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G = D725C2order 288 = 25·32

5th semidirect product of D72 and C2 acting faithfully

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D725C2, Q163D9, D18.3D4, C8.10D18, C24.13D6, C72.8C22, Q8.10D18, C36.10C23, Dic9.14D4, D36.5C22, (C8×D9)⋊3C2, C94(C4○D8), (C9×Q16)⋊3C2, C6.98(S3×D4), C2.24(D4×D9), C9⋊C8.8C22, Q82D94C2, Q83D93C2, C18.36(C2×D4), (C3×Q8).30D6, (C3×Q16).4S3, C3.(D24⋊C2), C4.10(C22×D9), (Q8×C9).5C22, C12.49(C22×S3), (C4×D9).12C22, SmallGroup(288,129)

Series: Derived Chief Lower central Upper central

C1C36 — D725C2
C1C3C9C18C36C4×D9Q83D9 — D725C2
C9C18C36 — D725C2
C1C2C4Q16

Generators and relations for D725C2
 G = < a,b,c | a72=b2=c2=1, bab=a-1, cac=a17, cbc=a52b >

Subgroups: 488 in 93 conjugacy classes, 34 normal (22 characteristic)
C1, C2, C2 [×3], C3, C4, C4 [×3], C22 [×3], S3 [×3], C6, C8, C8, C2×C4 [×3], D4 [×4], Q8 [×2], C9, Dic3, C12, C12 [×2], D6 [×3], C2×C8, D8, SD16 [×2], Q16, C4○D4 [×2], D9 [×3], C18, C3⋊C8, C24, C4×S3 [×3], D12 [×4], C3×Q8 [×2], C4○D8, Dic9, C36, C36 [×2], D18, D18 [×2], S3×C8, D24, Q82S3 [×2], C3×Q16, Q83S3 [×2], C9⋊C8, C72, C4×D9, C4×D9 [×2], D36 [×2], D36 [×2], Q8×C9 [×2], D24⋊C2, C8×D9, D72, Q82D9 [×2], C9×Q16, Q83D9 [×2], D725C2
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D6 [×3], C2×D4, D9, C22×S3, C4○D8, D18 [×3], S3×D4, C22×D9, D24⋊C2, D4×D9, D725C2

Smallest permutation representation of D725C2
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 72)(2 71)(3 70)(4 69)(5 68)(6 67)(7 66)(8 65)(9 64)(10 63)(11 62)(12 61)(13 60)(14 59)(15 58)(16 57)(17 56)(18 55)(19 54)(20 53)(21 52)(22 51)(23 50)(24 49)(25 48)(26 47)(27 46)(28 45)(29 44)(30 43)(31 42)(32 41)(33 40)(34 39)(35 38)(36 37)(73 84)(74 83)(75 82)(76 81)(77 80)(78 79)(85 144)(86 143)(87 142)(88 141)(89 140)(90 139)(91 138)(92 137)(93 136)(94 135)(95 134)(96 133)(97 132)(98 131)(99 130)(100 129)(101 128)(102 127)(103 126)(104 125)(105 124)(106 123)(107 122)(108 121)(109 120)(110 119)(111 118)(112 117)(113 116)(114 115)
(1 133)(2 78)(3 95)(4 112)(5 129)(6 74)(7 91)(8 108)(9 125)(10 142)(11 87)(12 104)(13 121)(14 138)(15 83)(16 100)(17 117)(18 134)(19 79)(20 96)(21 113)(22 130)(23 75)(24 92)(25 109)(26 126)(27 143)(28 88)(29 105)(30 122)(31 139)(32 84)(33 101)(34 118)(35 135)(36 80)(37 97)(38 114)(39 131)(40 76)(41 93)(42 110)(43 127)(44 144)(45 89)(46 106)(47 123)(48 140)(49 85)(50 102)(51 119)(52 136)(53 81)(54 98)(55 115)(56 132)(57 77)(58 94)(59 111)(60 128)(61 73)(62 90)(63 107)(64 124)(65 141)(66 86)(67 103)(68 120)(69 137)(70 82)(71 99)(72 116)

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

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

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

42 conjugacy classes

class 1 2A2B2C2D 3 4A4B4C4D4E 6 8A8B8C8D9A9B9C12A12B12C18A18B18C24A24B36A36B36C36D···36I72A···72F
order1222234444468888999121212181818242436363636···3672···72
size111836362244992221818222488222444448···84···4

42 irreducible representations

dim1111112222222224444
type++++++++++++++++++
imageC1C2C2C2C2C2S3D4D4D6D6D9C4○D8D18D18S3×D4D24⋊C2D4×D9D725C2
kernelD725C2C8×D9D72Q82D9C9×Q16Q83D9C3×Q16Dic9D18C24C3×Q8Q16C9C8Q8C6C3C2C1
# reps1112121111234361236

Matrix representation of D725C2 in GL4(𝔽73) generated by

04100
164100
00423
007045
,
04100
57000
003145
00342
,
465400
462700
007045
00423
G:=sub<GL(4,GF(73))| [0,16,0,0,41,41,0,0,0,0,42,70,0,0,3,45],[0,57,0,0,41,0,0,0,0,0,31,3,0,0,45,42],[46,46,0,0,54,27,0,0,0,0,70,42,0,0,45,3] >;

D725C2 in GAP, Magma, Sage, TeX

D_{72}\rtimes_5C_2
% in TeX

G:=Group("D72:5C2");
// GroupNames label

G:=SmallGroup(288,129);
// by ID

G=gap.SmallGroup(288,129);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,253,120,422,135,100,346,185,80,6725,292,9414]);
// Polycyclic

G:=Group<a,b,c|a^72=b^2=c^2=1,b*a*b=a^-1,c*a*c=a^17,c*b*c=a^52*b>;
// generators/relations

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