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G = C62.75D4order 288 = 25·32

59th non-split extension by C62 of D4 acting via D4/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C12 — C62.75D4
 Chief series C1 — C3 — C32 — C3×C6 — C3×C12 — C32⋊4Q8 — C2×C32⋊4Q8 — C62.75D4
 Lower central C32 — C3×C6 — C3×C12 — C62.75D4
 Upper central C1 — C2 — C2×C4 — C4○D4

Generators and relations for C62.75D4
G = < a,b,c,d | a6=b6=1, c4=d2=b3, ab=ba, cac-1=a-1b3, dad-1=a-1, cbc-1=dbd-1=b-1, dcd-1=c3 >

Subgroups: 556 in 180 conjugacy classes, 65 normal (21 characteristic)
C1, C2, C2 [×2], C3 [×4], C4 [×2], C4 [×3], C22, C22, C6 [×4], C6 [×8], C8 [×2], C2×C4, C2×C4 [×2], D4, D4, Q8, Q8 [×3], C32, Dic3 [×8], C12 [×8], C12 [×4], C2×C6 [×4], C2×C6 [×4], M4(2), SD16 [×2], Q16 [×2], C2×Q8, C4○D4, C3×C6, C3×C6 [×2], C3⋊C8 [×8], Dic6 [×12], C2×Dic3 [×4], C2×C12 [×4], C2×C12 [×4], C3×D4 [×4], C3×D4 [×4], C3×Q8 [×4], C8.C22, C3⋊Dic3 [×2], C3×C12 [×2], C3×C12, C62, C62, C4.Dic3 [×4], D4.S3 [×8], C3⋊Q16 [×8], C2×Dic6 [×4], C3×C4○D4 [×4], C324C8 [×2], C324Q8 [×2], C324Q8, C2×C3⋊Dic3, C6×C12, C6×C12, D4×C32, D4×C32, Q8×C32, Q8.14D6 [×4], C12.58D6, C329SD16 [×2], C327Q16 [×2], C2×C324Q8, C32×C4○D4, C62.75D4
Quotients: C1, C2 [×7], C22 [×7], S3 [×4], D4 [×2], C23, D6 [×12], C2×D4, C3⋊S3, C3⋊D4 [×8], C22×S3 [×4], C8.C22, C2×C3⋊S3 [×3], C2×C3⋊D4 [×4], C327D4 [×2], C22×C3⋊S3, Q8.14D6 [×4], C2×C327D4, C62.75D4

