direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C72⋊C2, C8⋊8D18, C4.7D36, C72⋊9C22, C18⋊1SD16, C24.76D6, C36.30D4, C12.41D12, C36.29C23, D36.6C22, C22.13D36, Dic18⋊3C22, (C2×C8)⋊5D9, (C2×C72)⋊7C2, C9⋊1(C2×SD16), (C2×C24).18S3, (C2×D36).4C2, C6.39(C2×D12), C2.12(C2×D36), C18.10(C2×D4), (C2×C6).26D12, (C2×C4).81D18, (C2×C18).17D4, C6.4(C24⋊C2), (C2×Dic18)⋊5C2, (C2×C12).371D6, C4.27(C22×D9), (C2×C36).90C22, C12.180(C22×S3), C3.(C2×C24⋊C2), SmallGroup(288,113)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C72⋊C2
G = < a,b,c | a2=b72=c2=1, ab=ba, ac=ca, cbc=b35 >
Subgroups: 592 in 102 conjugacy classes, 44 normal (24 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, D4, Q8, C23, C9, Dic3, C12, D6, C2×C6, C2×C8, SD16, C2×D4, C2×Q8, D9, C18, C18, C24, Dic6, D12, C2×Dic3, C2×C12, C22×S3, C2×SD16, Dic9, C36, D18, C2×C18, C24⋊C2, C2×C24, C2×Dic6, C2×D12, C72, Dic18, Dic18, D36, D36, C2×Dic9, C2×C36, C22×D9, C2×C24⋊C2, C72⋊C2, C2×C72, C2×Dic18, C2×D36, C2×C72⋊C2
Quotients: C1, C2, C22, S3, D4, C23, D6, SD16, C2×D4, D9, D12, C22×S3, C2×SD16, D18, C24⋊C2, C2×D12, D36, C22×D9, C2×C24⋊C2, C72⋊C2, C2×D36, C2×C72⋊C2
►On 144 points
Generators in S144
(1 115)(2 116)(3 117)(4 118)(5 119)(6 120)(7 121)(8 122)(9 123)(10 124)(11 125)(12 126)(13 127)(14 128)(15 129)(16 130)(17 131)(18 132)(19 133)(20 134)(21 135)(22 136)(23 137)(24 138)(25 139)(26 140)(27 141)(28 142)(29 143)(30 144)(31 73)(32 74)(33 75)(34 76)(35 77)(36 78)(37 79)(38 80)(39 81)(40 82)(41 83)(42 84)(43 85)(44 86)(45 87)(46 88)(47 89)(48 90)(49 91)(50 92)(51 93)(52 94)(53 95)(54 96)(55 97)(56 98)(57 99)(58 100)(59 101)(60 102)(61 103)(62 104)(63 105)(64 106)(65 107)(66 108)(67 109)(68 110)(69 111)(70 112)(71 113)(72 114)
(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)
(1 115)(2 78)(3 113)(4 76)(5 111)(6 74)(7 109)(8 144)(9 107)(10 142)(11 105)(12 140)(13 103)(14 138)(15 101)(16 136)(17 99)(18 134)(19 97)(20 132)(21 95)(22 130)(23 93)(24 128)(25 91)(26 126)(27 89)(28 124)(29 87)(30 122)(31 85)(32 120)(33 83)(34 118)(35 81)(36 116)(37 79)(38 114)(39 77)(40 112)(41 75)(42 110)(43 73)(44 108)(45 143)(46 106)(47 141)(48 104)(49 139)(50 102)(51 137)(52 100)(53 135)(54 98)(55 133)(56 96)(57 131)(58 94)(59 129)(60 92)(61 127)(62 90)(63 125)(64 88)(65 123)(66 86)(67 121)(68 84)(69 119)(70 82)(71 117)(72 80)
G:=sub<Sym(144)| (1,115)(2,116)(3,117)(4,118)(5,119)(6,120)(7,121)(8,122)(9,123)(10,124)(11,125)(12,126)(13,127)(14,128)(15,129)(16,130)(17,131)(18,132)(19,133)(20,134)(21,135)(22,136)(23,137)(24,138)(25,139)(26,140)(27,141)(28,142)(29,143)(30,144)(31,73)(32,74)(33,75)(34,76)(35,77)(36,78)(37,79)(38,80)(39,81)(40,82)(41,83)(42,84)(43,85)(44,86)(45,87)(46,88)(47,89)(48,90)(49,91)(50,92)(51,93)(52,94)(53,95)(54,96)(55,97)(56,98)(57,99)(58,100)(59,101)(60,102)(61,103)(62,104)(63,105)(64,106)(65,107)(66,108)(67,109)(68,110)(69,111)(70,112)(71,113)(72,114), (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), (1,115)(2,78)(3,113)(4,76)(5,111)(6,74)(7,109)(8,144)(9,107)(10,142)(11,105)(12,140)(13,103)(14,138)(15,101)(16,136)(17,99)(18,134)(19,97)(20,132)(21,95)(22,130)(23,93)(24,128)(25,91)(26,126)(27,89)(28,124)(29,87)(30,122)(31,85)(32,120)(33,83)(34,118)(35,81)(36,116)(37,79)(38,114)(39,77)(40,112)(41,75)(42,110)(43,73)(44,108)(45,143)(46,106)(47,141)(48,104)(49,139)(50,102)(51,137)(52,100)(53,135)(54,98)(55,133)(56,96)(57,131)(58,94)(59,129)(60,92)(61,127)(62,90)(63,125)(64,88)(65,123)(66,86)(67,121)(68,84)(69,119)(70,82)(71,117)(72,80)>;
