direct product, metabelian, supersoluble, monomial, A-group
Aliases: C2×C32⋊4C8, C12.57D6, C62.6C4, C12.8Dic3, C6⋊(C3⋊C8), (C3×C6)⋊4C8, C32⋊9(C2×C8), (C3×C12).9C4, (C6×C12).12C2, (C2×C12).16S3, (C2×C6).7Dic3, C4.3(C3⋊Dic3), C6.12(C2×Dic3), (C3×C12).48C22, C22.2(C3⋊Dic3), C3⋊2(C2×C3⋊C8), C4.14(C2×C3⋊S3), (C2×C4).5(C3⋊S3), (C3×C6).31(C2×C4), C2.1(C2×C3⋊Dic3), SmallGroup(144,90)
Series: Derived ►Chief ►Lower central ►Upper central
| C32 — C2×C32⋊4C8 |
Generators and relations for C2×C32⋊4C8
G = < a,b,c,d | a2=b3=c3=d8=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c-1 >
Subgroups: 114 in 66 conjugacy classes, 51 normal (13 characteristic)
C1, C2, C2, C3, C4, C22, C6, C8, C2×C4, C32, C12, C2×C6, C2×C8, C3×C6, C3×C6, C3⋊C8, C2×C12, C3×C12, C62, C2×C3⋊C8, C32⋊4C8, C6×C12, C2×C32⋊4C8
Quotients: C1, C2, C4, C22, S3, C8, C2×C4, Dic3, D6, C2×C8, C3⋊S3, C3⋊C8, C2×Dic3, C3⋊Dic3, C2×C3⋊S3, C2×C3⋊C8, C32⋊4C8, C2×C3⋊Dic3, C2×C32⋊4C8
►Regular action on 144 points
Generators in S144
(1 66)(2 67)(3 68)(4 69)(5 70)(6 71)(7 72)(8 65)(9 57)(10 58)(11 59)(12 60)(13 61)(14 62)(15 63)(16 64)(17 83)(18 84)(19 85)(20 86)(21 87)(22 88)(23 81)(24 82)(25 45)(26 46)(27 47)(28 48)(29 41)(30 42)(31 43)(32 44)(33 112)(34 105)(35 106)(36 107)(37 108)(38 109)(39 110)(40 111)(49 127)(50 128)(51 121)(52 122)(53 123)(54 124)(55 125)(56 126)(73 115)(74 116)(75 117)(76 118)(77 119)(78 120)(79 113)(80 114)(89 139)(90 140)(91 141)(92 142)(93 143)(94 144)(95 137)(96 138)(97 130)(98 131)(99 132)(100 133)(101 134)(102 135)(103 136)(104 129)
(1 91 119)(2 120 92)(3 93 113)(4 114 94)(5 95 115)(6 116 96)(7 89 117)(8 118 90)(9 86 98)(10 99 87)(11 88 100)(12 101 81)(13 82 102)(14 103 83)(15 84 104)(16 97 85)(17 62 136)(18 129 63)(19 64 130)(20 131 57)(21 58 132)(22 133 59)(23 60 134)(24 135 61)(25 110 50)(26 51 111)(27 112 52)(28 53 105)(29 106 54)(30 55 107)(31 108 56)(32 49 109)(33 122 47)(34 48 123)(35 124 41)(36 42 125)(37 126 43)(38 44 127)(39 128 45)(40 46 121)(65 76 140)(66 141 77)(67 78 142)(68 143 79)(69 80 144)(70 137 73)(71 74 138)(72 139 75)
(1 24 31)(2 32 17)(3 18 25)(4 26 19)(5 20 27)(6 28 21)(7 22 29)(8 30 23)(9 122 73)(10 74 123)(11 124 75)(12 76 125)(13 126 77)(14 78 127)(15 128 79)(16 80 121)(33 137 98)(34 99 138)(35 139 100)(36 101 140)(37 141 102)(38 103 142)(39 143 104)(40 97 144)(41 72 88)(42 81 65)(43 66 82)(44 83 67)(45 68 84)(46 85 69)(47 70 86)(48 87 71)(49 62 120)(50 113 63)(51 64 114)(52 115 57)(53 58 116)(54 117 59)(55 60 118)(56 119 61)(89 133 106)(90 107 134)(91 135 108)(92 109 136)(93 129 110)(94 111 130)(95 131 112)(96 105 132)
(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)
G:=sub<Sym(144)| (1,66)(2,67)(3,68)(4,69)(5,70)(6,71)(7,72)(8,65)(9,57)(10,58)(11,59)(12,60)(13,61)(14,62)(15,63)(16,64)(17,83)(18,84)(19,85)(20,86)(21,87)(22,88)(23,81)(24,82)(25,45)(26,46)(27,47)(28,48)(29,41)(30,42)(31,43)(32,44)(33,112)(34,105)(35,106)(36,107)(37,108)(38,109)(39,110)(40,111)(49,127)(50,128)(51,121)(52,122)(53,123)(54,124)(55,125)(56,126)(73,115)(74,116)(75,117)(76,118)(77,119)(78,120)(79,113)(80,114)(89,139)(90,140)(91,141)(92,142)(93,143)(94,144)(95,137)(96,138)(97,130)(98,131)(99,132)(100,133)(101,134)(102,135)(103,136)(104,129), (1,91,119)(2,120,92)(3,93,113)(4,114,94)(5,95,115)(6,116,96)(7,89,117)(8,118,90)(9,86,98)(10,99,87)(11,88,100)(