(69,164,80,175)(70,163,81,174)(71,162,82,173)(72,161,83,172)(73,160,84,171)(74,159,85,170)(75,158,86,169)(76,157,87,168)(77,156,88,167), (1,23)(2,33)(3,32)(4,31)(5,30)(6,29)(7,28)(8,27)(9,26)(10,25)(11,24)(12,34)(13,44)(14,43)(15,42)(16,41)(17,40)(18,39)(19,38)(20,37)(21,36)(22,35)(45,67)(46,77)(47,76)(48,75)(49,74)(50,73)(51,72)(52,71)(53,70)(54,69)(55,68)(56,78)(57,88)(58,87)(59,86)(60,85)(61,84)(62,83)(63,82)(64,81)(65,80)(66,79)(89,122)(90,132)(91,131)(92,130)(93,129)(94,128)(95,127)(96,126)(97,125)(98,124)(99,123)(100,111)(101,121)(102,120)(103,119)(104,118)(105,117)(106,116)(107,115)(108,114)(109,113)(110,112)(133,166)(134,176)(135,175)(136,174)(137,173)(138,172)(139,171)(140,170)(141,169)(142,168)(143,167)(144,155)(145,165)(146,164)(147,163)(148,162)(149,161)(150,160)(151,159)(152,158)(153,157)(154,156)>;
G:=Group( (1,67,23,45)(2,68,24,46)(3,69,25,47)(4,70,26,48)(5,71,27,49)(6,72,28,50)(7,73,29,51)(8,74,30,52)(9,75,31,53)(10,76,32,54)(11,77,33,55)(12,78,34,56)(13,79,35,57)(14,80,36,58)(15,81,37,59)(16,82,38,60)(17,83,39,61)(18,84,40,62)(19,85,41,63)(20,86,42,64)(21,87,43,65)(22,88,44,66)(89,155,111,133)(90,156,112,134)(91,157,113,135)(92,158,114,136)(93,159,115,137)(94,160,116,138)(95,161,117,139)(96,162,118,140)(97,163,119,141)(98,164,120,142)(99,165,121,143)(100,166,122,144)(101,167,123,145)(102,168,124,146)(103,169,125,147)(104,170,126,148)(105,171,127,149)(106,172,128,150)(107,173,129,151)(108,174,130,152)(109,175,131,153)(110,176,132,154), (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), (1,89,12,100)(2,99,13,110)(3,98,14,109)(4,97,15,108)(5,96,16,107)(6,95,17,106)(7,94,18,105)(8,93,19,104)(9,92,20,103)(10,91,21,102)(11,90,22,101)(23,111,34,122)(24,121,35,132)(25,120,36,131)(26,119,37,130)(27,118,38,129)(28,117,39,128)(29,116,40,127)(30,115,41,126)(31,114,42,125)(32,113,43,124)(33,112,44,123)(45,133,56,144)(46,143,57,154)(47,142,58,153)(48,141,59,152)(49,140,60,151)(50,139,61,150)(51,138,62,149)(52,137,63,148)(53,136,64,147)(54,135,65,146)(55,134,66,145)(67,155,78,166)(68,165,79,176)(69,164,80,175)(70,163,81,174)(71,162,82,173)(72,161,83,172)(73,160,84,171)(74,159,85,170)(75,158,86,169)(76,157,87,168)(77,156,88,167), (1,23)(2,33)(3,32)(4,31)(5,30)(6,29)(7,28)(8,27)(9,26)(10,25)(11,24)(12,34)(13,44)(14,43)(15,42)(16,41)(17,40)(18,39)(19,38)(20,37)(21,36)(22,35)(45,67)(46,77)(47,76)(48,75)(49,74)(50,73)(51,72)(52,71)(53,70)(54,69)(55,68)(56,78)(57,88)(58,87)(59,86)(60,85)(61,84)(62,83)(63,82)(64,81)(65,80)(66,79)(89,122)(90,132)(91,131)(92,130)(93,129)(94,128)(95,127)(96,126)(97,125)(98,124)(99,123)(100,111)(101,121)(102,120)(103,119)(104,118)(105,117)(106,116)(107,115)(108,114)(109,113)(110,112)(133,166)(134,176)(135,175)(136,174)(137,173)(138,172)(139,171)(140,170)(141,169)(142,168)(143,167)(144,155)(145,165)(146,164)(147,163)(148,162)(149,161)(150,160)(151,159)(152,158)(153,157)(154,156) );
G=PermutationGroup([[(1,67,23,45),(2,68,24,46),(3,69,25,47),(4,70,26,48),(5,71,27,49),(6,72,28,50),(7,73,29,51),(8,74,30,52),(9,75,31,53),(10,76,32,54),(11,77,33,55),(12,78,34,56),(13,79,35,57),(14,80,36,58),(15,81,37,59),(16,82,38,60),(17,83,39,61),(18,84,40,62),(19,85,41,63),(20,86,42,64),(21,87,43,65),(22,88,44,66),(89,155,111,133),(90,156,112,134),(91,157,113,135),(92,158,114,136),(93,159,115,137),(94,160,116,138),(95,161,117,139),(96,162,118,140),(97,163,119,141),(98,164,120,142),(99,165,121,143),(100,166,122,144),(101,167,123