direct product, metabelian, supersoluble, monomial
Aliases: C32×C24⋊C2, C12.8C62, C33⋊19SD16, C24⋊6(C3×S3), C24⋊2(C3×C6), (C3×C24)⋊18S3, (C3×C24)⋊12C6, C8⋊2(S3×C32), (C32×C24)⋊5C2, Dic6⋊1(C3×C6), D12.1(C3×C6), (C3×C6).84D12, C6.37(C3×D12), C12.112(S3×C6), C6.1(D4×C32), (C3×Dic6)⋊10C6, (C3×D12).10C6, (C3×C12).231D6, C3⋊1(C32×SD16), (C32×C6).52D4, C2.3(C32×D12), (C32×D12).6C2, C32⋊10(C3×SD16), (C32×Dic6)⋊16C2, (C32×C12).79C22, C4.8(S3×C3×C6), (C3×C6).44(C3×D4), (C3×C12).83(C2×C6), SmallGroup(432,466)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C32×C24⋊C2
G = < a,b,c,d | a3=b3=c24=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c11 >
Subgroups: 376 in 152 conjugacy classes, 66 normal (22 characteristic)
C1, C2, C2, C3, C3, C3, C4, C4, C22, S3, C6, C6, C6, C8, D4, Q8, C32, C32, C32, Dic3, C12, C12, C12, D6, C2×C6, SD16, C3×S3, C3×C6, C3×C6, C3×C6, C24, C24, C24, Dic6, D12, C3×D4, C3×Q8, C33, C3×Dic3, C3×C12, C3×C12, C3×C12, S3×C6, C62, C24⋊C2, C3×SD16, S3×C32, C32×C6, C3×C24, C3×C24, C3×C24, C3×Dic6, C3×D12, D4×C32, Q8×C32, C32×Dic3, C32×C12, S3×C3×C6, C3×C24⋊C2, C32×SD16, C32×C24, C32×Dic6, C32×D12, C32×C24⋊C2
Quotients: C1, C2, C3, C22, S3, C6, D4, C32, D6, C2×C6, SD16, C3×S3, C3×C6, D12, C3×D4, S3×C6, C62, C24⋊C2, C3×SD16, S3×C32, C3×D12, D4×C32, S3×C3×C6, C3×C24⋊C2, C32×SD16, C32×D12, C32×C24⋊C2
►On 144 points
Generators in S144
(1 51 74)(2 52 75)(3 53 76)(4 54 77)(5 55 78)(6 56 79)(7 57 80)(8 58 81)(9 59 82)(10 60 83)(11 61 84)(12 62 85)(13 63 86)(14 64 87)(15 65 88)(16 66 89)(17 67 90)(18 68 91)(19 69 92)(20 70 93)(21 71 94)(22 72 95)(23 49 96)(24 50 73)(25 135 113)(26 136 114)(27 137 115)(28 138 116)(29 139 117)(30 140 118)(31 141 119)(32 142 120)(33 143 97)(34 144 98)(35 121 99)(36 122 100)(37 123 101)(38 124 102)(39 125 103)(40 126 104)(41 127 105)(42 128 106)(43 129 107)(44 130 108)(45 131 109)(46 132 110)(47 133 111)(48 134 112)
(1 9 17)(2 10 18)(3 11 19)(4 12 20)(5 13 21)(6 14 22)(7 15 23)(8 16 24)(25 41 33)(26 42 34)(27 43 35)(28 44 36)(29 45 37)(30 46 38)(31 47 39)(32 48 40)(49 57 65)(50 58 66)(51 59 67)(52 60 68)(53 61 69)(54 62 70)(55 63 71)(56 64 72)(73 81 89)(74 82 90)(75 83 91)(76 84 92)(77 85 93)(78 86 94)(79 87 95)(80 88 96)(97 113 105)(98 114 106)(99 115 107)(100 116 108)(101 117 109)(102 118 110)(103 119 111)(104 120 112)(121 137 129)(122 138 130)(123 139 131)(124 140 132)(125 141 133)(126 142 134)(127 143 135)(128 144 136)
(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)
