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G = D727C2order 288 = 25·32

The semidirect product of D72 and C2 acting through Inn(D72)

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D727C2, C8.17D18, C24.77D6, C36.36D4, C4.20D36, Dic367C2, C12.56D12, C22.1D36, C72.18C22, C36.31C23, D36.7C22, Dic18.7C22, (C2×C8)⋊4D9, (C2×C72)⋊6C2, C91(C4○D8), C72⋊C27C2, C3.(C4○D24), (C2×C24).15S3, C2.14(C2×D36), C6.41(C2×D12), (C2×C6).28D12, (C2×C4).83D18, C18.12(C2×D4), (C2×C18).19D4, D365C21C2, (C2×C12).394D6, C4.29(C22×D9), (C2×C36).97C22, C12.182(C22×S3), SmallGroup(288,115)

Series: Derived Chief Lower central Upper central

C1C36 — D727C2
C1C3C9C18C36D36D365C2 — D727C2
C9C18C36 — D727C2
C1C4C2×C4C2×C8

Generators and relations for D727C2
 G = < a,b,c | a72=b2=c2=1, bab=a-1, ac=ca, cbc=a36b >

Subgroups: 488 in 93 conjugacy classes, 38 normal (30 characteristic)
C1, C2, C2 [×3], C3, C4 [×2], C4 [×2], C22, C22 [×2], S3 [×2], C6, C6, C8 [×2], C2×C4, C2×C4 [×2], D4 [×4], Q8 [×2], C9, Dic3 [×2], C12 [×2], D6 [×2], C2×C6, C2×C8, D8, SD16 [×2], Q16, C4○D4 [×2], D9 [×2], C18, C18, C24 [×2], Dic6 [×2], C4×S3 [×2], D12 [×2], C3⋊D4 [×2], C2×C12, C4○D8, Dic9 [×2], C36 [×2], D18 [×2], C2×C18, C24⋊C2 [×2], D24, Dic12, C2×C24, C4○D12 [×2], C72 [×2], Dic18 [×2], C4×D9 [×2], D36 [×2], C9⋊D4 [×2], C2×C36, C4○D24, Dic36, C72⋊C2 [×2], D72, C2×C72, D365C2 [×2], D727C2
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D6 [×3], C2×D4, D9, D12 [×2], C22×S3, C4○D8, D18 [×3], C2×D12, D36 [×2], C22×D9, C4○D24, C2×D36, D727C2

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

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

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

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

78 conjugacy classes

class 1 2A2B2C2D 3 4A4B4C4D4E6A6B6C8A8B8C8D9A9B9C12A12B12C12D18A···18I24A···24H36A···36L72A···72X
order1222234444466688889991212121218···1824···2436···3672···72
size112363621123636222222222222222···22···22···22···2

78 irreducible representations

dim111111222222222222222
type++++++++++++++++++
imageC1C2C2C2C2C2S3D4D4D6D6D9D12D12C4○D8D18D18D36D36C4○D24D727C2
kernelD727C2Dic36C72⋊C2D72C2×C72D365C2C2×C24C36C2×C18C24C2×C12C2×C8C12C2×C6C9C8C2×C4C4C22C3C1
# reps1121121112132246366824

Matrix representation of D727C2 in GL2(𝔽73) generated by

2263
1032
,
1948
2954
,
4313
6030
G:=sub<GL(2,GF(73))| [22,10,63,32],[19,29,48,54],[43,60,13,30] >;

D727C2 in GAP, Magma, Sage, TeX

D_{72}\rtimes_7C_2
% in TeX

G:=Group("D72:7C2");
// GroupNames label

G:=SmallGroup(288,115);
// by ID

G=gap.SmallGroup(288,115);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,120,254,58,675,80,6725,292,9414]);
// Polycyclic

G:=Group<a,b,c|a^72=b^2=c^2=1,b*a*b=a^-1,a*c=c*a,c*b*c=a^36*b>;
// generators/relations

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