12,101,81)(13,82,102)(14,103,83)(15,84,104)(16,97,85)(17,62,136)(18,129,63)(19,64,130)(20,131,57)(21,58,132)(22,133,59)(23,60,134)(24,135,61)(25,110,50)(26,51,111)(27,112,52)(28,53,105)(29,106,54)(30,55,107)(31,108,56)(32,49,109)(33,122,47)(34,48,123)(35,124,41)(36,42,125)(37,126,43)(38,44,127)(39,128,45)(40,46,121)(65,76,140)(66,141,77)(67,78,142)(68,143,79)(69,80,144)(70,137,73)(71,74,138)(72,139,75), (1,24,31)(2,32,17)(3,18,25)(4,26,19)(5,20,27)(6,28,21)(7,22,29)(8,30,23)(9,122,73)(10,74,123)(11,124,75)(12,76,125)(13,126,77)(14,78,127)(15,128,79)(16,80,121)(33,137,98)(34,99,138)(35,139,100)(36,101,140)(37,141,102)(38,103,142)(39,143,104)(40,97,144)(41,72,88)(42,81,65)(43,66,82)(44,83,67)(45,68,84)(46,85,69)(47,70,86)(48,87,71)(49,62,120)(50,113,63)(51,64,114)(52,115,57)(53,58,116)(54,117,59)(55,60,118)(56,119,61)(89,133,106)(90,107,134)(91,135,108)(92,109,136)(93,129,110)(94,111,130)(95,131,112)(96,105,132), (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)>;
G:=Group( (1,66)(2,67)(3,68)(4,69)(5,70)(6,71)(7,72)(8,65)(9,57)(10,58)(11,59)(12,60)(13,61)(14,62)(15,63)(16,64)(17,83)(18,84)(19,85)(20,86)(21,87)(22,88)(23,81)(24,82)(25,45)(26,46)(27,47)(28,48)(29,41)(30,42)(31,43)(32,44)(33,112)(34,105)(35,106)(36,107)(37,108)(38,109)(39,110)(40,111)(49,127)(50,128)(51,121)(52,122)(53,123)(54,124)(55,125)(56,126)(73,115)(74,116)(75,117)(76,118)(77,119)(78,120)(79,113)(80,114)(89,139)(90,140)(91,141)(92,142)(93,143)(94,144)(95,137)(96,138)(97,130)(98,131)(99,132)(100,133)(101,134)(102,135)(103,136)(104,129), (1,91,119)(2,120,92)(3,93,113)(4,114,94)(5,95,115)(6,116,96)(7,89,117)(8,118,90)(9,86,98)(10,99,87)(11,88,100)(12,101,81)(13,82,102)(14,103,83)(15,84,104)(16,97,85)(17,62,136)(18,129,63)(19,64,130)(20,131,57)(21,58,132)(22,133,59)(23,60,134)(24,135,61)(25,110,50)(26,51,111)(27,112,52)(28,53,105)(29,106,54)(30,55,107)(31,108,56)(32,49,109)(33,122,47)(34,48,123)(35,124,41)(36,42,125)(37,126,43)(38,44,127)(39,128,45)(40,46,121)(65,76,140)(66,141,77)(67,78,142)(68,143,79)(69,80,144)(70,137,73)(71,74,138)(72,139,75), (1,24,31)(2,32,17)(3,18,25)(4,26,19)(5,20,27)(6,28,21)(7,22,29)(8,30,23)(9,122,73)(10,74,123)(11,124,75)(12,76,125)(13,126,77)(14,78,127)(15,128,79)(16,80,121)(33,137,98)(34,99,138)(35,139,100)(36,101,140)(37,141,102)(38,103,142)(39,143,104)(40,97,144)(41,72,88)(42,81,65)(43,66,82)(44,83,67)(45,68,84)(46,85,69)(47,70,86)(48,87,71)(49,62,120)(50,113,63)(51,64,114)(52,115,57)(53,58,116)(54,117,59)(55,60,118)(56,119,61)(89,133,106)(90,107,134)(91,135,108)(92,109,136)(93,129,110)(94,111,130)(95,131,112)(96,105,132), (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) );
G=PermutationGroup([[(1,66),(2,67),(3,68),(4,69),(5,70),(6,71),(7,72),(8,65),(9,57),(10,58),(11,59),(12,60),(13,61),(14,62),(15,63),(16,64),(17,83),(18,84),(19,85),(20,86),(21,87),(22,88),(23,81),(24,82),(25,45),(26,46),(27,47),(28,48),(29,41),(30,42),(31,43),(32,44),(33,112),(34,105),(35,106),(36,107),(37,108),(38,109),(39,110),(40,111),(49,127),(50,128),(51,121),(52,122),(53,123),(54,124),(55,125),(56,126),(73,115),(74,116),(75,117),(76,118),(77,119),(78,120),(79,113),(80,114),(89,139),(90,140),(91,141),(92,142),(93,143),(94,144),(95,137),(96,138),(97,130),(98,131),(99,132),(100,133),(101,134),(102,135),(103,136),(104,129)], [(1,91,119),(2,120,92),