Smallest permutation representation of C62.75D4
On 144 points
Generators in S144
```(1 97 141 27 38 55)(2 52 39 32 142 102)(3 99 143 29 40 49)(4 54 33 26 144 104)(5 101 137 31 34 51)(6 56 35 28 138 98)(7 103 139 25 36 53)(8 50 37 30 140 100)(9 78 96 64 118 123)(10 128 119 61 89 75)(11 80 90 58 120 125)(12 122 113 63 91 77)(13 74 92 60 114 127)(14 124 115 57 93 79)(15 76 94 62 116 121)(16 126 117 59 95 73)(17 84 68 135 109 43)(18 48 110 132 69 81)(19 86 70 129 111 45)(20 42 112 134 71 83)(21 88 72 131 105 47)(22 44 106 136 65 85)(23 82 66 133 107 41)(24 46 108 130 67 87)
(1 110 58 5 106 62)(2 63 107 6 59 111)(3 112 60 7 108 64)(4 57 109 8 61 105)(9 29 83 13 25 87)(10 88 26 14 84 30)(11 31 85 15 27 81)(12 82 28 16 86 32)(17 37 75 21 33 79)(18 80 34 22 76 38)(19 39 77 23 35 73)(20 74 36 24 78 40)(41 56 95 45 52 91)(42 92 53 46 96 49)(43 50 89 47 54 93)(44 94 55 48 90 51)(65 121 141 69 125 137)(66 138 126 70 142 122)(67 123 143 71 127 139)(68 140 128 72 144 124)(97 132 120 101 136 116)(98 117 129 102 113 133)(99 134 114 103 130 118)(100 119 131 104 115 135)
(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 4 5 8)(2 7 6 3)(9 12 13 16)(10 15 14 11)(17 18 21 22)(19 24 23 20)(25 28 29 32)(26 31 30 27)(33 34 37 38)(35 40 39 36)(41 42 45 46)(43 48 47 44)(49 52 53 56)(50 55 54 51)(57 58 61 62)(59 64 63 60)(65 68 69 72)(66 71 70 67)(73 78 77 74)(75 76 79 80)(81 88 85 84)(82 83 86 87)(89 94 93 90)(91 92 95 96)(97 104 101 100)(98 99 102 103)(105 106 109 110)(107 112 111 108)(113 114 117 118)(115 120 119 116)(121 124 125 128)(122 127 126 123)(129 130 133 134)(131 136 135 132)(137 140 141 144)(138 143 142 139)```

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

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

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

51 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 3D 4A 4B 4C 4D 4E 6A 6B 6C 6D 6E ··· 6P 8A 8B 12A ··· 12H 12I ··· 12T order 1 2 2 2 3 3 3 3 4 4 4 4 4 6 6 6 6 6 ··· 6 8 8 12 ··· 12 12 ··· 12 size 1 1 2 4 2 2 2 2 2 2 4 36 36 2 2 2 2 4 ··· 4 36 36 2 ··· 2 4 ··· 4

51 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + - - image C1 C2 C2 C2 C2 C2 S3 D4 D4 D6 D6 D6 C3⋊D4 C3⋊D4 C8.C22 Q8.14D6 kernel C62.75D4 C12.58D6 C32⋊9SD16 C32⋊7Q16 C2×C32⋊4Q8 C32×C4○D4 C3×C4○D4 C3×C12 C62 C2×C12 C3×D4 C3×Q8 C12 C2×C6 C32 C3 # reps 1 1 2 2 1 1 4 1 1 4 4 4 8 8 1 8

Matrix representation of C62.75D4 in GL8(𝔽73)

 1 1 0 0 0 0 0 0 72 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 72 72 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0
,
 72 72 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 72 72 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 72 0 0 0 0 0 0 0 0 72 0 0 0 0 0 0 0 0 72 0 0 0 0 0 0 0 0 72
,
 51 10 0 0 0 0 0 0 32 22 0 0 0 0 0 0 0 0 42 70 0 0 0 0 0 0 28 31 0 0 0 0 0 0 0 0 37 36 45 45 0 0 0 0 37 37 28 45 0 0 0 0 28 28 36 37 0 0 0 0 45 28 36 36
,
 51 10 0 0 0 0 0 0 32 22 0 0 0 0 0 0 0 0 42 70 0 0 0 0 0 0 28 31 0 0 0 0 0 0 0 0 37 36 45 45 0 0 0 0 36 36 45 28 0 0 0 0 45 45 37 36 0 0 0 0 45 28 36 36

`G:=sub<GL(8,GF(73))| [1,72,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,1,72,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[72,1,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,72,1,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72],[51,32,0,0,0,0,0,0,10,22,0,0,0,0,0,0,0,0,42,28,0,0,0,0,0,0,70,31,0,0,0,0,0,0,0,0,37,37,28,45,0,0,0,0,36,37,28,28,0,0,0,0,45,28,36,36,0,0,0,0,45,45,37,36],[51,32,0,0,0,0,0,0,10,22,0,0,0,0,0,0,0,0,42,28,0,0,0,0,0,0,70,31,0,0,0,0,0,0,0,0,37,36,45,45,0,0,0,0,36,36,45,28,0,0,0,0,45,45,37,36,0,0,0,0,45,28,36,36] >;`

C62.75D4 in GAP, Magma, Sage, TeX

`C_6^2._{75}D_4`
`% in TeX`

`G:=Group("C6^2.75D4");`
`// GroupNames label`

`G:=SmallGroup(288,808);`
`// by ID`

`G=gap.SmallGroup(288,808);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,112,254,219,675,185,80,2693,9414]);`
`// Polycyclic`

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

׿
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