direct product, metabelian, supersoluble, monomial
Aliases: C2×C32⋊4Q8, C6⋊2Dic6, C12.49D6, C62.31C22, (C3×C6)⋊4Q8, C32⋊6(C2×Q8), (C2×C12).9S3, (C6×C12).5C2, C3⋊3(C2×Dic6), (C2×C6).36D6, C6.30(C22×S3), (C3×C6).29C23, (C3×C12).35C22, C3⋊Dic3.16C22, C4.11(C2×C3⋊S3), (C2×C4).4(C3⋊S3), C22.8(C2×C3⋊S3), C2.3(C22×C3⋊S3), (C2×C3⋊Dic3).9C2, SmallGroup(144,168)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C32⋊4Q8
G = < a,b,c,d,e | a2=b3=c3=d4=1, e2=d2, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, ebe-1=b-1, cd=dc, ece-1=c-1, ede-1=d-1 >
Subgroups: 290 in 114 conjugacy classes, 59 normal (9 characteristic)
C1, C2, C2, C3, C4, C4, C22, C6, C2×C4, C2×C4, Q8, C32, Dic3, C12, C2×C6, C2×Q8, C3×C6, C3×C6, Dic6, C2×Dic3, C2×C12, C3⋊Dic3, C3×C12, C62, C2×Dic6, C32⋊4Q8, C2×C3⋊Dic3, C6×C12, C2×C32⋊4Q8
Quotients: C1, C2, C22, S3, Q8, C23, D6, C2×Q8, C3⋊S3, Dic6, C22×S3, C2×C3⋊S3, C2×Dic6, C32⋊4Q8, C22×C3⋊S3, C2×C32⋊4Q8
►Regular action on 144 points
Generators in S144
(1 76)(2 73)(3 74)(4 75)(5 68)(6 65)(7 66)(8 67)(9 121)(10 122)(11 123)(12 124)(13 62)(14 63)(15 64)(16 61)(17 111)(18 112)(19 109)(20 110)(21 47)(22 48)(23 45)(24 46)(25 132)(26 129)(27 130)(28 131)(29 55)(30 56)(31 53)(32 54)(33 50)(34 51)(35 52)(36 49)(37 70)(38 71)(39 72)(40 69)(41 86)(42 87)(43 88)(44 85)(57 93)(58 94)(59 95)(60 96)(77 113)(78 114)(79 115)(80 116)(81 117)(82 118)(83 119)(84 120)(89 125)(90 126)(91 127)(92 128)(97 133)(98 134)(99 135)(100 136)(101 137)(102 138)(103 139)(104 140)(105 141)(106 142)(107 143)(108 144)
(1 16 33)(2 13 34)(3 14 35)(4 15 36)(5 44 29)(6 41 30)(7 42 31)(8 43 32)(9 138 118)(10 139 119)(11 140 120)(12 137 117)(17 114 134)(18 115 135)(19 116 136)(20 113 133)(21 58 40)(22 59 37)(23 60 38)(24 57 39)(25 128 143)(26 125 144)(27 126 141)(28 127 142)(45 96 71)(46 93 72)(47 94 69)(48 95 70)(49 75 64)(50 76 61)(51 73 62)(52 74 63)(53 66 87)(54 67 88)(55 68 85)(56 65 86)(77 97 110)(78 98 111)(79 99 112)(80 100 109)(81 124 101)(82 121 102)(83 122 103)(84 123 104)(89 108 129)(90 105 130)(91 106 131)(92 107 132)
(1 21 29)(2 22 30)(3 23 31)(4 24 32)(5 16 58)(6 13 59)(7 14 60)(8 15 57)(9 141 115)(10 142 116)(11 143 113)(12 144 114)(17 117 125)(18 118 126)(19 119 127)(20 120 128)(25 133 140)(26 134 137)(27 135 138)(28 136 139)(33 40 44)(34 37 41)(35 38 42)(36 39 43)(45 53 74)(46 54 75)(47 55 76)(48 56 73)(49 72 88)(50 69 85)(51 70 86)(52 71 87)(61 94 68)(62 95 65)(63 96 66)(64 93 67)(77 123 107)(78 124 108)(79 121 105)(80 122 106)(81 89 111)(82 90 112)(83 91 109)(84 92 110)(97 104 132)(98 101 129)(99 102 130)(100 103 131)
(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)
(1 112 3 110)(2 111 4 109)(5 102 7 104)(6 101 8 103)(9 87 11 85)(10 86 12 88)(13 98 15 100)(14 97 16 99)(17 75 19 73)(18 74 20 76)(21 90 23 92)(22 89 24 91)(25 94 27 96)(26 93 28 95)(29 82 31 84)(30 81 32 83)(33 79 35 77)(34 78 36 80)(37 108 39 106)(38 107 40 105)(41 124 43 122)(42 123 44 121)(45 128 47 126)(46 127 48 125)(49 116 51 114)(50 115 52 113)(53 120 55 118)(54 119 56 117)(57 131 59 129)(58 130 60 132)(61 135 63 133)(62 134 64 136)(65 137 67 139)(66 140 68 138)(69 141 71 143)(70 144 72 142)
