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## G = C62.69D4order 288 = 25·32

### 53rd non-split extension by C62 of D4 acting via D4/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C62 — C62.69D4
 Chief series C1 — C3 — C32 — C3×C6 — C62 — C22×C3⋊S3 — C2×C32⋊7D4 — C62.69D4
 Lower central C32 — C62 — C62.69D4
 Upper central C1 — C22 — C22⋊C4

Generators and relations for C62.69D4
G = < a,b,c,d | a6=b6=c4=1, d2=b3, ab=ba, cac-1=ab3, dad-1=a-1, bc=cb, dbd-1=b-1, dcd-1=b3c-1 >

Subgroups: 908 in 234 conjugacy classes, 77 normal (15 characteristic)
C1, C2, C2 [×2], C2 [×3], C3 [×4], C4 [×5], C22, C22 [×2], C22 [×5], S3 [×4], C6 [×12], C6 [×8], C2×C4 [×2], C2×C4 [×5], D4 [×2], C23, C23, C32, Dic3 [×12], C12 [×8], D6 [×12], C2×C6 [×12], C2×C6 [×8], C22⋊C4, C22⋊C4 [×2], C4⋊C4 [×2], C22×C4, C2×D4, C3⋊S3, C3×C6, C3×C6 [×2], C3×C6 [×2], C2×Dic3 [×20], C3⋊D4 [×8], C2×C12 [×8], C22×S3 [×4], C22×C6 [×4], C22.D4, C3⋊Dic3 [×3], C3×C12 [×2], C2×C3⋊S3 [×3], C62, C62 [×2], C62 [×2], C4⋊Dic3 [×8], D6⋊C4 [×8], C3×C22⋊C4 [×4], C22×Dic3 [×4], C2×C3⋊D4 [×4], C2×C3⋊Dic3, C2×C3⋊Dic3 [×2], C2×C3⋊Dic3 [×2], C327D4 [×2], C6×C12 [×2], C22×C3⋊S3, C2×C62, C23.21D6 [×4], C12⋊Dic3 [×2], C6.11D12 [×2], C32×C22⋊C4, C22×C3⋊Dic3, C2×C327D4, C62.69D4
Quotients: C1, C2 [×7], C22 [×7], S3 [×4], D4 [×2], C23, D6 [×12], C2×D4, C4○D4 [×2], C3⋊S3, D12 [×8], C22×S3 [×4], C22.D4, C2×C3⋊S3 [×3], C2×D12 [×4], D42S3 [×8], C12⋊S3 [×2], C22×C3⋊S3, C23.21D6 [×4], C2×C12⋊S3, C12.D6 [×2], C62.69D4

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

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

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

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

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 ··· 6T 12A ··· 12P 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 36 2 2 2 2 4 4 18 18 18 18 36 2 ··· 2 4 ··· 4 4 ··· 4

54 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 4 type + + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 S3 D4 D6 D6 C4○D4 D12 D4⋊2S3 kernel C62.69D4 C12⋊Dic3 C6.11D12 C32×C22⋊C4 C22×C3⋊Dic3 C2×C32⋊7D4 C3×C22⋊C4 C62 C2×C12 C22×C6 C3×C6 C2×C6 C6 # reps 1 2 2 1 1 1 4 2 8 4 4 16 8

Matrix representation of C62.69D4 in GL6(𝔽13)

 1 12 0 0 0 0 1 0 0 0 0 0 0 0 0 12 0 0 0 0 1 12 0 0 0 0 0 0 0 5 0 0 0 0 8 0
,
 0 12 0 0 0 0 1 12 0 0 0 0 0 0 0 12 0 0 0 0 1 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12
,
 3 7 0 0 0 0 6 10 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 0 1 0 0 0 0 1 0
,
 3 7 0 0 0 0 10 10 0 0 0 0 0 0 12 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 12 0

`G:=sub<GL(6,GF(13))| [1,1,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,12,12,0,0,0,0,0,0,0,8,0,0,0,0,5,0],[0,1,0,0,0,0,12,12,0,0,0,0,0,0,0,1,0,0,0,0,12,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[3,6,0,0,0,0,7,10,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[3,10,0,0,0,0,7,10,0,0,0,0,0,0,12,0,0,0,0,0,1,1,0,0,0,0,0,0,0,12,0,0,0,0,1,0] >;`

C62.69D4 in GAP, Magma, Sage, TeX

`C_6^2._{69}D_4`
`% in TeX`

`G:=Group("C6^2.69D4");`
`// GroupNames label`

`G:=SmallGroup(288,743);`
`// by ID`

`G=gap.SmallGroup(288,743);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,253,254,219,142,2693,9414]);`
`// 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*b^3,d*a*d^-1=a^-1,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=b^3*c^-1>;`
`// generators/relations`

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