direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C11×C4.4D4, C42⋊5C22, C44.39D4, (C4×C44)⋊12C2, (Q8×C22)⋊9C2, (C2×Q8)⋊2C22, C2.8(D4×C22), C4.4(D4×C11), C22⋊C4⋊5C22, (C2×D4).5C22, C22.71(C2×D4), (D4×C22).12C2, C23.2(C2×C22), C22.44(C4○D4), (C2×C22).79C23, (C2×C44).66C22, (C22×C22).2C22, C22.14(C22×C22), (C2×C4).6(C2×C22), C2.7(C11×C4○D4), (C11×C22⋊C4)⋊13C2, SmallGroup(352,159)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C11×C4.4D4
G = < a,b,c,d | a11=b4=c4=1, d2=b2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=b2c-1 >
Subgroups: 116 in 76 conjugacy classes, 44 normal (16 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C2×C4, C2×C4, D4, Q8, C23, C11, C42, C22⋊C4, C2×D4, C2×Q8, C22, C22, C22, C4.4D4, C44, C44, C2×C22, C2×C22, C2×C44, C2×C44, D4×C11, Q8×C11, C22×C22, C4×C44, C11×C22⋊C4, D4×C22, Q8×C22, C11×C4.4D4
Quotients: C1, C2, C22, D4, C23, C11, C2×D4, C4○D4, C22, C4.4D4, C2×C22, D4×C11, C22×C22, D4×C22, C11×C4○D4, C11×C4.4D4
►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 40 59 14)(2 41 60 15)(3 42 61 16)(4 43 62 17)(5 44 63 18)(6 34 64 19)(7 35 65 20)(8 36 66 21)(9 37 56 22)(10 38 57 12)(11 39 58 13)(23 77 174 45)(24 67 175 46)(25 68 176 47)(26 69 166 48)(27 70 167 49)(28 71 168 50)(29 72 169 51)(30 73 170 52)(31 74 171 53)(32 75 172 54)(33 76 173 55)(78 146 113 136)(79 147 114 137)(80 148 115 138)(81 149 116 139)(82 150 117 140)(83 151 118 141)(84 152 119 142)(85 153 120 143)(86 154 121 133)(87 144 111 134)(88 145 112 135)(89 165 110 130)(90 155 100 131)(91 156 101 132)(92 157 102 122)(93 158 103 123)(94 159 104 124)(95 160 105 125)(96 161 106 126)(97 162 107 127)(98 163 108 128)(99 164 109 129)
(1 137 70 158)(2 138 71 159)(3 139 72 160)(4 140 73 161)(5 141 74 162)(6 142 75 163)(7 143 76 164)(8 133 77 165)(9 134 67 155)(10 135 68 156)(11 136 69 157)(12 112 25 91)(13 113 26 92)(14 114 27 93)(15 115 28 94)(16 116 29 95)(17 117 30 96)(18 118 31 97)(19 119 32 98)(20 120 33 99)(21 121 23 89)(22 111 24 90)(34 84 172 108)(35 85 173 109)(36 86 174 110)(37 87 175 100)(38 88 176 101)(39 78 166 102)(40 79 167 103)(41 80 168 104)(42 81 169 105)(43 82 170 106)(44 83 171 107)(45 130 66 154)(46 131 56 144)(47 132 57 145)(48 122 58 146)(49 123 59 147)(50 124 60 148)(51 125 61 149)(52 126 62 150)(53 127 63 151)(54 128 64 152)(55 129 65 153)
(1 103 59 93)(2 104 60 94)(3 105 61 95)(4 106 62 96)(5 107 63 97)(6 108 64 98)(7 109 65 99)(8 110 66 89)(9 100 56 90)(10 101 57 91)(11 102 58 92)(12 132 38 156)(13 122 39 157)(14 123 40 158)(15 124 41 159)(16 125 42 160)(17 126 43 161)(18 127 44 162)(19 128 34 163)(20 129 35 164)(21 130 36 165)(22 131 37 155)(23 154 174 133)(24 144 175 134)(25 145 176 135)(26 146 166 136)(27 147 167 137)(28 148 168 138)(29 149 169 139)(30 150 170 140)(31 151 171 141)(32 152 172 142)(33 153 173 143)(45 121 77 86)(46 111 67 87)(47 112 68 88)(48 113 69 78)(49 114 70 79)(50 115 71 80)(51 116 72 81)(52 117 73 82)(53 118 74 83)(54 119 75 84)(55 120 76 85)