G:=sub<Sym(144)| (1,76)(2,73)(3,74)(4,75)(5,68)(6,65)(7,66)(8,67)(9,121)(10,122)(11,123)(12,124)(13,62)(14,63)(15,64)(16,61)(17,111)(18,112)(19,109)(20,110)(21,47)(22,48)(23,45)(24,46)(25,132)(26,129)(27,130)(28,131)(29,55)(30,56)(31,53)(32,54)(33,50)(34,51)(35,52)(36,49)(37,70)(38,71)(39,72)(40,69)(41,86)(42,87)(43,88)(44,85)(57,93)(58,94)(59,95)(60,96)(77,113)(78,114)(79,115)(80,116)(81,117)(82,118)(83,119)(84,120)(89,125)(90,126)(91,127)(92,128)(97,133)(98,134)(99,135)(100,136)(101,137)(102,138)(103,139)(104,140)(105,141)(106,142)(107,143)(108,144), (1,16,33)(2,13,34)(3,14,35)(4,15,36)(5,44,29)(6,41,30)(7,42,31)(8,43,32)(9,138,118)(10,139,119)(11,140,120)(12,137,117)(17,114,134)(18,115,135)(19,116,136)(20,113,133)(21,58,40)(22,59,37)(23,60,38)(24,57,39)(25,128,143)(26,125,144)(27,126,141)(28,127,142)(45,96,71)(46,93,72)(47,94,69)(48,95,70)(49,75,64)(50,76,61)(51,73,62)(52,74,63)(53,66,87)(54,67,88)(55,68,85)(56,65,86)(77,97,110)(78,98,111)(79,99,112)(80,100,109)(81,124,101)(82,121,102)(83,122,103)(84,123,104)(89,108,129)(90,105,130)(91,106,131)(92,107,132), (1,21,29)(2,22,30)(3,23,31)(4,24,32)(5,16,58)(6,13,59)(7,14,60)(8,15,57)(9,141,115)(10,142,116)(11,143,113)(12,144,114)(17,117,125)(18,118,126)(19,119,127)(20,120,128)(25,133,140)(26,134,137)(27,135,138)(28,136,139)(33,40,44)(34,37,41)(35,38,42)(36,39,43)(45,53,74)(46,54,75)(47,55,76)(48,56,73)(49,72,88)(50,69,85)(51,70,86)(52,71,87)(61,94,68)(62,95,65)(63,96,66)(64,93,67)(77,123,107)(78,124,108)(79,121,105)(80,122,106)(81,89,111)(82,90,112)(83,91,109)(84,92,110)(97,104,132)(98,101,129)(99,102,130)(100,103,131), (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), (1,112,3,110)(2,111,4,109)(5,102,7,104)(6,101,8,103)(9,87,11,85)(10,86,12,88)(13,98,15,100)(14,97,16,99)(17,75,19,73)(18,74,20,76)(21,90,23,92)(22,89,24,91)(25,94,27,96)(26,93,28,95)(29,82,31,84)(30,81,32,83)(33,79,35,77)(34,78,36,80)(37,108,39,106)(38,107,40,105)(41,124,43,122)(42,123,44,121)(45,128,47,126)(46,127,48,125)(49,116,51,114)(50,115,52,113)(53,120,55,118)(54,119,56,117)(57,131,59,129)(58,130,60,132)(61,135,63,133)(62,134,64,136)(65,137,67,139)(66,140,68,138)(69,141,71,143)(70,144,72,142)>;
G:=Group( (1,76)(2,73)(3,74)(4,75)(5,68)(6,65)(7,66)(8,67)(9,121)(10,122)(11,123)(12,124)(13,62)(14,63)(15,64)(16,61)(17,111)(18,112)(19,109)(20,110)(21,47)(22,48)(23,45)(24,46)(25,132)(26,129)(27,130)(28,131)(29,55)(30,56)(31,53)(32,54)(33,50)(34,51)(35,52)(36,49)(37,70)(38,71)(39,72)(40,69)(41,86)(42,87)(43,88)(44,85)(57,93)(58,94)(59,95)(60,96)(77,113)(78,114)(79,115)(80,116)(81,117)(82,118)(83,119)(84,120)(89,125)(90,126)(91,127)(92,128)(97,133)(98,134)(99,135)(100,136)(101,137)(102,138)(103,139)(104,140)(105,141)(106,142)(107,143)(108,144), (1,16,33)(2,13,34)(3,14,35)(4,15,36)(5,44,29)(6,41,30)(7,42,31)(8,43,32)(9,138,118)(10,139,119)(11,140,120)(12,137,117)(17,114,134)(18,115,135)(19,116,136)(20,113,133)(21,58,40)(22,59,37)(23,60,38)(24,57,39)(25,128,143)(26,125,144)(27,126,141)(28,127,142)(45,96