direct product, metacyclic, supersoluble, monomial
Aliases: C3×C72⋊C2, C24⋊6D9, C72⋊10C6, D36.3C6, C6.20D36, Dic18⋊7C6, C12.72D18, C8⋊2(C3×D9), (C3×C72)⋊4C2, C4.9(C6×D9), (C3×C9)⋊8SD16, C24.8(C3×S3), C9⋊5(C3×SD16), C6.2(C3×D12), C2.4(C3×D36), C12.65(S3×C6), (C3×C24).18S3, C36.32(C2×C6), (C3×D36).2C2, C18.18(C3×D4), (C3×C18).25D4, (C3×C6).50D12, (C3×Dic18)⋊2C2, (C3×C12).203D6, (C3×C36).56C22, C32.4(C24⋊C2), C3.1(C3×C24⋊C2), SmallGroup(432,107)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C72⋊C2
G = < a,b,c | a3=b72=c2=1, ab=ba, ac=ca, cbc=b35 >
Subgroups: 326 in 68 conjugacy classes, 30 normal (all characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C8, D4, Q8, C9, C9, C32, Dic3, C12, C12, D6, C2×C6, SD16, D9, C18, C18, C3×S3, C3×C6, C24, C24, Dic6, D12, C3×D4, C3×Q8, C3×C9, Dic9, C36, C36, D18, C3×Dic3, C3×C12, S3×C6, C24⋊C2, C3×SD16, C3×D9, C3×C18, C72, C72, Dic18, D36, C3×C24, C3×Dic6, C3×D12, C3×Dic9, C3×C36, C6×D9, C72⋊C2, C3×C24⋊C2, C3×C72, C3×Dic18, C3×D36, C3×C72⋊C2
Quotients: C1, C2, C3, C22, S3, C6, D4, D6, C2×C6, SD16, D9, C3×S3, D12, C3×D4, D18, S3×C6, C24⋊C2, C3×SD16, C3×D9, D36, C3×D12, C6×D9, C72⋊C2, C3×C24⋊C2, C3×D36, C3×C72⋊C2
►On 144 points
Generators in S144
(1 49 25)(2 50 26)(3 51 27)(4 52 28)(5 53 29)(6 54 30)(7 55 31)(8 56 32)(9 57 33)(10 58 34)(11 59 35)(12 60 36)(13 61 37)(14 62 38)(15 63 39)(16 64 40)(17 65 41)(18 66 42)(19 67 43)(20 68 44)(21 69 45)(22 70 46)(23 71 47)(24 72 48)(73 97 121)(74 98 122)(75 99 123)(76 100 124)(77 101 125)(78 102 126)(79 103 127)(80 104 128)(81 105 129)(82 106 130)(83 107 131)(84 108 132)(85 109 133)(86 110 134)(87 111 135)(88 112 136)(89 113 137)(90 114 138)(91 115 139)(92 116 140)(93 117 141)(94 118 142)(95 119 143)(96 120 144)
(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 94)(2 129)(3 92)(4 127)(5 90)(6 125)(7 88)(8 123)(9 86)(10 121)(11 84)(12 119)(13 82)(14 117)(15 80)(16 115)(17 78)(18 113)(19 76)(20 111)(21 74)(22 109)(23 144)(24 107)(25 142)(26 105)(27 140)(28 103)(29 138)(30 101)(31 136)(32 99)(33 134)(34 97)(35 132)(36 95)(37 130)(38 93)(39 128)(40 91)(41 126)(42 89)(43 124)(44 87)(45 122)(46 85)(47 120)(48 83)(49 118)(50 81)(51 116)(52 79)(53 114)(54 77)(55 112)(56 75)(57 110)(58 73)(59 108)(60 143)(61 106)(62 141)(63 104)(64 139)(65 102)(66 137)(67 100)(68 135)(69 98)(70 133)(71 96)(72 131)
