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## G = C2.D72order 288 = 25·32

### 2nd central extension by C2 of D72

Series: Derived Chief Lower central Upper central

 Derived series C1 — C36 — C2.D72
 Chief series C1 — C3 — C9 — C18 — C36 — C2×C36 — C2×D36 — C2.D72
 Lower central C9 — C18 — C36 — C2.D72
 Upper central C1 — C22 — C2×C4 — C2×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 2A 2B 2C 2D 2E 3 4A 4B 4C 4D 6A 6B 6C 8A 8B 8C 8D 9A 9B 9C 12A 12B 12C 12D 18A ··· 18I 24A ··· 24H 36A ··· 36L 72A ··· 72X order 1 2 2 2 2 2 3 4 4 4 4 6 6 6 8 8 8 8 9 9 9 12 12 12 12 18 ··· 18 24 ··· 24 36 ··· 36 72 ··· 72 size 1 1 1 1 36 36 2 2 2 36 36 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 ··· 2 2 ··· 2 2 ··· 2 2 ··· 2

78 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + + + + + + + + + + + image C1 C2 C2 C2 C4 S3 D4 D4 D6 D8 SD16 D9 C4×S3 C3⋊D4 D12 D18 C24⋊C2 D24 C4×D9 C9⋊D4 D36 C72⋊C2 D72 kernel C2.D72 C4⋊Dic9 C2×C72 C2×D36 D36 C2×C24 C36 C2×C18 C2×C12 C18 C18 C2×C8 C12 C12 C2×C6 C2×C4 C6 C6 C4 C4 C22 C2 C2 # reps 1 1 1 1 4 1 1 1 1 2 2 3 2 2 2 3 4 4 6 6 6 12 12

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

 72 0 0 0 1 0 0 0 1
,
 27 0 0 0 11 71 0 2 13
,
 46 0 0 0 11 71 0 60 62
`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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