,71)(46,93,72)(47,94,69)(48,95,70)(49,75,64)(50,76,61)(51,73,62)(52,74,63)(53,66,87)(54,67,88)(55,68,85)(56,65,86)(77,97,110)(78,98,111)(79,99,112)(80,100,109)(81,124,101)(82,121,102)(83,122,103)(84,123,104)(89,108,129)(90,105,130)(91,106,131)(92,107,132), (1,21,29)(2,22,30)(3,23,31)(4,24,32)(5,16,58)(6,13,59)(7,14,60)(8,15,57)(9,141,115)(10,142,116)(11,143,113)(12,144,114)(17,117,125)(18,118,126)(19,119,127)(20,120,128)(25,133,140)(26,134,137)(27,135,138)(28,136,139)(33,40,44)(34,37,41)(35,38,42)(36,39,43)(45,53,74)(46,54,75)(47,55,76)(48,56,73)(49,72,88)(50,69,85)(51,70,86)(52,71,87)(61,94,68)(62,95,65)(63,96,66)(64,93,67)(77,123,107)(78,124,108)(79,121,105)(80,122,106)(81,89,111)(82,90,112)(83,91,109)(84,92,110)(97,104,132)(98,101,129)(99,102,130)(100,103,131), (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), (1,112,3,110)(2,111,4,109)(5,102,7,104)(6,101,8,103)(9,87,11,85)(10,86,12,88)(13,98,15,100)(14,97,16,99)(17,75,19,73)(18,74,20,76)(21,90,23,92)(22,89,24,91)(25,94,27,96)(26,93,28,95)(29,82,31,84)(30,81,32,83)(33,79,35,77)(34,78,36,80)(37,108,39,106)(38,107,40,105)(41,124,43,122)(42,123,44,121)(45,128,47,126)(46,127,48,125)(49,116,51,114)(50,115,52,113)(53,120,55,118)(54,119,56,117)(57,131,59,129)(58,130,60,132)(61,135,63,133)(62,134,64,136)(65,137,67,139)(66,140,68,138)(69,141,71,143)(70,144,72,142) );
G=PermutationGroup([[(1,76),(2,73),(3,74),(4,75),(5,68),(6,65),(7,66),(8,67),(9,121),(10,122),(11,123),(12,124),(13,62),(14,63),(15,64),(16,61),(17,111),(18,112),(19,109),(20,110),(21,47),(22,48),(23,45),(24,46),(25,132),(26,129),(27,130),(28,131),(29,55),(30,56),(31,53),(32,54),(33,50),(34,51),(35,52),(36,49),(37,70),(38,71),(39,72),(40,69),(41,86),(42,87),(43,88),(44,85),(57,93),(58,94),(59,95),(60,96),(77,113),(78,114),(79,115),(80,116),(81,117),(82,118),(83,119),(84,120),(89,125),(90,126),(91,127),(92,128),(97,133),(98,134),(99,135),(100,136),(101,137),(102,138),(103,139),(104,140),(105,141),(106,142),(107,143),(108,144)], [(1,16,33),(2,13,34),(3,14,35),(4,15,36),(5,44,29),(6,41,30),(7,42,31),(8,43,32),(9,138,118),(10,139,119),(11,140,120),(12,137,117),(17,114,134),(18,115,135),(19,116,136),(20,113,133),(21,58,40),(22,59,37),(23,60,38),(24,57,39),(25,128,143),(26,125,144),(27,126,141),(28,127,142),(45,96,71),(46,93,72),(47,94,69),(48,95,70),(49,75,64),(50,76,61),(51,73,62),(52,74,63),(53,66,87),(54,67,88),(55,68,85),(56,65,86),(77,97,110),(78,98,111),(79,99,112),(80,100,109),(81,124,101),(82,121,102),(83,122,103),(84,123,104),(89,108,129),(90,105,130),(91,106,131),(92,107,132)], [(1,21,29),(2,22,30),(3,23,31),(4,24,32),(5,16,58),(6,13,59),(7,14,60),(8,15,57),(9,141,115),(10,142,116),(11,143,113),(12,144,114),(17,117,125),(18,118,126),(19,119,127),(20,120,128),(25,133,140),(26,134,137),(27,135,138),(28,136,139),(33,40,44),(34,37,41),(35,38,42),(36,39,43),(45,53,74),(46,54,75),(47,55,76),(48,56,73),(49,72,88),(50,69,85),(51,70,86),(52,71,87),(61,94,68),(62,95,65),(63,96,66),(64,93,67),(77,123,107),(78,124,108),(79,121,105),(80,122,106),(81,89,111),(82,90,112),(83,91,109),(