G:=sub<Sym(144)| (1,49,25)(2,50,26)(3,51,27)(4,52,28)(5,53,29)(6,54,30)(7,55,31)(8,56,32)(9,57,33)(10,58,34)(11,59,35)(12,60,36)(13,61,37)(14,62,38)(15,63,39)(16,64,40)(17,65,41)(18,66,42)(19,67,43)(20,68,44)(21,69,45)(22,70,46)(23,71,47)(24,72,48)(73,97,121)(74,98,122)(75,99,123)(76,100,124)(77,101,125)(78,102,126)(79,103,127)(80,104,128)(81,105,129)(82,106,130)(83,107,131)(84,108,132)(85,109,133)(86,110,134)(87,111,135)(88,112,136)(89,113,137)(90,114,138)(91,115,139)(92,116,140)(93,117,141)(94,118,142)(95,119,143)(96,120,144), (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), (1,94)(2,129)(3,92)(4,127)(5,90)(6,125)(7,88)(8,123)(9,86)(10,121)(11,84)(12,119)(13,82)(14,117)(15,80)(16,115)(17,78)(18,113)(19,76)(20,111)(21,74)(22,109)(23,144)(24,107)(25,142)(26,105)(27,140)(28,103)(29,138)(30,101)(31,136)(32,99)(33,134)(34,97)(35,132)(36,95)(37,130)(38,93)(39,128)(40,91)(41,126)(42,89)(43,124)(44,87)(45,122)(46,85)(47,120)(48,83)(49,118)(50,81)(51,116)(52,79)(53,114)(54,77)(55,112)(56,75)(57,110)(58,73)(59,108)(60,143)(61,106)(62,141)(63,104)(64,139)(65,102)(66,137)(67,100)(68,135)(69,98)(70,133)(71,96)(72,131)>;
G:=Group( (1,49,25)(2,50,26)(3,51,27)(4,52,28)(5,53,29)(6,54,30)(7,55,31)(8,56,32)(9,57,33)(10,58,34)(11,59,35)(12,60,36)(13,61,37)(14,62,38)(15,63,39)(16,64,40)(17,65,41)(18,66,42)(19,67,43)(20,68,44)(21,69,45)(22,70,46)(23,71,47)(24,72,48)(73,97,121)(74,98,122)(75,99,123)(76,100,124)(77,101,125)(78,102,126)(79,103,127)(80,104,128)(81,105,129)(82,106,130)(83,107,131)(84,108,132)(85,109,133)(86,110,134)(87,111,135)(88,112,136)(89,113,137)(90,114,138)(91,115,139)(92,116,140)(93,117,141)(94,118,142)(95,119,143)(96,120,144), (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), (1,94)(2,129)(3,92)(4,127)(5,90)(6,125)(7,88)(8,123)(9,86)(10,121)(11,84)(12,119)(13,82)(14,117)(15,80)(16,115)(17,78)(18,113)(19,76)(20,111)(21,74)(22,109)(23,144)(24,107)(25,142)(26,105)(27,140)(28,103)(29,138)(30,101)(31,136)(32,99)(33,134)(34,97)(35,132)(36,95)(37,130)(38,93)(39,128)(40,91)(41,126)(42,89)(43,124)(44,87)(45,122)(46,85)(47,120)(48,83)(49,118)(50,81)(51,116)(52,79)(53,114)(54,77)(55,112)(56,75)(57,110)(58,73)(59,108)(60,143)(61,106)(62,141)(63,104)(64,139)(65,102)(66,137)(67,100)(68,135)(69,98)(70,133)(71,96)(72,131) );
G=PermutationGroup([[(1,49,25),(2,50,26),(3,51,27),(4,52,28),(5,53,29),(6,54,30),(7,55,31),(8,56,32),(9,57,33),(10,58,34),(11,59,35),(12,60,36),(13,61,37),(14,62,38),(15,63,39),(16,64,40),(17,65,41),(18,66,42),(19,67,43),(20,68,44),(21,69,45),(22,70,46),(23,71,47),(24,72,48),(73,97,121),(74,98,122),(75,99,123),(76,100,124),(77,101,125),(78,102,126),(79,103,127),(80,104,128),(81,105,129),(82,106,130),(83,107,131),(84,108,132),(85,109,133),(86,110,134),(87,111,135),(88,112,136),(89,113,137