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## G = D72⋊5C2order 288 = 25·32

### 5th semidirect product of D72 and C2 acting faithfully

Series: Derived Chief Lower central Upper central

 Derived series C1 — C36 — D72⋊5C2
 Chief series C1 — C3 — C9 — C18 — C36 — C4×D9 — Q8⋊3D9 — D72⋊5C2
 Lower central C9 — C18 — C36 — D72⋊5C2
 Upper central C1 — C2 — C4 — Q16

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 2A 2B 2C 2D 3 4A 4B 4C 4D 4E 6 8A 8B 8C 8D 9A 9B 9C 12A 12B 12C 18A 18B 18C 24A 24B 36A 36B 36C 36D ··· 36I 72A ··· 72F order 1 2 2 2 2 3 4 4 4 4 4 6 8 8 8 8 9 9 9 12 12 12 18 18 18 24 24 36 36 36 36 ··· 36 72 ··· 72 size 1 1 18 36 36 2 2 4 4 9 9 2 2 2 18 18 2 2 2 4 8 8 2 2 2 4 4 4 4 4 8 ··· 8 4 ··· 4

42 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 S3 D4 D4 D6 D6 D9 C4○D8 D18 D18 S3×D4 D24⋊C2 D4×D9 D72⋊5C2 kernel D72⋊5C2 C8×D9 D72 Q8⋊2D9 C9×Q16 Q8⋊3D9 C3×Q16 Dic9 D18 C24 C3×Q8 Q16 C9 C8 Q8 C6 C3 C2 C1 # reps 1 1 1 2 1 2 1 1 1 1 2 3 4 3 6 1 2 3 6

Matrix representation of D725C2 in GL4(𝔽73) generated by

 0 41 0 0 16 41 0 0 0 0 42 3 0 0 70 45
,
 0 41 0 0 57 0 0 0 0 0 31 45 0 0 3 42
,
 46 54 0 0 46 27 0 0 0 0 70 45 0 0 42 3
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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