Copied to
clipboard

G = C2×C72⋊C2order 288 = 25·32

Direct product of C2 and C72⋊C2

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C2×C72⋊C2, C88D18, C4.7D36, C729C22, C181SD16, C24.76D6, C36.30D4, C12.41D12, C36.29C23, D36.6C22, C22.13D36, Dic183C22, (C2×C8)⋊5D9, (C2×C72)⋊7C2, C91(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

C1C36 — C2×C72⋊C2
C1C3C9C18C36D36C2×D36 — C2×C72⋊C2
C9C18C36 — C2×C72⋊C2
C1C22C2×C4C2×C8

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 [×2], C2 [×2], C3, C4 [×2], C4 [×2], C22, C22 [×4], S3 [×2], C6, C6 [×2], C8 [×2], C2×C4, C2×C4, D4 [×3], Q8 [×3], C23, C9, Dic3 [×2], C12 [×2], D6 [×4], C2×C6, C2×C8, SD16 [×4], C2×D4, C2×Q8, D9 [×2], C18, C18 [×2], C24 [×2], Dic6 [×3], D12 [×3], C2×Dic3, C2×C12, C22×S3, C2×SD16, Dic9 [×2], C36 [×2], D18 [×4], C2×C18, C24⋊C2 [×4], C2×C24, C2×Dic6, C2×D12, C72 [×2], Dic18 [×2], Dic18, D36 [×2], D36, C2×Dic9, C2×C36, C22×D9, C2×C24⋊C2, C72⋊C2 [×4], C2×C72, C2×Dic18, C2×D36, C2×C72⋊C2
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D6 [×3], SD16 [×2], C2×D4, D9, D12 [×2], C22×S3, C2×SD16, D18 [×3], C24⋊C2 [×2], C2×D12, D36 [×2], C22×D9, C2×C24⋊C2, C72⋊C2 [×2], C2×D36, C2×C72⋊C2

Smallest permutation representation of C2×C72⋊C2
On 144 points
Generators in S144
(1 92)(2 93)(3 94)(4 95)(5 96)(6 97)(7 98)(8 99)(9 100)(10 101)(11 102)(12 103)(13 104)(14 105)(15 106)(16 107)(17 108)(18 109)(19 110)(20 111)(21 112)(22 113)(23 114)(24 115)(25 116)(26 117)(27 118)(28 119)(29 120)(30 121)(31 122)(32 123)(33 124)(34 125)(35 126)(36 127)(37 128)(38 129)(39 130)(40 131)(41 132)(42 133)(43 134)(44 135)(45 136)(46 137)(47 138)(48 139)(49 140)(50 141)(51 142)(52 143)(53 144)(54 73)(55 74)(56 75)(57 76)(58 77)(59 78)(60 79)(61 80)(62 81)(63 82)(64 83)(65 84)(66 85)(67 86)(68 87)(69 88)(70 89)(71 90)(72 91)
(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 92)(2 127)(3 90)(4 125)(5 88)(6 123)(7 86)(8 121)(9 84)(10 119)(11 82)(12 117)(13 80)(14 115)(15 78)(16 113)(17 76)(18 111)(19 74)(20 109)(21 144)(22 107)(23 142)(24 105)(25 140)(26 103)(27 138)(28 101)(29 136)(30 99)(31 134)(32 97)(33 132)(34 95)(35 130)(36 93)(37 128)(38 91)(39 126)(40 89)(41 124)(42 87)(43 122)(44 85)(45 120)(46 83)(47 118)(48 81)(49 116)(50 79)(51 114)(52 77)(53 112)(54 75)(55 110)(56 73)(57 108)(58 143)(59 106)(60 141)(61 104)(62 139)(63 102)(64 137)(65 100)(66 135)(67 98)(68 133)(69 96)(70 131)(71 94)(72 129)

G:=sub<Sym(144)| (1,92)(2,93)(3,94)(4,95)(5,96)(6,97)(7,98)(8,99)(9,100)(10,101)(11,102)(12,103)(13,104)(14,105)(15,106)(16,107)(17,108)(18,109)(19,110)(20,111)(21,112)(22,113)(23,114)(24,115)(25,116)(26,117)(27,118)(28,119)(29,120)(30,121)(31,122)(32,123)(33,124)(34,125)(35,126)(36,127)(37,128)(38,129)(39,130)(40,131)(41,132)(42,133)(43,134)(44,135)(45,136)(46,137)(47,138)(48,139)(49,140)(50,141)(51,142)(52,143)(53,144)(54,73)(55,74)(56,75)(57,76)(58,77)(59,78)(60,79)(61,80)(62,81)(63,82)(64,83)(65,84)(66,85)(67,86)(68,87)(69,88)(70,89)(71,90)(72,91), (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), (1,92)(2,127)(3,90)(4,125)(5,88)(6,123)(7,86)(8,121)(9,84)(10,119)(11,82)(12,117)(13,80)(14,115)(15,78)(16,113)(17,76)(18,111)(19,74)(20,109)(21,144)(22,107)(23,142)(24,105)(25,140)(26,103)(27,138)(28,101)(29,136)(30,99)(31,134)(32,97)(33,132)(34,95)(35,130)(36,93)(37,128)(38,91)(39,126)(40,89)(41,124)(42,87)(43,122)(44,85)(45,120)(46,83)(47,118)(48,81)(49,116)(50,79)(51,114)(52,77)(53,112)(54,75)(55,110)(56,73)(57,108)(58,143)(59,106)(60,141)(61,104)(62,139)(63,102)(64,137)(65,100)(66,135)(67,98)(68,133)(69,96)(70,131)(71,94)(72,129)>;

