direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C11×C8○D4, D4.C44, Q8.C44, M4(2)⋊5C22, C44.54C23, C88.30C22, (C2×C8)⋊7C22, (C2×C88)⋊15C2, C4.5(C2×C44), C8.7(C2×C22), C44.32(C2×C4), C4○D4.3C22, (D4×C11).2C4, (Q8×C11).2C4, C22.1(C2×C44), C2.7(C22×C44), C4.12(C22×C22), C22.35(C22×C4), (C11×M4(2))⋊11C2, (C2×C44).128C22, (C2×C22).8(C2×C4), (C2×C4).24(C2×C22), (C11×C4○D4).6C2, SmallGroup(352,166)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C11×C8○D4
G = < a,b,c,d | a11=b8=d2=1, c2=b4, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=b4c >
Subgroups: 68 in 62 conjugacy classes, 56 normal (14 characteristic)
C1, C2, C2, C4, C4, C22, C8, C8, C2×C4, D4, Q8, C11, C2×C8, M4(2), C4○D4, C22, C22, C8○D4, C44, C44, C2×C22, C88, C88, C2×C44, D4×C11, Q8×C11, C2×C88, C11×M4(2), C11×C4○D4, C11×C8○D4
Quotients: C1, C2, C4, C22, C2×C4, C23, C11, C22×C4, C22, C8○D4, C44, C2×C22, C2×C44, C22×C22, C22×C44, C11×C8○D4
►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)
(1 159 71 149 61 123 47 133)(2 160 72 150 62 124 48 134)(3 161 73 151 63 125 49 135)(4 162 74 152 64 126 50 136)(5 163 75 153 65 127 51 137)(6 164 76 154 66 128 52 138)(7 165 77 144 56 129 53 139)(8 155 67 145 57 130 54 140)(9 156 68 146 58 131 55 141)(10 157 69 147 59 132 45 142)(11 158 70 148 60 122 46 143)(12 110 172 84 31 94 34 120)(13 100 173 85 32 95 35 121)(14 101 174 86 33 96 36 111)(15 102 175 87 23 97 37 112)(16 103 176 88 24 98 38 113)(17 104 166 78 25 99 39 114)(18 105 167 79 26 89 40 115)(19 106 168 80 27 90 41 116)(20 107 169 81 28 91 42 117)(21 108 170 82 29 92 43 118)(22 109 171 83 30 93 44 119)
(1 105 61 89)(2 106 62 90)(3 107 63 91)(4 108 64 92)(5 109 65 93)(6 110 66 94)(7 100 56 95)(8 101 57 96)(9 102 58 97)(10 103 59 98)(11 104 60 99)(12 154 31 138)(13 144 32 139)(14 145 33 140)(15 146 23 141)(16 147 24 142)(17 148 25 143)(18 149 26 133)(19 150 27 134)(20 151 28 135)(21 152 29 136)(22 153 30 137)(34 164 172 128)(35 165 173 129)(36 155 174 130)(37 156 175 131)(38 157 176 132)(39 158 166 122)(40 159 167 123)(41 160 168 124)(42 161 169 125)(43 162 170 126)(44 163 171 127)(45 113 69 88)(46 114 70 78)(47 115 71 79)(48 116 72 80)(49 117 73 81)(50 118 74 82)(51 119 75 83)(52 120 76 84)(53 121 77 85)(54 111 67 86)(55 112 68 87)
(1 89)(2 90)(3 91)(4 92)(5 93)(6 94)(7 95)(8 96)(9 97)(10 98)(11 99)(12 154)(13 144)(14 145)(15 146)(16 147)(17 148)(18 149)(19 150)(20 151)(21 152)(22 153)(23 141)(24 142)(25 143)(26 133)(27 134)(28 135)(29 136)(30 137)(31 138)(32 139)(33 140)(34 164)(35 165)(36 155)(37 156)(38 157)(39 158)(40 159)(41 160)(42 161)(43 162)(44 163)(45 88)(46 78)(47 79)(48 80)(49 81)(50 82)(51 83)(52 84)(53 85)(54 86)(55 87)(56 100)(57 101)(58 102)(59 103)(60 104)(61 105)(62 106)(63 107)(64 108)(65 109)(66 110)(67 111)(68 112)(69 113)(70 114)(71 115)(72 116)(73 117)(74 118)(75 119)(76 120)(77 121)(122 166)(123 167)(124 168)(125 169)(126 170)(127 171)(128 172)(129 173)(130 174)(131 175)(132 176)