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), (1,40,59,14)(2,41,60,15)(3,42,61,16)(4,43,62,17)(5,44,63,18)(6,34,64,19)(7,35,65,20)(8,36,66,21)(9,37,56,22)(10,38,57,12)(11,39,58,13)(23,77,174,45)(24,67,175,46)(25,68,176,47)(26,69,166,48)(27,70,167,49)(28,71,168,50)(29,72,169,51)(30,73,170,52)(31,74,171,53)(32,75,172,54)(33,76,173,55)(78,146,113,136)(79,147,114,137)(80,148,115,138)(81,149,116,139)(82,150,117,140)(83,151,118,141)(84,152,119,142)(85,153,120,143)(86,154,121,133)(87,144,111,134)(88,145,112,135)(89,165,110,130)(90,155,100,131)(91,156,101,132)(92,157,102,122)(93,158,103,123)(94,159,104,124)(95,160,105,125)(96,161,106,126)(97,162,107,127)(98,163,108,128)(99,164,109,129), (1,137,70,158)(2,138,71,159)(3,139,72,160)(4,140,73,161)(5,141,74,162)(6,142,75,163)(7,143,76,164)(8,133,77,165)(9,134,67,155)(10,135,68,156)(11,136,69,157)(12,112,25,91)(13,113,26,92)(14,114,27,93)(15,115,28,94)(16,116,29,95)(17,117,30,96)(18,118,31,97)(19,119,32,98)(20,120,33,99)(21,121,23,89)(22,111,24,90)(34,84,172,108)(35,85,173,109)(36,86,174,110)(37,87,175,100)(38,88,176,101)(39,78,166,102)(40,79,167,103)(41,80,168,104)(42,81,169,105)(43,82,170,106)(44,83,171,107)(45,130,66,154)(46,131,56,144)(47,132,57,145)(48,122,58,146)(49,123,59,147)(50,124,60,148)(51,125,61,149)(52,126,62,150)(53,127,63,151)(54,128,64,152)(55,129,65,153), (1,103,59,93)(2,104,60,94)(3,105,61,95)(4,106,62,96)(5,107,63,97)(6,108,64,98)(7,109,65,99)(8,110,66,89)(9,100,56,90)(10,101,57,91)(11,102,58,92)(12,132,38,156)(13,122,39,157)(14,123,40,158)(15,124,41,159)(16,125,42,160)(17,126,43,161)(18,127,44,162)(19,128,34,163)(20,129,35,164)(21,130,36,165)(22,131,37,155)(23,154,174,133)(24,144,175,134)(25,145,176,135)(26,146,166,136)(27,147,167,137)(28,148,168,138)(29,149,169,139)(30,150,170,140)(31,151,171,141)(32,152,172,142)(33,153,173,143)(45,121,77,86)(46,111,67,87)(47,112,68,88)(48,113,69,78)(49,114,70,79)(50,115,71,80)(51,116,72,81)(52,117,73,82)(53,118,74,83)(54,119,75,84)(55,120,76,85)>;
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), (1,40,59,14)(2,41,60,15)(3,42,61,16)(4,43,62,17)(5,44,63,18)(6,34,64,19)(7,35,65,20)(8,36,66,21)(9,37,56,22)(10,38,57,12)(11,39,58,13)(23,77,174,45)(24,67,175,46)(25,68,176,47)(26,69,166,48)(27,70,167,49)(28,71,168,50)(29,72,169,51)(30,73,170,52)(31,74,171,53)(32,75,172,54)(33,76,173,55)(78,146,113,136)(79,147,114,137)(80,148,115,138)(81,149,116,139)(82,150,117,140)(83,151,118,141)(84,152,119,142)(85,153,120