(3,93,113),(4,114,94),(5,95,115),(6,116,96),(7,89,117),(8,118,90),(9,86,98),(10,99,87),(11,88,100),(12,101,81),(13,82,102),(14,103,83),(15,84,104),(16,97,85),(17,62,136),(18,129,63),(19,64,130),(20,131,57),(21,58,132),(22,133,59),(23,60,134),(24,135,61),(25,110,50),(26,51,111),(27,112,52),(28,53,105),(29,106,54),(30,55,107),(31,108,56),(32,49,109),(33,122,47),(34,48,123),(35,124,41),(36,42,125),(37,126,43),(38,44,127),(39,128,45),(40,46,121),(65,76,140),(66,141,77),(67,78,142),(68,143,79),(69,80,144),(70,137,73),(71,74,138),(72,139,75)], [(1,24,31),(2,32,17),(3,18,25),(4,26,19),(5,20,27),(6,28,21),(7,22,29),(8,30,23),(9,122,73),(10,74,123),(11,124,75),(12,76,125),(13,126,77),(14,78,127),(15,128,79),(16,80,121),(33,137,98),(34,99,138),(35,139,100),(36,101,140),(37,141,102),(38,103,142),(39,143,104),(40,97,144),(41,72,88),(42,81,65),(43,66,82),(44,83,67),(45,68,84),(46,85,69),(47,70,86),(48,87,71),(49,62,120),(50,113,63),(51,64,114),(52,115,57),(53,58,116),(54,117,59),(55,60,118),(56,119,61),(89,133,106),(90,107,134),(91,135,108),(92,109,136),(93,129,110),(94,111,130),(95,131,112),(96,105,132)], [(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)]])
C2×C32⋊4C8 is a maximal subgroup of
Dic3×C3⋊C8 C3⋊C8⋊Dic3 C12.77D12 D12⋊3Dic3 Dic6⋊Dic3 C12.81D12 C12.6Dic6 C12.8Dic6 C62.5Q8 C122.C2 C12.57D12 C12.9Dic6 C12.10Dic6 C62.113D4 C62.114D4 C8×C3⋊Dic3 C12.30Dic6 C24⋊Dic3 C12.60D12 C62.8Q8 C62⋊7C8 C62.116D4 C62.117D4 C62.4C8 C2×S3×C3⋊C8 D12.Dic3 D12.30D6 C2×C8×C3⋊S3 C24.47D6 D4.(C3⋊Dic3) C62.74D4
C2×C32⋊4C8 is a maximal quotient of
C12.57D12 C24.94D6 C62⋊7C8
48 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 4A | 4B | 4C | 4D | 6A | ··· | 6L | 8A | ··· | 8H | 12A | ··· | 12P |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 8 | ··· | 8 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 9 | ··· | 9 | 2 | ··· | 2 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | - | ||||
| image | C1 | C2 | C2 | C4 | C4 | C8 | S3 | Dic3 | D6 | Dic3 | C3⋊C8 |
| kernel | C2×C32⋊4C8 | C32⋊4C8 | C6×C12 | C3×C12 | C62 | C3×C6 | C2×C12 | C12 | C12 | C2×C6 | C6 |
| # reps | 1 | 2 | 1 | 2 | 2 | 8 | 4 | 4 | 4 | 4 | 16 |
Matrix representation of C2×C32⋊4C8 ►in GL5(𝔽73)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 0 | 72 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 72 | 72 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 1 |
| 0 | 0 | 0 | 72 | 0 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 72 | 72 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 51 | 0 | 0 | 0 | 0 |
| 0 | 39 | 47 | 0 | 0 |
| 0 | 8 | 34 | 0 | 0 |
| 0 | 0 | 0 | 46 | 27 |
| 0 | 0 | 0 | 0 | 27 |
G:=sub<GL(5,GF(73))| [1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,72,0,0,0,0,0,72],[1,0,0,0,0,0,72,1,0,0,0,72,0,0,0,0,0,0,72,72,0,0,0,1,0],[1,0,0,0,0,0,0,72,0,0,0,1,72,0,0,0,0,0,1,0,0,0,0,0,1],[51,0,0,0,0,0,39,8,0,0,0,47,34,0,0,0,0,0,46,0,0,0,0,27,27] >;
C2×C32⋊4C8 in GAP, Magma, Sage, TeX
C_2\times C_3^2\rtimes_4C_8
% in TeX
G:=Group("C2xC3^2:4C8"); // GroupNames label
G:=SmallGroup(144,90);
// by ID
G=gap.SmallGroup(144,90);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-3,-3,24,50,964,3461]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^3=c^3=d^8=1,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=c^-1>;
// generators/relations