84,92,110),(97,104,132),(98,101,129),(99,102,130),(100,103,131)], [(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)], [(1,112,3,110),(2,111,4,109),(5,102,7,104),(6,101,8,103),(9,87,11,85),(10,86,12,88),(13,98,15,100),(14,97,16,99),(17,75,19,73),(18,74,20,76),(21,90,23,92),(22,89,24,91),(25,94,27,96),(26,93,28,95),(29,82,31,84),(30,81,32,83),(33,79,35,77),(34,78,36,80),(37,108,39,106),(38,107,40,105),(41,124,43,122),(42,123,44,121),(45,128,47,126),(46,127,48,125),(49,116,51,114),(50,115,52,113),(53,120,55,118),(54,119,56,117),(57,131,59,129),(58,130,60,132),(61,135,63,133),(62,134,64,136),(65,137,67,139),(66,140,68,138),(69,141,71,143),(70,144,72,142)]])
C2×C32⋊4Q8 is a maximal subgroup of
C12.71D12 C12.73D12 C62.114D4 C6.4Dic12 C12.20D12 C3⋊Dic3.D4 D12.29D6 C62.10C23 Dic3⋊6Dic6 D6⋊7Dic6 C12.27D12 Dic3⋊Dic6 D6⋊3Dic6 C62.83C23 C12⋊6Dic6 C122⋊6C2 C62⋊6Q8 C62.229C23 C62.231C23 C12⋊2Dic6 C62.240C23 C12.31D12 C24.5D6 C62⋊10Q8 C62.254C23 C62.259C23 C62.75D4 C2×S3×Dic6 D12.34D6 C2×Q8×C3⋊S3 C32⋊92- 1+4
C2×C32⋊4Q8 is a maximal quotient of
C12⋊6Dic6 C12.25Dic6 C62⋊6Q8 C12⋊2Dic6 C62.234C23 C62⋊10Q8
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 4A | 4B | 4C | 4D | 4E | 4F | 6A | ··· | 6L | 12A | ··· | 12P |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 18 | 18 | 18 | 18 | 2 | ··· | 2 | 2 | ··· | 2 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | + | + | - |
| image | C1 | C2 | C2 | C2 | S3 | Q8 | D6 | D6 | Dic6 |
| kernel | C2×C32⋊4Q8 | C32⋊4Q8 | C2×C3⋊Dic3 | C6×C12 | C2×C12 | C3×C6 | C12 | C2×C6 | C6 |
| # reps | 1 | 4 | 2 | 1 | 4 | 2 | 8 | 4 | 16 |
Matrix representation of C2×C32⋊4Q8 ►in GL6(𝔽13)
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 1 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 12 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 12 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 3 | 6 | 0 | 0 | 0 | 0 |
| 7 | 10 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 6 | 0 | 0 |
| 0 | 0 | 7 | 10 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 4 | 2 | 0 | 0 | 0 | 0 |
| 11 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 11 | 2 | 0 | 0 |
| 0 | 0 | 4 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 10 | 6 |
| 0 | 0 | 0 | 0 | 3 | 3 |
G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,12,0,0,0,0,1,0],[0,12,0,0,0,0,1,12,0,0,0,0,0,0,0,12,0,0,0,0,1,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[3,7,0,0,0,0,6,10,0,0,0,0,0,0,3,7,0,0,0,0,6,10,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[4,11,0,0,0,0,2,9,0,0,0,0,0,0,11,4,0,0,0,0,2,2,0,0,0,0,0,0,10,3,0,0,0,0,6,3] >;
C2×C32⋊4Q8 in GAP, Magma, Sage, TeX
C_2\times C_3^2\rtimes_4Q_8
% in TeX
G:=Group("C2xC3^2:4Q8"); // GroupNames label
G:=SmallGroup(144,168);
// by ID
G=gap.SmallGroup(144,168);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-3,-3,48,218,50,964,3461]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^3=c^3=d^4=1,e^2=d^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,e*b*e^-1=b^-1,c*d=d*c,e*c*e^-1=c^-1,e*d*e^-1=d^-1>;
// generators/relations