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G = C2×D72order 288 = 25·32

Direct product of C2 and D72

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

Aliases: C2×D72, C181D8, C87D18, C4.8D36, C6.6D24, C728C22, C24.71D6, C36.31D4, C12.42D12, D363C22, C36.30C23, C22.14D36, C91(C2×D8), (C2×C8)⋊3D9, C3.(C2×D24), (C2×C72)⋊5C2, (C2×D36)⋊5C2, (C2×C24).11S3, C6.40(C2×D12), C18.11(C2×D4), C2.13(C2×D36), (C2×C6).27D12, (C2×C18).18D4, (C2×C4).82D18, (C2×C12).372D6, C4.28(C22×D9), (C2×C36).91C22, C12.181(C22×S3), SmallGroup(288,114)

Series: Derived Chief Lower central Upper central

C1C36 — C2×D72
C1C3C9C18C36D36C2×D36 — C2×D72
C9C18C36 — C2×D72
C1C22C2×C4C2×C8

Generators and relations for C2×D72
 G = < a,b,c | a2=b72=c2=1, ab=ba, ac=ca, cbc=b-1 >

Subgroups: 800 in 114 conjugacy classes, 44 normal (22 characteristic)
C1, C2, C2 [×2], C2 [×4], C3, C4 [×2], C22, C22 [×8], S3 [×4], C6, C6 [×2], C8 [×2], C2×C4, D4 [×6], C23 [×2], C9, C12 [×2], D6 [×8], C2×C6, C2×C8, D8 [×4], C2×D4 [×2], D9 [×4], C18, C18 [×2], C24 [×2], D12 [×6], C2×C12, C22×S3 [×2], C2×D8, C36 [×2], D18 [×8], C2×C18, D24 [×4], C2×C24, C2×D12 [×2], C72 [×2], D36 [×4], D36 [×2], C2×C36, C22×D9 [×2], C2×D24, D72 [×4], C2×C72, C2×D36 [×2], C2×D72
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D6 [×3], D8 [×2], C2×D4, D9, D12 [×2], C22×S3, C2×D8, D18 [×3], D24 [×2], C2×D12, D36 [×2], C22×D9, C2×D24, D72 [×2], C2×D36, C2×D72

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

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

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

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

78 conjugacy classes

class 1 2A2B2C2D2E2F2G 3 4A4B6A6B6C8A8B8C8D9A9B9C12A12B12C12D18A···18I24A···24H36A···36L72A···72X
order1222222234466688889991212121218···1824···2436···3672···72
size111136363636222222222222222222···22···22···22···2

78 irreducible representations

dim1111222222222222222
type+++++++++++++++++++
imageC1C2C2C2S3D4D4D6D6D8D9D12D12D18D18D24D36D36D72
kernelC2×D72D72C2×C72C2×D36C2×C24C36C2×C18C24C2×C12C18C2×C8C12C2×C6C8C2×C4C6C4C22C2
# reps14121112143226386624

Matrix representation of C2×D72 in GL4(𝔽73) generated by

72000
07200
0010
0001
,
254400
295400
006341
003222
,
33100
287000
00331
002870
G:=sub<GL(4,GF(73))| [72,0,0,0,0,72,0,0,0,0,1,0,0,0,0,1],[25,29,0,0,44,54,0,0,0,0,63,32,0,0,41,22],[3,28,0,0,31,70,0,0,0,0,3,28,0,0,31,70] >;

C2×D72 in GAP, Magma, Sage, TeX

C_2\times D_{72}
% in TeX

G:=Group("C2xD72");
// GroupNames label

G:=SmallGroup(288,114);
// by ID

G=gap.SmallGroup(288,114);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,254,142,675,80,6725,292,9414]);
// 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^-1>;
// generators/relations

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