),(90,114,138),(91,115,139),(92,116,140),(93,117,141),(94,118,142),(95,119,143),(96,120,144)], [(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)], [(1,94),(2,129),(3,92),(4,127),(5,90),(6,125),(7,88),(8,123),(9,86),(10,121),(11,84),(12,119),(13,82),(14,117),(15,80),(16,115),(17,78),(18,113),(19,76),(20,111),(21,74),(22,109),(23,144),(24,107),(25,142),(26,105),(27,140),(28,103),(29,138),(30,101),(31,136),(32,99),(33,134),(34,97),(35,132),(36,95),(37,130),(38,93),(39,128),(40,91),(41,126),(42,89),(43,124),(44,87),(45,122),(46,85),(47,120),(48,83),(49,118),(50,81),(51,116),(52,79),(53,114),(54,77),(55,112),(56,75),(57,110),(58,73),(59,108),(60,143),(61,106),(62,141),(63,104),(64,139),(65,102),(66,137),(67,100),(68,135),(69,98),(70,133),(71,96),(72,131)]])
117 conjugacy classes
| class | 1 | 2A | 2B | 3A | 3B | 3C | 3D | 3E | 4A | 4B | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 8A | 8B | 9A | ··· | 9I | 12A | ··· | 12H | 12I | 12J | 18A | ··· | 18I | 24A | ··· | 24P | 36A | ··· | 36R | 72A | ··· | 72AJ |
| order | 1 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 9 | ··· | 9 | 12 | ··· | 12 | 12 | 12 | 18 | ··· | 18 | 24 | ··· | 24 | 36 | ··· | 36 | 72 | ··· | 72 |
| size | 1 | 1 | 36 | 1 | 1 | 2 | 2 | 2 | 2 | 36 | 1 | 1 | 2 | 2 | 2 | 36 | 36 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 36 | 36 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
117 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 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 | D9 | C3×S3 | C3×D4 | D12 | D18 | S3×C6 | C3×SD16 | C24⋊C2 | C3×D9 | D36 | C3×D12 | C6×D9 | C72⋊C2 | C3×C24⋊C2 | C3×D36 | C3×C72⋊C2 |
| kernel | C3×C72⋊C2 | C3×C72 | C3×Dic18 | C3×D36 | C72⋊C2 | C72 | Dic18 | D36 | C3×C24 | C3×C18 | C3×C12 | C3×C9 | C24 | C24 | C18 | C3×C6 | C12 | C12 | C9 | C32 | C8 | C6 | C6 | C4 | C3 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 2 | 3 | 2 | 2 | 2 | 3 | 2 | 4 | 4 | 6 | 6 | 4 | 6 | 12 | 8 | 12 | 24 |
Matrix representation of C3×C72⋊C2 ►in GL2(𝔽73) generated by
| 64 | 0 |
| 0 | 64 |
| 5 | 0 |
| 5 | 29 |
| 41 | 7 |
| 52 | 32 |
G:=sub<GL(2,GF(73))| [64,0,0,64],[5,5,0,29],[41,52,7,32] >;
C3×C72⋊C2 in GAP, Magma, Sage, TeX
C_3\times C_{72}\rtimes C_2 % in TeX
G:=Group("C3xC72:C2"); // GroupNames label
G:=SmallGroup(432,107);
// by ID
G=gap.SmallGroup(432,107);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-2,-3,-3,197,92,1011,80,10085,292,14118]);
// Polycyclic
G:=Group<a,b,c|a^3=b^72=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^35>;
// generators/relations