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), (1,159,71,149,61,123,47,133)(2,160,72,150,62,124,48,134)(3,161,73,151,63,125,49,135)(4,162,74,152,64,126,50,136)(5,163,75,153,65,127,51,137)(6,164,76,154,66,128,52,138)(7,165,77,144,56,129,53,139)(8,155,67,145,57,130,54,140)(9,156,68,146,58,131,55,141)(10,157,69,147,59,132,45,142)(11,158,70,148,60,122,46,143)(12,110,172,84,31,94,34,120)(13,100,173,85,32,95,35,121)(14,101,174,86,33,96,36,111)(15,102,175,87,23,97,37,112)(16,103,176,88,24,98,38,113)(17,104,166,78,25,99,39,114)(18,105,167,79,26,89,40,115)(19,106,168,80,27,90,41,116)(20,107,169,81,28,91,42,117)(21,108,170,82,29,92,43,118)(22,109,171,83,30,93,44,119), (1,105,61,89)(2,106,62,90)(3,107,63,91)(4,108,64,92)(5,109,65,93)(6,110,66,94)(7,100,56,95)(8,101,57,96)(9,102,58,97)(10,103,59,98)(11,104,60,99)(12,154,31,138)(13,144,32,139)(14,145,33,140)(15,146,23,141)(16,147,24,142)(17,148,25,143)(18,149,26,133)(19,150,27,134)(20,151,28,135)(21,152,29,136)(22,153,30,137)(34,164,172,128)(35,165,173,129)(36,155,174,130)(37,156,175,131)(38,157,176,132)(39,158,166,122)(40,159,167,123)(41,160,168,124)(42,161,169,125)(43,162,170,126)(44,163,171,127)(45,113,69,88)(46,114,70,78)(47,115,71,79)(48,116,72,80)(49,117,73,81)(50,118,74,82)(51,119,75,83)(52,120,76,84)(53,121,77,85)(54,111,67,86)(55,112,68,87), (1,89)(2,90)(3,91)(4,92)(5,93)(6,94)(7,95)(8,96)(9,97)(10,98)(11,99)(12,154)(13,144)(14,145)(15,146)(16,147)(17,148)(18,149)(19,150)(20,151)(21,152)(22,153)(23,141)(24,142)(25,143)(26,133)(27,134)(28,135)(29,136)(30,137)(31,138)(32,139)(33,140)(34,164)(35,165)(36,155)(37,156)(38,157)(39,158)(40,159)(41,160)(42,161)(43,162)(44,163)(45,88)(46,78)(47,79)(48,80)(49,81)(50,82)(51,83)(52,84)(53,85)(54,86)(55,87)(56,100)(57,101)(58,102)(59,103)(60,104)(61,105)(62,106)(63,107)(64,108)(65,109)(66,110)(67,111)(68,112)(69,113)(70,114)(71,115)(72,116)(73,117)(74,118)(75,119)(76,120)(77,121)(122,166)(123,167)(124,168)(125,169)(126,170)(127,171)(128,172)(129,173)(130,174)(131,175)(132,176)>;
