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## G = C62.134D4order 288 = 25·32

### 39th non-split extension by C62 of D4 acting via D4/C22=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C12 — C62.134D4
 Chief series C1 — C3 — C32 — C3×C6 — C3×C12 — C12⋊S3 — C12.59D6 — C62.134D4
 Lower central C32 — C3×C6 — C3×C12 — C62.134D4
 Upper central C1 — C2 — C2×C4 — C2×Q8

Generators and relations for C62.134D4
G = < a,b,c,d | a6=b6=d2=1, c4=b3, ab=ba, cac-1=dad=a-1b3, cbc-1=dbd=b-1, dcd=c3 >

Subgroups: 620 in 180 conjugacy classes, 65 normal (19 characteristic)
C1, C2, C2 [×2], C3 [×4], C4 [×2], C4 [×3], C22, C22, S3 [×4], C6 [×4], C6 [×4], C8 [×2], C2×C4, C2×C4 [×2], D4 [×2], Q8 [×2], Q8 [×2], C32, Dic3 [×4], C12 [×8], C12 [×8], D6 [×4], C2×C6 [×4], M4(2), SD16 [×2], Q16 [×2], C2×Q8, C4○D4, C3⋊S3, C3×C6, C3×C6, C3⋊C8 [×8], Dic6 [×4], C4×S3 [×4], D12 [×4], C3⋊D4 [×4], C2×C12 [×4], C2×C12 [×4], C3×Q8 [×8], C3×Q8 [×4], C8.C22, C3⋊Dic3, C3×C12 [×2], C3×C12 [×2], C2×C3⋊S3, C62, C4.Dic3 [×4], Q82S3 [×8], C3⋊Q16 [×8], C4○D12 [×4], C6×Q8 [×4], C324C8 [×2], C324Q8, C4×C3⋊S3, C12⋊S3, C327D4, C6×C12, C6×C12, Q8×C32 [×2], Q8×C32, Q8.11D6 [×4], C12.58D6, C3211SD16 [×2], C327Q16 [×2], C12.59D6, Q8×C3×C6, C62.134D4
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, Q8.11D6 [×4], C2×C327D4, C62.134D4

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

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

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

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

51 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 3D 4A 4B 4C 4D 4E 6A ··· 6L 8A 8B 12A ··· 12X order 1 2 2 2 3 3 3 3 4 4 4 4 4 6 ··· 6 8 8 12 ··· 12 size 1 1 2 36 2 2 2 2 2 2 4 4 36 2 ··· 2 36 36 4 ··· 4

51 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 S3 D4 D4 D6 D6 C3⋊D4 C3⋊D4 C8.C22 Q8.11D6 kernel C62.134D4 C12.58D6 C32⋊11SD16 C32⋊7Q16 C12.59D6 Q8×C3×C6 C6×Q8 C3×C12 C62 C2×C12 C3×Q8 C12 C2×C6 C32 C3 # reps 1 1 2 2 1 1 4 1 1 4 8 8 8 1 8

Matrix representation of C62.134D4 in GL6(𝔽73)

 0 72 0 0 0 0 1 72 0 0 0 0 0 0 60 30 0 0 0 0 43 30 0 0 0 0 0 0 60 30 0 0 0 0 43 30
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 72 0 0 0 0 0 0 0 1 1 0 0 0 0 72 0
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 51 62 45 12 0 0 11 22 40 28 0 0 63 46 19 0 0 0 56 10 54 54
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 25 25 72 0 0 0 0 48 1 1

`G:=sub<GL(6,GF(73))| [0,1,0,0,0,0,72,72,0,0,0,0,0,0,60,43,0,0,0,0,30,30,0,0,0,0,0,0,60,43,0,0,0,0,30,30],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,72,0,0,0,0,1,0,0,0,0,0,0,0,1,72,0,0,0,0,1,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,51,11,63,56,0,0,62,22,46,10,0,0,45,40,19,54,0,0,12,28,0,54],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,25,0,0,0,1,0,25,48,0,0,0,0,72,1,0,0,0,0,0,1] >;`

C62.134D4 in GAP, Magma, Sage, TeX

`C_6^2._{134}D_4`
`% in TeX`

`G:=Group("C6^2.134D4");`
`// GroupNames label`

`G:=SmallGroup(288,799);`
`// by ID`

`G=gap.SmallGroup(288,799);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,120,254,219,100,675,185,80,2693,9414]);`
`// Polycyclic`

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

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