(1 123)(2 134)(3 121)(4 132)(5 143)(6 130)(7 141)(8 128)(9 139)(10 126)(11 137)(12 124)(13 135)(14 122)(15 133)(16 144)(17 131)(18 142)(19 129)(20 140)(21 127)(22 138)(23 125)(24 136)(25 86)(26 73)(27 84)(28 95)(29 82)(30 93)(31 80)(32 91)(33 78)(34 89)(35 76)(36 87)(37 74)(38 85)(39 96)(40 83)(41 94)(42 81)(43 92)(44 79)(45 90)(46 77)(47 88)(48 75)(49 103)(50 114)(51 101)(52 112)(53 99)(54 110)(55 97)(56 108)(57 119)(58 106)(59 117)(60 104)(61 115)(62 102)(63 113)(64 100)(65 111)(66 98)(67 109)(68 120)(69 107)(70 118)(71 105)(72 116)
G:=sub<Sym(144)| (1,51,74)(2,52,75)(3,53,76)(4,54,77)(5,55,78)(6,56,79)(7,57,80)(8,58,81)(9,59,82)(10,60,83)(11,61,84)(12,62,85)(13,63,86)(14,64,87)(15,65,88)(16,66,89)(17,67,90)(18,68,91)(19,69,92)(20,70,93)(21,71,94)(22,72,95)(23,49,96)(24,50,73)(25,135,113)(26,136,114)(27,137,115)(28,138,116)(29,139,117)(30,140,118)(31,141,119)(32,142,120)(33,143,97)(34,144,98)(35,121,99)(36,122,100)(37,123,101)(38,124,102)(39,125,103)(40,126,104)(41,127,105)(42,128,106)(43,129,107)(44,130,108)(45,131,109)(46,132,110)(47,133,111)(48,134,112), (1,9,17)(2,10,18)(3,11,19)(4,12,20)(5,13,21)(6,14,22)(7,15,23)(8,16,24)(25,41,33)(26,42,34)(27,43,35)(28,44,36)(29,45,37)(30,46,38)(31,47,39)(32,48,40)(49,57,65)(50,58,66)(51,59,67)(52,60,68)(53,61,69)(54,62,70)(55,63,71)(56,64,72)(73,81,89)(74,82,90)(75,83,91)(76,84,92)(77,85,93)(78,86,94)(79,87,95)(80,88,96)(97,113,105)(98,114,106)(99,115,107)(100,116,108)(101,117,109)(102,118,110)(103,119,111)(104,120,112)(121,137,129)(122,138,130)(123,139,131)(124,140,132)(125,141,133)(126,142,134)(127,143,135)(128,144,136), (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), (1,123)(2,134)(3,121)(4,132)(5,143)(6,130)(7,141)(8,128)(9,139)(10,126)(11,137)(12,124)(13,135)(14,122)(15,133)(16,144)(17,131)(18,142)(19,129)(20,140)(21,127)(22,138)(23,125)(24,136)(25,86)(26,73)(27,84)(28,95)(29,82)(30,93)(31,80)(32,91)(33,78)(34,89)(35,76)(36,87)(37,74)(38,85)(39,96)(40,83)(41,94)(42,81)(43,92)(44,79)(45,90)(46,77)(47,88)(48,75)(49,103)(50,114)(51,101)(52,112)(53,99)(54,110)(55,97)(56,108)(57,119)(58,106)(59,117)(60,104)(61,115)(62,102)(63,113)(64,100)(65,111)(66,98)(67,109)(68,120)(69,107)(70,118)(71,105)(72,116)>;
G:=Group( (1,51,74)(2,52,75)(3,53,76)(4,54,77)(5,55,78)(6,56,79)(7,57,80)(8,58,81)(9,59,82)(10,60,83)(11,61,84)(12,62,85)(13,63,86)(14,64,87)(15,65,88)(16,66,89)(17,67,90)(18,68,91)(19,69,92)(20,70,93)(21,71,94)(22,72,95)(23,49,96)(24,50,73)(25,135,113)(26,136,114)(27,137,115)(28,138,116)(29,139,117)(30,140,118)(31,141,119)(32,142,120)(33,143,97)(34,144,98)(35,121,99)(36,122,100)(37,123,101)(38,124,102)(39,125,