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), (1,159,71,149,61,123,47,133)(2,160,72,150,62,124,48,134)(3,161,73,151,63,125,49,135)(4,162,74,152,64,126,50,136)(5,163,75,153,65,127,51,137)(6,164,76,154,66,128,52,138)(7,165,77,144,56,129,53,139)(8,155,67,145,57,130,54,140)(9,156,68,146,58,131,55,141)(10,157,69,147,59,132,45,142)(11,158,70,148,60,122,46,143)(12,110,172,84,31,94,34,120)(13,100,173,85,32,95,35,121)(14,101,174,86,33,96,36,111)(15,102,175,87,23,97,37,112)(16,103,176,88,24,98,38,113)(17,104,166,78,25,99,39,114)(18,105,167,79,26,89,40,115)(19,106,168,80,27,90,41,116)(20,107,169,81,28,91,42,117)(21,108,170,82,29,92,43,118)(22,109,171,83,30,93,44,119), (1,105,61,89)(2,106,62,90)(3,107,63,91)(4,108,64,92)(5,109,65,93)(6,110,66,94)(7,100,56,95)(8,101,57,96)(9,102,58,97)(10,103,59,98)(11,104,60,99)(12,154,31,138)(13,144,32,139)(14,145,33,140)(15,146,23,141)(16,147,24,142)(17,148,25,143)(18,149,26,133)(19,150,27,134)(20,151,28,135)(21,152,29,136)(22,153,30,137)(34,164,172,128)(35,165,173,129)(36,155,174,130)(37,156,175,131)(38,157,176,132)(39,158,166,122)(40,159,167,123)(41,160,168,124)(42,161,169,125)(43,162,170,126)(44,163,171,127)(45,113,69,88)(46,114,70,78)(47,115,71,79)(48,116,72,80)(49,117,73,81)(50,118,74,82)(51,119,75,83)(52,120,76,84)(53,121,77,85)(54,111,67,86)(55,112,68,87), (1,89)(2,90)(3,91)(4,92)(5,93)(6,94)(7,95)(8,96)(9,97)(10,98)(11,99)(12,154)(13,144)(14,145)(15,146)(16,147)(17,148)(18,149)(19,150)(20,151)(21,152)(22,153)(23,141)(24,142)(25,143)(26,133)(27,134)(28,135)(29,136)(30,137)(31,138)(32,139)(33,140)(34,164)(35,165)(36,155)(37,156)(38,157)(39,158)(40,159)(41,160)(42,161)(43,162)(44,163)(45,88)(46,78)(47,79)(48,80)(49,81)(50,82)(51,83)(52,84)(53,85)(54,86)(55,87)(56,100)(57,101)(58,102)(59,103)(60,104)(61,105)(62,106)(63,107)(64,108)(65,109)(66,110)(67,111)(68,112)(69,113)(70,114)(71,115)(72,116)(73,117)(74,118)(75,119)(76,120)(77,121)(122,166)(123,167)(124,168)(125,169)(126,170)(127,171)(128,172)(129,173)(130,174)(131,175)(132,176) );
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)], [(1,159,71,149,61,123,47,133),(2,160,72,150,62,124,48,134),(3,161,73,151,63,125,49,135),(4,162,74,152,64,126,50,136),(5,163,75,153,65,127,51,137),(6,164,76,154,66,128,52,138),(7,165,77,144,56,129,53,139),(8,155,67,145,57,130,54,140),(9,156,68,146,58,131,55,141),(10,157,69,147,59,132,45,142),(11,158,70,148,60,122,46,143),(12,110,172,84,31,94,34,120),(13,100,173,85,32,95,35,121),(14,101,174,86,33,96,36,111),(15,102,175,87,23,97,37,112),(16,103,176,88,24,98,38,113),(17,104,166,78,25,99,39,114),(18,105,167,79,26,89,40,115),(19,106,168,80,27,90,41,116),(20,107,169,81,28,91,42,117),(21,108,170,82,29,92,43,118),(22,109,171,83,30,93,44,119)], [(1,105,61,89),(2,106,62,90),(3,107,63,91),(4,108,64,92),(5,109,65,93),(6,110,66,94),(7,100,56,95),(8,101,57,96),(9,102,58,97),(10,103,59,98),(11,104,60,99),(12,154,31,138),(13,144,32,139),(14,145,33,140),(15,146,23,141),(16,147,24,142),(17,148,25,143),