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## G = C62.72D4order 288 = 25·32

### 56th non-split extension by C62 of D4 acting via D4/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C62 — C62.72D4
 Chief series C1 — C3 — C32 — C3×C6 — C62 — C2×C3⋊Dic3 — C22×C3⋊Dic3 — C62.72D4
 Lower central C32 — C62 — C62.72D4
 Upper central C1 — C22 — C2×D4

Generators and relations for C62.72D4
G = < a,b,c,d | a6=b6=c4=1, d2=b3, ab=ba, cac-1=a-1b3, dad-1=a-1, cbc-1=dbd-1=b-1, dcd-1=b3c-1 >

Subgroups: 716 in 234 conjugacy classes, 77 normal (17 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, C6, C6, C2×C4, C2×C4, D4, C23, C32, Dic3, C12, C2×C6, C2×C6, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C3×C6, C3×C6, C3×C6, C2×Dic3, C2×C12, C3×D4, C22×C6, C22.D4, C3⋊Dic3, C3×C12, C62, C62, C62, Dic3⋊C4, C6.D4, C22×Dic3, C6×D4, C2×C3⋊Dic3, C2×C3⋊Dic3, C6×C12, D4×C32, C2×C62, C23.23D6, C6.Dic6, C625C4, C625C4, C22×C3⋊Dic3, D4×C3×C6, C62.72D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, C3⋊S3, C3⋊D4, C22×S3, C22.D4, C2×C3⋊S3, D42S3, C2×C3⋊D4, C327D4, C22×C3⋊S3, C23.23D6, C12.D6, C2×C327D4, C62.72D4

Smallest permutation representation of C62.72D4
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 38 110 23 91 97)(2 39 111 24 92 98)(3 40 112 19 93 99)(4 41 113 20 94 100)(5 42 114 21 95 101)(6 37 109 22 96 102)(7 115 105 58 65 124)(8 116 106 59 66 125)(9 117 107 60 61 126)(10 118 108 55 62 121)(11 119 103 56 63 122)(12 120 104 57 64 123)(13 67 46 33 53 27)(14 68 47 34 54 28)(15 69 48 35 49 29)(16 70 43 36 50 30)(17 71 44 31 51 25)(18 72 45 32 52 26)(73 139 137 83 88 128)(74 140 138 84 89 129)(75 141 133 79 90 130)(76 142 134 80 85 131)(77 143 135 81 86 132)(78 144 136 82 87 127)
(1 81 27 104)(2 76 28 122)(3 79 29 108)(4 74 30 126)(5 83 25 106)(6 78 26 124)(7 102 144 52)(8 114 139 71)(9 100 140 50)(10 112 141 69)(11 98 142 54)(12 110 143 67)(13 120 38 135)(14 63 39 131)(15 118 40 133)(16 61 41 129)(17 116 42 137)(18 65 37 127)(19 75 48 121)(20 84 43 107)(21 73 44 125)(22 82 45 105)(23 77 46 123)(24 80 47 103)(31 66 95 128)(32 115 96 136)(33 64 91 132)(34 119 92 134)(35 62 93 130)(36 117 94 138)(49 55 99 90)(51 59 101 88)(53 57 97 86)(56 111 85 68)(58 109 87 72)(60 113 89 70)
(1 123 23 104)(2 122 24 103)(3 121 19 108)(4 126 20 107)(5 125 21 106)(6 124 22 105)(7 109 58 102)(8 114 59 101)(9 113 60 100)(10 112 55 99)(11 111 56 98)(12 110 57 97)(13 132 33 135)(14 131 34 134)(15 130 35 133)(16 129 36 138)(17 128 31 137)(18 127 32 136)(25 73 44 83)(26 78 45 82)(27 77 46 81)(28 76 47 80)(29 75 48 79)(30 74 43 84)(37 65 96 115)(38 64 91 120)(39 63 92 119)(40 62 93 118)(41 61 94 117)(42 66 95 116)(49 141 69 90)(50 140 70 89)(51 139 71 88)(52 144 72 87)(53 143 67 86)(54 142 68 85)```

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

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

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

54 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 3A 3B 3C 3D 4A 4B 4C 4D 4E 4F 4G 6A ··· 6L 6M ··· 6AB 12A ··· 12H order 1 2 2 2 2 2 2 3 3 3 3 4 4 4 4 4 4 4 6 ··· 6 6 ··· 6 12 ··· 12 size 1 1 1 1 2 2 4 2 2 2 2 4 18 18 18 18 36 36 2 ··· 2 4 ··· 4 4 ··· 4

54 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 4 type + + + + + + + + + - image C1 C2 C2 C2 C2 S3 D4 D6 D6 C4○D4 C3⋊D4 D4⋊2S3 kernel C62.72D4 C6.Dic6 C62⋊5C4 C22×C3⋊Dic3 D4×C3×C6 C6×D4 C62 C2×C12 C22×C6 C3×C6 C2×C6 C6 # reps 1 2 3 1 1 4 2 4 8 4 16 8

Matrix representation of C62.72D4 in GL6(𝔽13)

 3 0 0 0 0 0 0 9 0 0 0 0 0 0 1 0 0 0 0 0 8 12 0 0 0 0 0 0 9 0 0 0 0 0 0 3
,
 3 0 0 0 0 0 0 9 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 0 1 0 0 0 0 12 0 0 0 0 0 0 0 8 11 0 0 0 0 0 5 0 0 0 0 0 0 0 1 0 0 0 0 1 0
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 8 0 0 0 0 0 0 8 0 0 0 0 0 0 0 1 0 0 0 0 1 0

`G:=sub<GL(6,GF(13))| [3,0,0,0,0,0,0,9,0,0,0,0,0,0,1,8,0,0,0,0,0,12,0,0,0,0,0,0,9,0,0,0,0,0,0,3],[3,0,0,0,0,0,0,9,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,12,0,0,0,0,1,0,0,0,0,0,0,0,8,0,0,0,0,0,11,5,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,8,0,0,0,0,0,0,8,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;`

C62.72D4 in GAP, Magma, Sage, TeX

`C_6^2._{72}D_4`
`% in TeX`

`G:=Group("C6^2.72D4");`
`// GroupNames label`

`G:=SmallGroup(288,792);`
`// by ID`

`G=gap.SmallGroup(288,792);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,112,254,219,2693,9414]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^6=b^6=c^4=1,d^2=b^3,a*b=b*a,c*a*c^-1=a^-1*b^3,d*a*d^-1=a^-1,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=b^3*c^-1>;`
`// generators/relations`

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