103)(40,126,104)(41,127,105)(42,128,106)(43,129,107)(44,130,108)(45,131,109)(46,132,110)(47,133,111)(48,134,112), (1,9,17)(2,10,18)(3,11,19)(4,12,20)(5,13,21)(6,14,22)(7,15,23)(8,16,24)(25,41,33)(26,42,34)(27,43,35)(28,44,36)(29,45,37)(30,46,38)(31,47,39)(32,48,40)(49,57,65)(50,58,66)(51,59,67)(52,60,68)(53,61,69)(54,62,70)(55,63,71)(56,64,72)(73,81,89)(74,82,90)(75,83,91)(76,84,92)(77,85,93)(78,86,94)(79,87,95)(80,88,96)(97,113,105)(98,114,106)(99,115,107)(100,116,108)(101,117,109)(102,118,110)(103,119,111)(104,120,112)(121,137,129)(122,138,130)(123,139,131)(124,140,132)(125,141,133)(126,142,134)(127,143,135)(128,144,136), (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), (1,123)(2,134)(3,121)(4,132)(5,143)(6,130)(7,141)(8,128)(9,139)(10,126)(11,137)(12,124)(13,135)(14,122)(15,133)(16,144)(17,131)(18,142)(19,129)(20,140)(21,127)(22,138)(23,125)(24,136)(25,86)(26,73)(27,84)(28,95)(29,82)(30,93)(31,80)(32,91)(33,78)(34,89)(35,76)(36,87)(37,74)(38,85)(39,96)(40,83)(41,94)(42,81)(43,92)(44,79)(45,90)(46,77)(47,88)(48,75)(49,103)(50,114)(51,101)(52,112)(53,99)(54,110)(55,97)(56,108)(57,119)(58,106)(59,117)(60,104)(61,115)(62,102)(63,113)(64,100)(65,111)(66,98)(67,109)(68,120)(69,107)(70,118)(71,105)(72,116) );
G=PermutationGroup([[(1,51,74),(2,52,75),(3,53,76),(4,54,77),(5,55,78),(6,56,79),(7,57,80),(8,58,81),(9,59,82),(10,60,83),(11,61,84),(12,62,85),(13,63,86),(14,64,87),(15,65,88),(16,66,89),(17,67,90),(18,68,91),(19,69,92),(20,70,93),(21,71,94),(22,72,95),(23,49,96),(24,50,73),(25,135,113),(26,136,114),(27,137,115),(28,138,116),(29,139,117),(30,140,118),(31,141,119),(32,142,120),(33,143,97),(34,144,98),(35,121,99),(36,122,100),(37,123,101),(38,124,102),(39,125,103),(40,126,104),(41,127,105),(42,128,106),(43,129,107),(44,130,108),(45,131,109),(46,132,110),(47,133,111),(48,134,112)], [(1,9,17),(2,10,18),(3,11,19),(4,12,20),(5,13,21),(6,14,22),(7,15,23),(8,16,24),(25,41,33),(26,42,34),(27,43,35),(28,44,36),(29,45,37),(30,46,38),(31,47,39),(32,48,40),(49,57,65),(50,58,66),(51,59,67),(52,60,68),(53,61,69),(54,62,70),(55,63,71),(56,64,72),(73,81,89),(74,82,90),(75,83,91),(76,84,92),(77,85,93),(78,86,94),(79,87,95),(80,88,96),(97,113,105),(98,114,106),(99,115,107),(100,116,108),(101,117,109),(102,118,110),(103,119,111),(104,120,112),(121,137,129),(122,138,130),(123,139,131),(124,140,132),(125,141,133),(126,142,134),(127,143,135),(128,144,136)], [(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)], [(1,123),(2,134),(3,121),(4,132),(5,143),(6,130),(7,141),(8,128),(9,139),(10,126),(11,