,145),(102,168,124,146),(103,169,125,147),(104,170,126,148),(105,171,127,149),(106,172,128,150),(107,173,129,151),(108,174,130,152),(109,175,131,153),(110,176,132,154)], [(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)], [(1,89,12,100),(2,99,13,110),(3,98,14,109),(4,97,15,108),(5,96,16,107),(6,95,17,106),(7,94,18,105),(8,93,19,104),(9,92,20,103),(10,91,21,102),(11,90,22,101),(23,111,34,122),(24,121,35,132),(25,120,36,131),(26,119,37,130),(27,118,38,129),(28,117,39,128),(29,116,40,127),(30,115,41,126),(31,114,42,125),(32,113,43,124),(33,112,44,123),(45,133,56,144),(46,143,57,154),(47,142,58,153),(48,141,59,152),(49,140,60,151),(50,139,61,150),(51,138,62,149),(52,137,63,148),(53,136,64,147),(54,135,65,146),(55,134,66,145),(67,155,78,166),(68,165,79,176),(69,164,80,175),(70,163,81,174),(71,162,82,173),(72,161,83,172),(73,160,84,171),(74,159,85,170),(75,158,86,169),(76,157,87,168),(77,156,88,167)], [(1,23),(2,33),(3,32),(4,31),(5,30),(6,29),(7,28),(8,27),(9,26),(10,25),(11,24),(12,34),(13,44),(14,43),(15,42),(16,41),(17,40),(18,39),(19,38),(20,37),(21,36),(22,35),(45,67),(46,77),(47,76),(48,75),(49,74),(50,73),(51,72),(52,71),(53,70),(54,69),(55,68),(56,78),(57,88),(58,87),(59,86),(60,85),(61,84),(62,83),(63,82),(64,81),(65,80),(66,79),(89,122),(90,132),(91,131),(92,130),(93,129),(94,128),(95,127),(96,126),(97,125),(98,124),(99,123),(100,111),(101,121),(102,120),(103,119),(104,118),(105,117),(106,116),(107,115),(108,114),(109,113),(110,112),(133,166),(134,176),(135,175),(136,174),(137,173),(138,172),(139,171),(140,170),(141,169),(142,168),(143,167),(144,155),(145,165),(146,164),(147,163),(148,162),(149,161),(150,160),(151,159),(152,158),(153,157),(154,156)]])
100 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | ··· | 4L | 11A | ··· | 11E | 22A | ··· | 22AI | 44A | ··· | 44AN |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 11 | ··· | 11 | 22 | ··· | 22 | 44 | ··· | 44 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 22 | 22 | 1 | 1 | 1 | 1 | 2 | 2 | 22 | ··· | 22 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
100 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | C4○D4 | D11 | D22 | D22 | C11⋊D4 | C4×D11 | D44⋊5C2 |
| kernel | C4×C11⋊D4 | C4×Dic11 | Dic11⋊C4 | D22⋊C4 | C23.D11 | C2×C4×D11 | C2×C11⋊D4 | C22×C44 | C11⋊D4 | C44 | C22 | C22×C4 | C2×C4 | C23 | C4 | C22 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 8 | 2 | 2 | 5 | 10 | 5 | 20 | 20 | 20 |
Matrix representation of C4×C11⋊D4 ►in GL4(𝔽89) generated by
| 55 | 0 | 0 | 0 |
| 0 | 55 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 88 | 0 | 0 |
| 1 | 47 | 0 | 0 |
| 0 | 0 | 29 | 88 |
| 0 | 0 | 29 | 42 |
| 88 | 42 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 3 | 74 |
| 0 | 0 | 60 | 86 |
| 88 | 42 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 11 | 18 |
| 0 | 0 | 23 | 78 |
G:=sub<GL(4,GF(89))| [55,0,0,0,0,55,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,88,47,0,0,0,0,29,29,0,0,88,42],[88,0,0,0,42,1,0,0,0,0,3,60,0,0,74,86],[88,0,0,0,42,1,0,0,0,0,11,23,0,0,18,78] >;
C4×C11⋊D4 in GAP, Magma, Sage, TeX
C_4\times C_{11}\rtimes D_4 % in TeX
G:=Group("C4xC11:D4"); // GroupNames label
G:=SmallGroup(352,123);
// by ID
G=gap.SmallGroup(352,123);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-11,217,50,11525]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^11=c^4=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations