direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C11×C22⋊Q8, C44.62D4, C4⋊C4⋊3C22, (C2×C22)⋊2Q8, C22⋊(Q8×C11), (C2×Q8)⋊1C22, (Q8×C22)⋊8C2, C2.6(D4×C22), C2.3(Q8×C22), C22.69(C2×D4), C4.13(D4×C11), C22.20(C2×Q8), C22⋊C4.1C22, (C22×C4).5C22, C23.9(C2×C22), C22.42(C4○D4), (C22×C44).15C2, (C2×C22).77C23, (C2×C44).124C22, (C22×C22).28C22, C22.12(C22×C22), (C11×C4⋊C4)⋊12C2, (C2×C4).4(C2×C22), C2.5(C11×C4○D4), (C11×C22⋊C4).4C2, SmallGroup(352,157)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C11×C22⋊Q8
G = < a,b,c,d,e | a11=b2=c2=d4=1, e2=d2, ab=ba, ac=ca, ad=da, ae=ea, ebe-1=bc=cb, bd=db, cd=dc, ce=ec, ede-1=d-1 >
Subgroups: 100 in 74 conjugacy classes, 48 normal (24 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C2×C4, C2×C4, C2×C4, Q8, C23, C11, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C2×Q8, C22, C22, C22⋊Q8, C44, C44, C2×C22, C2×C22, C2×C22, C2×C44, C2×C44, C2×C44, Q8×C11, C22×C22, C11×C22⋊C4, C11×C4⋊C4, C11×C4⋊C4, C22×C44, Q8×C22, C11×C22⋊Q8
Quotients: C1, C2, C22, D4, Q8, C23, C11, C2×D4, C2×Q8, C4○D4, C22, C22⋊Q8, C2×C22, D4×C11, Q8×C11, C22×C22, D4×C22, Q8×C22, C11×C4○D4, C11×C22⋊Q8
►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)
(12 30)(13 31)(14 32)(15 33)(16 23)(17 24)(18 25)(19 26)(20 27)(21 28)(22 29)(34 172)(35 173)(36 174)(37 175)(38 176)(39 166)(40 167)(41 168)(42 169)(43 170)(44 171)(122 165)(123 155)(124 156)(125 157)(126 158)(127 159)(128 160)(129 161)(130 162)(131 163)(132 164)(133 152)(134 153)(135 154)(136 144)(137 145)(138 146)(139 147)(140 148)(141 149)(142 150)(143 151)
(1 63)(2 64)(3 65)(4 66)(5 56)(6 57)(7 58)(8 59)(9 60)(10 61)(11 62)(12 30)(13 31)(14 32)(15 33)(16 23)(17 24)(18 25)(19 26)(20 27)(21 28)(22 29)(34 172)(35 173)(36 174)(37 175)(38 176)(39 166)(40 167)(41 168)(42 169)(43 170)(44 171)(45 76)(46 77)(47 67)(48 68)(49 69)(50 70)(51 71)(52 72)(53 73)(54 74)(55 75)(78 121)(79 111)(80 112)(81 113)(82 114)(83 115)(84 116)(85 117)(86 118)(87 119)(88 120)(89 108)(90 109)(91 110)(92 100)(93 101)(94 102)(95 103)(96 104)(97 105)(98 106)(99 107)(122 165)(123 155)(124 156)(125 157)(126 158)(127 159)(128 160)(129 161)(130 162)(131 163)(132 164)(133 152)(134 153)(135 154)(136 144)(137 145)(138 146)(139 147)(140 148)(141 149)(142 150)(143 151)
(1 79 47 105)(2 80 48 106)(3 81 49 107)(4 82 50 108)(5 83 51 109)(6 84 52 110)(7 85 53 100)(8 86 54 101)(9 87 55 102)(10 88 45 103)(11 78 46 104)(12 163 37 138)(13 164 38 139)(14 165 39 140)(15 155 40 141)(16 156 41 142)(17 157 42 143)(18 158 43 133)(19 159 44 134)(20 160 34 135)(21 161 35 136)(22 162 36 137)(23 124 168 150)(24 125 169 151)(25 126 170 152)(26 127 171 153)(27 128 172 154)(28 129 173 144)(29 130 174 145)(30 131 175 146)(31 132 176 147)(32 122 166 148)(33 123 167 149)(56 115 71 90)(57 116 72 91)(58 117 73 92)(59 118 74 93)(60 119 75 94)(61 120 76 95)(62 121 77 96)(63 111 67 97)(64 112 68 98)(65 113 69 99)(66 114 70 89)
(1 123 47 149)(2 124 48 150)(3 125 49 151)(4 126 50 152)(5 127 51 153)(6 128 52 154)(7 129 53 144)(8 130 54 145)(9 131 55 146)(10 132 45 147)(11 122 46 148)(12 94 37 119)(13 95 38 120)(14 96 39 121)(15 97 40 111)(16 98 41 112)(17 99 42 113)(18 89 43 114)(19 90 44 115)(20 91 34 116)(21 92 35 117)(22 93 36 118)(23 106 168 80)(24 107 169 81)(25 108 170 82)(26 109 171 83)(27 110 172 84)(28 100 173 85)(29 101 174 86)(30 102 175 87)(31 103 176 88)(32 104 166 78)(33 105 167 79)(56 159 71 134)(57 160 72 135)(58 161 73 136)(59 162 74 137)(60 163 75 138)(61 164 76 139)(62 165 77 140)(63 155 67 141)(64 156 68 142)(65 157 69 143)(66 158 70 133)
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), (12,30)(13,31)(14,32)(15,33)(16,23)(17,24)(18,25)(19,26)(20,27)(21,28)(22,29)(34,172)(35,173)(36,174)(37,175)(38,176)(39,166)(40,167)(41,168)(42,169)(43,170)(44,171)(122,165)(123,155)(124,156)(125,157)(126,158)(127,159)(128,160)(129,161)(130,162)(131,163)(132,164)(133,152)(134,153)(135,154)(136,144)(137,145)(138,146)(139,147)(140,148)(141,149)(142,150)(143,151), (1,63)(2,64)(3,65)(4,66)(5,56)(6,57)(7,58)(8,59)(9,60)(10,61)(11,62)(12,30)(13,31)(14,32)(15,33)(16,23)(17,24)(18,25)(19,26)(20,27)(21,28)(22,29)(34,172)(35,173)(36,174)(37,175)(38,176)(39,166)(40,167)(41,168)(42,169)(43,170)(44,171)(45,76)(46,77)(47,67)(48,68)(49,69)(50,70)(51,71)(52,72)(53,73)(54,74)(55,75)(78,121)(79,111)(80,112)(81,113)(82,114)(83,115)(84,116)(85,117)(86,118)(87,119)(88,120)(89,108)(90,109)(91,110)(92,100)(93,101)(94,102)(95,103)(96,104)(97,105)(98,106)(99,107)(122,165)(123,155)(124,156)(125,157)(126,158)(127,159)(128,160)(129,161)(130,162)(131,163)(132,164)(133,152)(134,153)(135,154)(136,144)(137,145)(138,146)(139,147)(140,148)(141,149)(142,150)(143,151), (1,79,47,105)(2,80,48,106)(3,81,49,107)(4,82,50,108)(5,83,51,109)(6,84,52,110)(7,85,53,100)(8,86,54,101)(9,87,55,102)(10,88,45,103)(11,78,46,104)(12,163,37,138)(13,164,38,139)(14,165,39,140)(15,155,40,141)(16,156,41,142)(17,157,42,143)(18,158,43,133)(19,159,44,134)(20,160,34,135)(21,161,35,136)(22,162,36,137)(23,124,168,150)(24,125,169,151)(25,126,170,152)(26,127,171,153)(27,128,172,154)(28,129,173,144)(29,130,174,145)(30,131,175,146)(31,132,176,147)(32,122,166,148)(33,123,167,149)(56,115,71,90)(57,116,72,91)(58,117,73,92)(59,118,74,93)(60,119,75,94)(61,120,76,95)(62,121,77,96)(63,111,67,97)(64,112,68,98)(65,113,69,99)(66,114,70,89), (1,123,47,149)(2,124,48,150)(3,125,49,151)(4,126,50,152)(5,127,51,153)(6,128,52,154)(7,129,53,144)(8,130,54,145)(9,131,55,146)(10,132,45,147)(11,122,46,148)(12,94,37,119)(13,95,38,120)(14,96,39,121)(15,97,40,111)(16,98,41,112)(17,99,42,113)(18,89,43,114)(19,90,44,115)(20,91,34,116)(21,92,35,117)(22,93,36,118)(23,106,168,80)(24,107,169,81)(25,108,170,82)(26,109,171,83)(27,110,172,84)(28,100,173,85)(29,101,174,86)(30,102,175,87)(31,103,176,88)(32,104,166,78)(33,105,167,79)(56,159,71,134)(57,160,72,135)(58,161,73,136)(59,162,74,137)(60,163,75,138)(61,164,76,139)(62,165,77,140)(63,155,67,141)(64,156,68,142)(65,157,69,143)(66,158,70,133)>;
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), (12,30)(13,31)(14,32)(15,33)(16,23)(17,24)(18,25)(19,26)(20,27)(21,28)(22,29)(34,172)(35,173)(36,174)(37,175)(38,176)(39,166)(40,167)(41,168)(42,169)(43,170)(44,171)(122,165)(123,155)(124,156)(125,157)(126,158)(127,159)(128,160)(129,161)(130,162)(131,163)(132,164)(133,152)(134,153)(135,154)(136,144)(137,145)(138,146)(139,147)(140,148)(141,149)(142,150)(143,151), (1,63)(2,64)(3,65)(4,66)(5,56)(6,57)(7,58)(8,59)(9,60)(10,61)(11,62)(12,30)(13,31)(14,32)(15,33)(16,23)(17,24)(18,25)(19,26)(20,27)(21,28)(22,29)(34,172)(35,173)(36,174)(37,175)(38,176)(39,166)(40,167)(41,168)(42,169)(43,170)(44,171)(45,76)(46,77)(47,67)(48,68)(49,69)(50,70)(51,71)(52,72)(53,73)(54,74)(55,75)(78,121)(79,111)(80,112)(81,113)(82,114)(83,115)(84,116)(85,117)(86,118)(87,119)(88,120)(89,108)(90,109)(91,110)(92,100)(93,101)(94,102)(95,103)(96,104)(97,105)(98,106)(99,107)(122,165)(123,155)(124,156)(125,157)(126,158)(127,159)(128,160)(129,161)(130,162)(131,163)(132,164)(133,152)(134,153)(135,154)(136,144)(137,145)(138,146)(139,147)(140,148)(141,149)(142,150)(143,151), (1,79,47,105)(2,80,48,106)(3,81,49,107)(4,82,50,108)(5,83,51,109)(6,84,52,110)(7,85,53,100)(8,86,54,101)(9,87,55,102)(10,88,45,103)(11,78,46,104)(12,163,37,138)(13,164,38,139)(14,165,39,140)(15,155,40,141)(16,156,41,142)(17,157,42,143)(18,158,43,133)(19,159,44,134)(20,160,34,135)(21,161,35,136)(22,162,36,137)(23,124,168,150)(24,125,169,151)(25,126,170,152)(26,127,171,153)(27,128,172,154)(28,129,173,144)(29,130,174,145)(30,131,175,146)(31,132,176,147)(32,122,166,148)(33,123,167,149)(56,115,71,90)(57,116,72,91)(58,117,73,92)(59,118,74,93)(60,119,75,94)(61,120,76,95)(62,121,77,96)(63,111,67,97)(64,112,68,98)(65,113,69,99)(66,114,70,89), (1,123,47,149)(2,124,48,150)(3,125,49,151)(4,126,50,152)(5,127,51,153)(6,128,52,154)(7,129,53,144)(8,130,54,145)(9,131,55,146)(10,132,45,147)(11,122,46,148)(12,94,37,119)(13,95,38,120)(14,96,39,121)(15,97,40,111)(16,98,41,112)(17,99,42,113)(18,89,43,114)(19,90,44,115)(20,91,34,116)(21,92,35,117)(22,93,36,118)(23,106,168,80)(24,107,169,81)(25,108,170,82)(26,109,171,83)(27,110,172,84)(28,100,173,85)(29,101,174,86)(30,102,175,87)(31,103,176,88)(32,104,166,78)(33,105,167,79)(56,159,71,134)(57,160,72,135)(58,161,73,136)(59,162,74,137)(60,163,75,138)(61,164,76,139)(62,165,77,140)(63,155,67,141)(64,156,68,142)(65,157,69,143)(66,158,70,133) );
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)], [(12,30),(13,31),(14,32),(15,33),(16,23),(17,24),(18,25),(19,26),(20,27),(21,28),(22,29),(34,172),(35,173),(36,174),(37,175),(38,176),(39,166),(40,167),(41,168),(42,169),(43,170),(44,171),(122,165),(123,155),(124,156),(125,157),(126,158),(127,159),(128,160),(129,161),(130,162),(131,163),(132,164),(133,152),(134,153),(135,154),(136,144),(137,145),(138,146),(139,147),(140,148),(141,149),(142,150),(143,151)], [(1,63),(2,64),(3,65),(4,66),(5,56),(6,57),(7,58),(8,59),(9,60),(10,61),(11,62),(12,30),(13,31),(14,32),(15,33),(16,23),(17,24),(18,25),(19,26),(20,27),(21,28),(22,29),(34,172),(35,173),(36,174),(37,175),(38,176),(39,166),(40,167),(41,168),(42,169),(43,170),(44,171),(45,76),(46,77),(47,67),(48,68),(49,69),(50,70),(51,71),(52,72),(53,73),(54,74),(55,75),(78,121),(79,111),(80,112),(81,113),(82,114),(83,115),(84,116),(85,117),(86,118),(87,119),(88,120),(89,108),(90,109),(91,110),(92,100),(93,101),(94,102),(95,103),(96,104),(97,105),(98,106),(99,107),(122,165),(123,155),(124,156),(125,157),(126,158),(127,159),(128,160),(129,161),(130,162),(131,163),(132,164),(133,152),(134,153),(135,154),(136,144),(137,145),(138,146),(139,147),(140,148),(141,149),(142,150),(143,151)], [(1,79,47,105),(2,80,48,106),(3,81