G:=Group( (1,92)(2,93)(3,94)(4,95)(5,96)(6,97)(7,98)(8,99)(9,100)(10,101)(11,102)(12,103)(13,104)(14,105)(15,106)(16,107)(17,108)(18,109)(19,110)(20,111)(21,112)(22,113)(23,114)(24,115)(25,116)(26,117)(27,118)(28,119)(29,120)(30,121)(31,122)(32,123)(33,124)(34,125)(35,126)(36,127)(37,128)(38,129)(39,130)(40,131)(41,132)(42,133)(43,134)(44,135)(45,136)(46,137)(47,138)(48,139)(49,140)(50,141)(51,142)(52,143)(53,144)(54,73)(55,74)(56,75)(57,76)(58,77)(59,78)(60,79)(61,80)(62,81)(63,82)(64,83)(65,84)(66,85)(67,86)(68,87)(69,88)(70,89)(71,90)(72,91), (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), (1,92)(2,127)(3,90)(4,125)(5,88)(6,123)(7,86)(8,121)(9,84)(10,119)(11,82)(12,117)(13,80)(14,115)(15,78)(16,113)(17,76)(18,111)(19,74)(20,109)(21,144)(22,107)(23,142)(24,105)(25,140)(26,103)(27,138)(28,101)(29,136)(30,99)(31,134)(32,97)(33,132)(34,95)(35,130)(36,93)(37,128)(38,91)(39,126)(40,89)(41,124)(42,87)(43,122)(44,85)(45,120)(46,83)(47,118)(48,81)(49,116)(50,79)(51,114)(52,77)(53,112)(54,75)(55,110)(56,73)(57,108)(58,143)(59,106)(60,141)(61,104)(62,139)(63,102)(64,137)(65,100)(66,135)(67,98)(68,133)(69,96)(70,131)(71,94)(72,129) );

G=PermutationGroup([(1,92),(2,93),(3,94),(4,95),(5,96),(6,97),(7,98),(8,99),(9,100),(10,101),(11,102),(12,103),(13,104),(14,105),(15,106),(16,107),(17,108),(18,109),(19,110),(20,111),(21,112),(22,113),(23,114),(24,115),(25,116),(26,117),(27,118),(28,119),(29,120),(30,121),(31,122),(32,123),(33,124),(34,125),(35,126),(36,127),(37,128),(38,129),(39,130),(40,131),(41,132),(42,133),(43,134),(44,135),(45,136),(46,137),(47,138),(48,139),(49,140),(50,141),(51,142),(52,143),(53,144),(54,73),(55,74),(56,75),(57,76),(58,77),(59,78),(60,79),(61,80),(62,81),(63,82),(64,83),(65,84),(66,85),(67,86),(68,87),(69,88),(70,89),(71,90),(72,91)], [(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)], [(1,92),(2,127),(3,90),(4,125),(5,88),(6,123),(7,86),(8,121),(9,84),(10,119),(11,82),(12,117),(13,80),(14,115),(15,78),(16,113),(17,76),(18,111),(19,74),(20,109),(21,144),(22,107),(23,142),(24,105),(25,140),(26,103),(27,138),(28,101),(29,136),(30,99),(31,134),(32,97),(33,132),(34,95),(35,130),(36,93),(37,128),(38,91),(39,126),(40,89),(41,124),(42,87),(43,122),(44,85),(45,120),(46,83),(47,118),(48,81),(49,116),(50,79),(51,114),(52,77),(53,112),(54,75),(55,110),(56,73),(57,108),(58,143),(59,106),(60,141),(61,104),(62,139),(63,102),(64,137),(65,100),(66,135),(67,98),(68,133),(69,96),(70,131),(71,94),(72,129)])

78 conjugacy classes

class 1 2A2B2C2D2E 3 4A4B4C4D6A6B6C8A8B8C8D9A9B9C12A12B12C12D18A···18I24A···24H36A···36L72A···72X
order1222223444466688889991212121218···1824···2436···3672···72
size111136362223636222222222222222···22···22···22···2

78 irreducible representations

dim11111222222222222222
type+++++++++++++++++
imageC1C2C2C2C2S3D4D4D6D6SD16D9D12D12D18D18C24⋊C2D36D36C72⋊C2
kernelC2×C72⋊C2C72⋊C2C2×C72C2×Dic18C2×D36C2×C24C36C2×C18C24C2×C12C18C2×C8C12C2×C6C8C2×C4C6C4C22C2
# reps141111112143226386624

Matrix representation of C2×C72⋊C2 in GL3(𝔽73) generated by

7200
0720
0072
,
7200
0545
06859
,
100
0072
0720
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

׿
×
𝔽