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G = C2.D72order 288 = 25·32

2nd central extension by C2 of D72

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D362C4, C2.2D72, C18.5D8, C6.5D24, C36.46D4, C18.3SD16, C22.10D36, (C2×C8)⋊2D9, (C2×C72)⋊2C2, C4.8(C4×D9), (C2×C24).4S3, C4⋊Dic91C2, C92(D4⋊C4), C12.56(C4×S3), C36.18(C2×C4), (C2×D36).1C2, (C2×C18).15D4, (C2×C4).76D18, (C2×C6).23D12, C3.(C2.D24), C6.3(C24⋊C2), C2.8(D18⋊C4), C6.14(D6⋊C4), (C2×C12).365D6, C2.3(C72⋊C2), C4.20(C9⋊D4), C18.7(C22⋊C4), (C2×C36).84C22, C12.107(C3⋊D4), SmallGroup(288,28)

Series: Derived Chief Lower central Upper central

C1C36 — C2.D72
C1C3C9C18C36C2×C36C2×D36 — C2.D72
C9C18C36 — C2.D72
C1C22C2×C4C2×C8

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

Subgroups: 488 in 75 conjugacy classes, 32 normal (30 characteristic)
C1, C2 [×3], C2 [×2], C3, C4 [×2], C4, C22, C22 [×4], S3 [×2], C6 [×3], C8, C2×C4, C2×C4, D4 [×3], C23, C9, Dic3, C12 [×2], D6 [×4], C2×C6, C4⋊C4, C2×C8, C2×D4, D9 [×2], C18 [×3], C24, D12 [×3], C2×Dic3, C2×C12, C22×S3, D4⋊C4, Dic9, C36 [×2], D18 [×4], C2×C18, C4⋊Dic3, C2×C24, C2×D12, C72, D36 [×2], D36, C2×Dic9, C2×C36, C22×D9, C2.D24, C4⋊Dic9, C2×C72, C2×D36, C2.D72
Quotients: C1, C2 [×3], C4 [×2], C22, S3, C2×C4, D4 [×2], D6, C22⋊C4, D8, SD16, D9, C4×S3, D12, C3⋊D4, D4⋊C4, D18, C24⋊C2, D24, D6⋊C4, C4×D9, D36, C9⋊D4, C2.D24, C72⋊C2, D72, D18⋊C4, C2.D72

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

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

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

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

78 conjugacy classes

class 1 2A2B2C2D2E 3 4A4B4C4D6A6B6C8A8B8C8D9A9B9C12A12B12C12D18A···18I24A···24H36A···36L72A···72X
order1222223444466688889991212121218···1824···2436···3672···72
size111136362223636222222222222222···22···22···22···2

78 irreducible representations

dim11111222222222222222222
type+++++++++++++++
imageC1C2C2C2C4S3D4D4D6D8SD16D9C4×S3C3⋊D4D12D18C24⋊C2D24C4×D9C9⋊D4D36C72⋊C2D72
kernelC2.D72C4⋊Dic9C2×C72C2×D36D36C2×C24C36C2×C18C2×C12C18C18C2×C8C12C12C2×C6C2×C4C6C6C4C4C22C2C2
# reps1111411112232223446661212

Matrix representation of C2.D72 in GL3(𝔽73) generated by

7200
010
001
,
2700
01171
0213
,
4600
01171
06062
G:=sub<GL(3,GF(73))| [72,0,0,0,1,0,0,0,1],[27,0,0,0,11,2,0,71,13],[46,0,0,0,11,60,0,71,62] >;

C2.D72 in GAP, Magma, Sage, TeX

C_2.D_{72}
% in TeX

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

G:=SmallGroup(288,28);
// by ID

G=gap.SmallGroup(288,28);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,85,92,422,100,6725,292,9414]);
// Polycyclic

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

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