(18,149,26,133),(19,150,27,134),(20,151,28,135),(21,152,29,136),(22,153,30,137),(34,164,172,128),(35,165,173,129),(36,155,174,130),(37,156,175,131),(38,157,176,132),(39,158,166,122),(40,159,167,123),(41,160,168,124),(42,161,169,125),(43,162,170,126),(44,163,171,127),(45,113,69,88),(46,114,70,78),(47,115,71,79),(48,116,72,80),(49,117,73,81),(50,118,74,82),(51,119,75,83),(52,120,76,84),(53,121,77,85),(54,111,67,86),(55,112,68,87)], [(1,89),(2,90),(3,91),(4,92),(5,93),(6,94),(7,95),(8,96),(9,97),(10,98),(11,99),(12,154),(13,144),(14,145),(15,146),(16,147),(17,148),(18,149),(19,150),(20,151),(21,152),(22,153),(23,141),(24,142),(25,143),(26,133),(27,134),(28,135),(29,136),(30,137),(31,138),(32,139),(33,140),(34,164),(35,165),(36,155),(37,156),(38,157),(39,158),(40,159),(41,160),(42,161),(43,162),(44,163),(45,88),(46,78),(47,79),(48,80),(49,81),(50,82),(51,83),(52,84),(53,85),(54,86),(55,87),(56,100),(57,101),(58,102),(59,103),(60,104),(61,105),(62,106),(63,107),(64,108),(65,109),(66,110),(67,111),(68,112),(69,113),(70,114),(71,115),(72,116),(73,117),(74,118),(75,119),(76,120),(77,121),(122,166),(123,167),(124,168),(125,169),(126,170),(127,171),(128,172),(129,173),(130,174),(131,175),(132,176)]])
220 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 8A | 8B | 8C | 8D | 8E | ··· | 8J | 11A | ··· | 11J | 22A | ··· | 22J | 22K | ··· | 22AN | 44A | ··· | 44T | 44U | ··· | 44AX | 88A | ··· | 88AN | 88AO | ··· | 88CV |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | ··· | 8 | 11 | ··· | 11 | 22 | ··· | 22 | 22 | ··· | 22 | 44 | ··· | 44 | 44 | ··· | 44 | 88 | ··· | 88 | 88 | ··· | 88 |
| size | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 |
220 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | + | ||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C11 | C22 | C22 | C22 | C44 | C44 | C8○D4 | C11×C8○D4 |
| kernel | C11×C8○D4 | C2×C88 | C11×M4(2) | C11×C4○D4 | D4×C11 | Q8×C11 | C8○D4 | C2×C8 | M4(2) | C4○D4 | D4 | Q8 | C11 | C1 |
| # reps | 1 | 3 | 3 | 1 | 6 | 2 | 10 | 30 | 30 | 10 | 60 | 20 | 4 | 40 |
Matrix representation of C11×C8○D4 ►in GL2(𝔽89) generated by
| 8 | 0 |
| 0 | 8 |
| 52 | 0 |
| 0 | 52 |
| 0 | 1 |
| 88 | 0 |
| 0 | 1 |
| 1 | 0 |
G:=sub<GL(2,GF(89))| [8,0,0,8],[52,0,0,52],[0,88,1,0],[0,1,1,0] >;
C11×C8○D4 in GAP, Magma, Sage, TeX
C_{11}\times C_8\circ D_4 % in TeX
G:=Group("C11xC8oD4"); // GroupNames label
G:=SmallGroup(352,166);
// by ID
G=gap.SmallGroup(352,166);
# by ID
G:=PCGroup([6,-2,-2,-2,-11,-2,-2,528,1628,88]);
// Polycyclic
G:=Group<a,b,c,d|a^11=b^8=d^2=1,c^2=b^4,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=b^4*c>;
// generators/relations