,143)(86,154,121,133)(87,144,111,134)(88,145,112,135)(89,165,110,130)(90,155,100,131)(91,156,101,132)(92,157,102,122)(93,158,103,123)(94,159,104,124)(95,160,105,125)(96,161,106,126)(97,162,107,127)(98,163,108,128)(99,164,109,129), (1,137,70,158)(2,138,71,159)(3,139,72,160)(4,140,73,161)(5,141,74,162)(6,142,75,163)(7,143,76,164)(8,133,77,165)(9,134,67,155)(10,135,68,156)(11,136,69,157)(12,112,25,91)(13,113,26,92)(14,114,27,93)(15,115,28,94)(16,116,29,95)(17,117,30,96)(18,118,31,97)(19,119,32,98)(20,120,33,99)(21,121,23,89)(22,111,24,90)(34,84,172,108)(35,85,173,109)(36,86,174,110)(37,87,175,100)(38,88,176,101)(39,78,166,102)(40,79,167,103)(41,80,168,104)(42,81,169,105)(43,82,170,106)(44,83,171,107)(45,130,66,154)(46,131,56,144)(47,132,57,145)(48,122,58,146)(49,123,59,147)(50,124,60,148)(51,125,61,149)(52,126,62,150)(53,127,63,151)(54,128,64,152)(55,129,65,153), (1,103,59,93)(2,104,60,94)(3,105,61,95)(4,106,62,96)(5,107,63,97)(6,108,64,98)(7,109,65,99)(8,110,66,89)(9,100,56,90)(10,101,57,91)(11,102,58,92)(12,132,38,156)(13,122,39,157)(14,123,40,158)(15,124,41,159)(16,125,42,160)(17,126,43,161)(18,127,44,162)(19,128,34,163)(20,129,35,164)(21,130,36,165)(22,131,37,155)(23,154,174,133)(24,144,175,134)(25,145,176,135)(26,146,166,136)(27,147,167,137)(28,148,168,138)(29,149,169,139)(30,150,170,140)(31,151,171,141)(32,152,172,142)(33,153,173,143)(45,121,77,86)(46,111,67,87)(47,112,68,88)(48,113,69,78)(49,114,70,79)(50,115,71,80)(51,116,72,81)(52,117,73,82)(53,118,74,83)(54,119,75,84)(55,120,76,85) );
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)], [(1,40,59,14),(2,41,60,15),(3,42,61,16),(4,43,62,17),(5,44,63,18),(6,34,64,19),(7,35,65,20),(8,36,66,21),(9,37,56,22),(10,38,57,12),(11,39,58,13),(23,77,174,45),(24,67,175,46),(25,68,176,47),(26,69,166,48),(27,70,167,49),(28,71,168,50),(29,72,169,51),(30,73,170,52),(31,74,171,53),(32,75,172,54),(33,76,173,55),(78,146,113,136),(79,147,114,137),(80,148,115,138),(81,149,116,139),(82,150,117,140),(83,151,118,141),(84,152,119,142),(85,153,120,143),(86,154,121,133),(87,144,111,134),(88,145,112,135),(89,165,110,130),(90,155,100,131),(91,156,101,132),(92,157,102,122),(93,158,103,123),(94,159,104,124),(95,160,105,125),(96,161,106,126),(97,162,107,127),(98,163,108,128),(99,164,109,129)], [(1,137,70,158),(2,138,71,159),(3,139,72,160),(4,140,73,161),(5,141,74,162),(6,142,75,163),(7,143,76,164),(8,133,77,165),(9,134,67,155),(10,135,68,156),(11,136,69,157),(12,112,25,91),(13,113,26,92),(14,114,27,93),(15,115,28,94),(16,116,29,95),(17,117,30,96),(18,118,31,97),(19,119,32,98),(20,120,33,99),(21,121,23,89),(22,111,24,90),(34,84,172,108),(35,85,173,109),(36,86,174,110),(37,87,175,100),(38,88,176,101),(39,78,166,102),(40,79,167