,49,107),(4,82,50,108),(5,83,51,109),(6,84,52,110),(7,85,53,100),(8,86,54,101),(9,87,55,102),(10,88,45,103),(11,78,46,104),(12,163,37,138),(13,164,38,139),(14,165,39,140),(15,155,40,141),(16,156,41,142),(17,157,42,143),(18,158,43,133),(19,159,44,134),(20,160,34,135),(21,161,35,136),(22,162,36,137),(23,124,168,150),(24,125,169,151),(25,126,170,152),(26,127,171,153),(27,128,172,154),(28,129,173,144),(29,130,174,145),(30,131,175,146),(31,132,176,147),(32,122,166,148),(33,123,167,149),(56,115,71,90),(57,116,72,91),(58,117,73,92),(59,118,74,93),(60,119,75,94),(61,120,76,95),(62,121,77,96),(63,111,67,97),(64,112,68,98),(65,113,69,99),(66,114,70,89)], [(1,123,47,149),(2,124,48,150),(3,125,49,151),(4,126,50,152),(5,127,51,153),(6,128,52,154),(7,129,53,144),(8,130,54,145),(9,131,55,146),(10,132,45,147),(11,122,46,148),(12,94,37,119),(13,95,38,120),(14,96,39,121),(15,97,40,111),(16,98,41,112),(17,99,42,113),(18,89,43,114),(19,90,44,115),(20,91,34,116),(21,92,35,117),(22,93,36,118),(23,106,168,80),(24,107,169,81),(25,108,170,82),(26,109,171,83),(27,110,172,84),(28,100,173,85),(29,101,174,86),(30,102,175,87),(31,103,176,88),(32,104,166,78),(33,105,167,79),(56,159,71,134),(57,160,72,135),(58,161,73,136),(59,162,74,137),(60,163,75,138),(61,164,76,139),(62,165,77,140),(63,155,67,141),(64,156,68,142),(65,157,69,143),(66,158,70,133)]])
154 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 11A | ··· | 11J | 22A | ··· | 22AD | 22AE | ··· | 22AX | 44A | ··· | 44AN | 44AO | ··· | 44CB |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 11 | ··· | 11 | 22 | ··· | 22 | 22 | ··· | 22 | 44 | ··· | 44 | 44 | ··· | 44 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
154 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | - | |||||||||
| image | C1 | C2 | C2 | C2 | C2 | C11 | C22 | C22 | C22 | C22 | D4 | Q8 | C4○D4 | D4×C11 | Q8×C11 | C11×C4○D4 |
| kernel | C11×C22⋊Q8 | C11×C22⋊C4 | C11×C4⋊C4 | C22×C44 | Q8×C22 | C22⋊Q8 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×Q8 | C44 | C2×C22 | C22 | C4 | C22 | C2 |
| # reps | 1 | 2 | 3 | 1 | 1 | 10 | 20 | 30 | 10 | 10 | 2 | 2 | 2 | 20 | 20 | 20 |
Matrix representation of C11×C22⋊Q8 ►in GL4(𝔽89) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 8 |
| 1 | 0 | 0 | 0 |
| 0 | 88 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 88 |
| 88 | 0 | 0 | 0 |
| 0 | 88 | 0 | 0 |
| 0 | 0 | 88 | 0 |
| 0 | 0 | 0 | 88 |
| 34 | 0 | 0 | 0 |
| 0 | 55 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 |
| 88 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
G:=sub<GL(4,GF(89))| [1,0,0,0,0,1,0,0,0,0,8,0,0,0,0,8],[1,0,0,0,0,88,0,0,0,0,1,0,0,0,0,88],[88,0,0,0,0,88,0,0,0,0,88,0,0,0,0,88],[34,0,0,0,0,55,0,0,0,0,1,0,0,0,0,1],[0,88,0,0,1,0,0,0,0,0,0,1,0,0,1,0] >;
C11×C22⋊Q8 in GAP, Magma, Sage, TeX
C_{11}\times C_2^2\rtimes Q_8 % in TeX
G:=Group("C11xC2^2:Q8"); // GroupNames label
G:=SmallGroup(352,157);
// by ID
G=gap.SmallGroup(352,157);
# by ID
G:=PCGroup([6,-2,-2,-2,-11,-2,-2,528,1081,535,3242]);
// Polycyclic
G:=Group<a,b,c,d,e|a^11=b^2=c^2=d^4=1,e^2=d^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,e*b*e^-1=b*c=c*b,b*d=d*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^-1>;
// generators/relations