G:=Group( (1,115)(2,116)(3,117)(4,118)(5,119)(6,120)(7,121)(8,122)(9,123)(10,124)(11,125)(12,126)(13,127)(14,128)(15,129)(16,130)(17,131)(18,132)(19,133)(20,134)(21,135)(22,136)(23,137)(24,138)(25,139)(26,140)(27,141)(28,142)(29,143)(30,144)(31,73)(32,74)(33,75)(34,76)(35,77)(36,78)(37,79)(38,80)(39,81)(40,82)(41,83)(42,84)(43,85)(44,86)(45,87)(46,88)(47,89)(48,90)(49,91)(50,92)(51,93)(52,94)(53,95)(54,96)(55,97)(56,98)(57,99)(58,100)(59,101)(60,102)(61,103)(62,104)(63,105)(64,106)(65,107)(66,108)(67,109)(68,110)(69,111)(70,112)(71,113)(72,114), (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), (1,115)(2,78)(3,113)(4,76)(5,111)(6,74)(7,109)(8,144)(9,107)(10,142)(11,105)(12,140)(13,103)(14,138)(15,101)(16,136)(17,99)(18,134)(19,97)(20,132)(21,95)(22,130)(23,93)(24,128)(25,91)(26,126)(27,89)(28,124)(29,87)(30,122)(31,85)(32,120)(33,83)(34,118)(35,81)(36,116)(37,79)(38,114)(39,77)(40,112)(41,75)(42,110)(43,73)(44,108)(45,143)(46,106)(47,141)(48,104)(49,139)(50,102)(51,137)(52,100)(53,135)(54,98)(55,133)(56,96)(57,131)(58,94)(59,129)(60,92)(61,127)(62,90)(63,125)(64,88)(65,123)(66,86)(67,121)(68,84)(69,119)(70,82)(71,117)(72,80) );
G=PermutationGroup([[(1,115),(2,116),(3,117),(4,118),(5,119),(6,120),(7,121),(8,122),(9,123),(10,124),(11,125),(12,126),(13,127),(14,128),(15,129),(16,130),(17,131),(18,132),(19,133),(20,134),(21,135),(22,136),(23,137),(24,138),(25,139),(26,140),(27,141),(28,142),(29,143),(30,144),(31,73),(32,74),(33,75),(34,76),(35,77),(36,78),(37,79),(38,80),(39,81),(40,82),(41,83),(42,84),(43,85),(44,86),(45,87),(46,88),(47,89),(48,90),(49,91),(50,92),(51,93),(52,94),(53,95),(54,96),(55,97),(56,98),(57,99),(58,100),(59,101),(60,102),(61,103),(62,104),(63,105),(64,106),(65,107),(66,108),(67,109),(68,110),(69,111),(70,112),(71,113),(72,114)], [(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)], [(1,115),(2,78),(3,113),(4,76),(5,111),(6,74),(7,109),(8,144),(9,107),(10,142),(11,105),(12,140),(13,103),(14,138),(15,101),(16,136),(17,99),(18,134),(19,97),(20,132),(21,95),(22,130),(23,93),(24,128),(25,91),(26,126),(27,89),(28,124),(29,87),(30,122),(31,85),(32,120),(33,83),(34,118),(35,81),(36,116),(37,79),(38,114),(39,77),(40,112),(41,75),(42,110),(43,73),(44,108),(45,143),(46,106),(47,141),(48,104),(49,139),(50,102),(51,137),(52,100),(53,135),(54,98),(55,133),(56,96),(57,131),(58,94),(59,129),(60,92),(61,127),(62,90),(63,125),(64,88),(65,123),(66,86),(67,121),(68,84),(69,119),(70,82),(71,117),(72,80)]])
78 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 9A | 9B | 9C | 12A | 12B | 12C | 12D | 18A | ··· | 18I | 24A | ··· | 24H | 36A | ··· | 36L | 72A | ··· | 72X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 9 | 9 | 9 | 12 | 12 | 12 | 12 | 18 | ··· | 18 | 24 | ··· | 24 | 36 | ··· | 36 | 72 | ··· | 72 |
| size | 1 | 1 | 1 | 1 | 36 | 36 | 2 | 2 | 2 | 36 | 36 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
78 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D6 | D6 | SD16 | D9 | D12 | D12 | D18 | D18 | C24⋊C2 | D36 | D36 | C72⋊C2 |
| kernel | C2×C72⋊C2 | C72⋊C2 | C2×C72 | C2×Dic18 | C2×D36 | C2×C24 | C36 | C2×C18 | C24 | C2×C12 | C18 | C2×C8 | C12 | C2×C6 | C8 | C2×C4 | C6 | C4 | C22 | C2 |
| # reps | 1 | 4 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 4 | 3 | 2 | 2 | 6 | 3 | 8 | 6 | 6 | 24 |
Matrix representation of C2×C72⋊C2 ►in GL3(𝔽73) generated by
| 72 | 0 | 0 |
| 0 | 72 | 0 |
| 0 | 0 | 72 |
| 72 | 0 | 0 |
| 0 | 54 | 5 |
| 0 | 68 | 59 |
| 1 | 0 | 0 |
| 0 | 0 | 72 |
| 0 | 72 | 0 |
G:=sub<GL(3,GF(73))| [72,0,0,0,72,0,0,0,72],[72,0,0,0,54,68,0,5,59],[1,0,0,0,0,72,0,72,0] >;
C2×C72⋊C2 in GAP, Magma, Sage, TeX
C_2\times C_{72}\rtimes C_2 % in TeX
G:=Group("C2xC72:C2"); // GroupNames label
G:=SmallGroup(288,113);
// by ID
G=gap.SmallGroup(288,113);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,254,58,675,80,6725,292,9414]);
// Polycyclic
G:=Group<a,b,c|a^2=b^72=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^35>;
// generators/relations