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G = C2×C72order 144 = 24·32

Abelian group of type [2,72]

direct product, abelian, monomial, 2-elementary

Aliases: C2×C72, SmallGroup(144,23)

Series: Derived Chief Lower central Upper central

C1 — C2×C72
C1C2C6C12C36C72 — C2×C72
C1 — C2×C72
C1 — C2×C72

Generators and relations for C2×C72
 G = < a,b | a2=b72=1, ab=ba >


Smallest permutation representation of C2×C72
Regular action on 144 points
Generators in S144
(1 111)(2 112)(3 113)(4 114)(5 115)(6 116)(7 117)(8 118)(9 119)(10 120)(11 121)(12 122)(13 123)(14 124)(15 125)(16 126)(17 127)(18 128)(19 129)(20 130)(21 131)(22 132)(23 133)(24 134)(25 135)(26 136)(27 137)(28 138)(29 139)(30 140)(31 141)(32 142)(33 143)(34 144)(35 73)(36 74)(37 75)(38 76)(39 77)(40 78)(41 79)(42 80)(43 81)(44 82)(45 83)(46 84)(47 85)(48 86)(49 87)(50 88)(51 89)(52 90)(53 91)(54 92)(55 93)(56 94)(57 95)(58 96)(59 97)(60 98)(61 99)(62 100)(63 101)(64 102)(65 103)(66 104)(67 105)(68 106)(69 107)(70 108)(71 109)(72 110)
(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)| (1,111)(2,112)(3,113)(4,114)(5,115)(6,116)(7,117)(8,118)(9,119)(10,120)(11,121)(12,122)(13,123)(14,124)(15,125)(16,126)(17,127)(18,128)(19,129)(20,130)(21,131)(22,132)(23,133)(24,134)(25,135)(26,136)(27,137)(28,138)(29,139)(30,140)(31,141)(32,142)(33,143)(34,144)(35,73)(36,74)(37,75)(38,76)(39,77)(40,78)(41,79)(42,80)(43,81)(44,82)(45,83)(46,84)(47,85)(48,86)(49,87)(50,88)(51,89)(52,90)(53,91)(54,92)(55,93)(56,94)(57,95)(58,96)(59,97)(60,98)(61,99)(62,100)(63,101)(64,102)(65,103)(66,104)(67,105)(68,106)(69,107)(70,108)(71,109)(72,110), (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( (1,111)(2,112)(3,113)(4,114)(5,115)(6,116)(7,117)(8,118)(9,119)(10,120)(11,121)(12,122)(13,123)(14,124)(15,125)(16,126)(17,127)(18,128)(19,129)(20,130)(21,131)(22,132)(23,133)(24,134)(25,135)(26,136)(27,137)(28,138)(29,139)(30,140)(31,141)(32,142)(33,143)(34,144)(35,73)(36,74)(37,75)(38,76)(39,77)(40,78)(41,79)(42,80)(43,81)(44,82)(45,83)(46,84)(47,85)(48,86)(49,87)(50,88)(51,89)(52,90)(53,91)(54,92)(55,93)(56,94)(57,95)(58,96)(59,97)(60,98)(61,99)(62,100)(63,101)(64,102)(65,103)(66,104)(67,105)(68,106)(69,107)(70,108)(71,109)(72,110), (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([(1,111),(2,112),(3,113),(4,114),(5,115),(6,116),(7,117),(8,118),(9,119),(10,120),(11,121),(12,122),(13,123),(14,124),(15,125),(16,126),(17,127),(18,128),(19,129),(20,130),(21,131),(22,132),(23,133),(24,134),(25,135),(26,136),(27,137),(28,138),(29,139),(30,140),(31,141),(32,142),(33,143),(34,144),(35,73),(36,74),(37,75),(38,76),(39,77),(40,78),(41,79),(42,80),(43,81),(44,82),(45,83),(46,84),(47,85),(48,86),(49,87),(50,88),(51,89),(52,90),(53,91),(54,92),(55,93),(56,94),(57,95),(58,96),(59,97),(60,98),(61,99),(62,100),(63,101),(64,102),(65,103),(66,104),(67,105),(68,106),(69,107),(70,108),(71,109),(72,110)], [(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×C72 is a maximal subgroup of
C36.C8  C72.C4  Dic9⋊C8  C72⋊C4  C36.45D4  C8⋊Dic9  C721C4  D18⋊C8  C2.D72  D36.2C4  D727C2

144 conjugacy classes

class 1 2A2B2C3A3B4A4B4C4D6A···6F8A···8H9A···9F12A···12H18A···18R24A···24P36A···36X72A···72AV
order12223344446···68···89···912···1218···1824···2436···3672···72
size11111111111···11···11···11···11···11···11···11···1

144 irreducible representations

dim111111111111111111
type+++
imageC1C2C2C3C4C4C6C6C8C9C12C12C18C18C24C36C36C72
kernelC2×C72C72C2×C36C2×C24C36C2×C18C24C2×C12C18C2×C8C12C2×C6C8C2×C4C6C4C22C2
# reps12122242864412616121248

Matrix representation of C2×C72 in GL2(𝔽73) generated by

720
01
,
100
014
G:=sub<GL(2,GF(73))| [72,0,0,1],[10,0,0,14] >;

C2×C72 in GAP, Magma, Sage, TeX

C_2\times C_{72}
% in TeX

G:=Group("C2xC72");
// GroupNames label

G:=SmallGroup(144,23);
// by ID

G=gap.SmallGroup(144,23);
# by ID

G:=PCGroup([6,-2,-2,-3,-2,-3,-2,72,122,165]);
// Polycyclic

G:=Group<a,b|a^2=b^72=1,a*b=b*a>;
// generators/relations

Export

Subgroup lattice of C2×C72 in TeX

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