,103),(41,80,168,104),(42,81,169,105),(43,82,170,106),(44,83,171,107),(45,130,66,154),(46,131,56,144),(47,132,57,145),(48,122,58,146),(49,123,59,147),(50,124,60,148),(51,125,61,149),(52,126,62,150),(53,127,63,151),(54,128,64,152),(55,129,65,153)], [(1,103,59,93),(2,104,60,94),(3,105,61,95),(4,106,62,96),(5,107,63,97),(6,108,64,98),(7,109,65,99),(8,110,66,89),(9,100,56,90),(10,101,57,91),(11,102,58,92),(12,132,38,156),(13,122,39,157),(14,123,40,158),(15,124,41,159),(16,125,42,160),(17,126,43,161),(18,127,44,162),(19,128,34,163),(20,129,35,164),(21,130,36,165),(22,131,37,155),(23,154,174,133),(24,144,175,134),(25,145,176,135),(26,146,166,136),(27,147,167,137),(28,148,168,138),(29,149,169,139),(30,150,170,140),(31,151,171,141),(32,152,172,142),(33,153,173,143),(45,121,77,86),(46,111,67,87),(47,112,68,88),(48,113,69,78),(49,114,70,79),(50,115,71,80),(51,116,72,81),(52,117,73,82),(53,118,74,83),(54,119,75,84),(55,120,76,85)]])
154 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | ··· | 4F | 4G | 4H | 11A | ··· | 11J | 22A | ··· | 22AD | 22AE | ··· | 22AX | 44A | ··· | 44BH | 44BI | ··· | 44CB |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 11 | ··· | 11 | 22 | ··· | 22 | 22 | ··· | 22 | 44 | ··· | 44 | 44 | ··· | 44 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 2 | ··· | 2 | 4 | 4 | 1 | ··· | 1 | 1 | ··· | 1 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
154 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | ||||||||
| image | C1 | C2 | C2 | C2 | C2 | C11 | C22 | C22 | C22 | C22 | D4 | C4○D4 | D4×C11 | C11×C4○D4 |
| kernel | C11×C4.4D4 | C4×C44 | C11×C22⋊C4 | D4×C22 | Q8×C22 | C4.4D4 | C42 | C22⋊C4 | C2×D4 | C2×Q8 | C44 | C22 | C4 | C2 |
| # reps | 1 | 1 | 4 | 1 | 1 | 10 | 10 | 40 | 10 | 10 | 2 | 4 | 20 | 40 |
Matrix representation of C11×C4.4D4 ►in GL4(𝔽89) generated by
| 64 | 0 | 0 | 0 |
| 0 | 64 | 0 | 0 |
| 0 | 0 | 45 | 0 |
| 0 | 0 | 0 | 45 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 66 | 68 |
| 0 | 0 | 21 | 23 |
| 34 | 87 | 0 | 0 |
| 0 | 55 | 0 | 0 |
| 0 | 0 | 19 | 87 |
| 0 | 0 | 2 | 70 |
| 34 | 87 | 0 | 0 |
| 88 | 55 | 0 | 0 |
| 0 | 0 | 55 | 0 |
| 0 | 0 | 66 | 34 |
G:=sub<GL(4,GF(89))| [64,0,0,0,0,64,0,0,0,0,45,0,0,0,0,45],[1,0,0,0,0,1,0,0,0,0,66,21,0,0,68,23],[34,0,0,0,87,55,0,0,0,0,19,2,0,0,87,70],[34,88,0,0,87,55,0,0,0,0,55,66,0,0,0,34] >;
C11×C4.4D4 in GAP, Magma, Sage, TeX
C_{11}\times C_4._4D_4 % in TeX
G:=Group("C11xC4.4D4"); // GroupNames label
G:=SmallGroup(352,159);
// by ID
G=gap.SmallGroup(352,159);
# by ID
G:=PCGroup([6,-2,-2,-2,-11,-2,-2,1081,1063,3242,410]);
// Polycyclic
G:=Group<a,b,c,d|a^11=b^4=c^4=1,d^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=b^2*c^-1>;
// generators/relations