137),(12,124),(13,135),(14,122),(15,133),(16,144),(17,131),(18,142),(19,129),(20,140),(21,127),(22,138),(23,125),(24,136),(25,86),(26,73),(27,84),(28,95),(29,82),(30,93),(31,80),(32,91),(33,78),(34,89),(35,76),(36,87),(37,74),(38,85),(39,96),(40,83),(41,94),(42,81),(43,92),(44,79),(45,90),(46,77),(47,88),(48,75),(49,103),(50,114),(51,101),(52,112),(53,99),(54,110),(55,97),(56,108),(57,119),(58,106),(59,117),(60,104),(61,115),(62,102),(63,113),(64,100),(65,111),(66,98),(67,109),(68,120),(69,107),(70,118),(71,105),(72,116)]])
135 conjugacy classes
| class | 1 | 2A | 2B | 3A | ··· | 3H | 3I | ··· | 3Q | 4A | 4B | 6A | ··· | 6H | 6I | ··· | 6Q | 6R | ··· | 6Y | 8A | 8B | 12A | ··· | 12Z | 12AA | ··· | 12AH | 24A | ··· | 24AZ |
| order | 1 | 2 | 2 | 3 | ··· | 3 | 3 | ··· | 3 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 6 | ··· | 6 | 8 | 8 | 12 | ··· | 12 | 12 | ··· | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 12 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | 12 | 1 | ··· | 1 | 2 | ··· | 2 | 12 | ··· | 12 | 2 | 2 | 2 | ··· | 2 | 12 | ··· | 12 | 2 | ··· | 2 |
135 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | ||||||||||||
| image | C1 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | S3 | D4 | D6 | SD16 | C3×S3 | D12 | C3×D4 | S3×C6 | C24⋊C2 | C3×SD16 | C3×D12 | C3×C24⋊C2 |
| kernel | C32×C24⋊C2 | C32×C24 | C32×Dic6 | C32×D12 | C3×C24⋊C2 | C3×C24 | C3×Dic6 | C3×D12 | C3×C24 | C32×C6 | C3×C12 | C33 | C24 | C3×C6 | C3×C6 | C12 | C32 | C32 | C6 | C3 |
| # reps | 1 | 1 | 1 | 1 | 8 | 8 | 8 | 8 | 1 | 1 | 1 | 2 | 8 | 2 | 8 | 8 | 4 | 16 | 16 | 32 |
Matrix representation of C32×C24⋊C2 ►in GL4(𝔽73) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 64 | 0 |
| 0 | 0 | 0 | 64 |
| 64 | 0 | 0 | 0 |
| 0 | 64 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 3 | 0 | 0 | 0 |
| 71 | 49 | 0 | 0 |
| 0 | 0 | 1 | 18 |
| 0 | 0 | 48 | 60 |
| 60 | 7 | 0 | 0 |
| 49 | 13 | 0 | 0 |
| 0 | 0 | 22 | 42 |
| 0 | 0 | 25 | 51 |
G:=sub<GL(4,GF(73))| [1,0,0,0,0,1,0,0,0,0,64,0,0,0,0,64],[64,0,0,0,0,64,0,0,0,0,1,0,0,0,0,1],[3,71,0,0,0,49,0,0,0,0,1,48,0,0,18,60],[60,49,0,0,7,13,0,0,0,0,22,25,0,0,42,51] >;
C32×C24⋊C2 in GAP, Magma, Sage, TeX
C_3^2\times C_{24}\rtimes C_2 % in TeX
G:=Group("C3^2xC24:C2"); // GroupNames label
G:=SmallGroup(432,466);
// by ID
G=gap.SmallGroup(432,466);
# by ID
G:=PCGroup([7,-2,-2,-3,-3,-2,-2,-3,533,260,3784,102,14118]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^3=c^24=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^11>;
// generators/relations