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## G = C42.285D6order 192 = 26·3

### 8th central extension by C42 of D6

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C42.285D6
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C2×C3⋊C8 — C4×C3⋊C8 — C42.285D6
 Lower central C3 — C6 — C42.285D6
 Upper central C1 — C42 — C2×C42

Generators and relations for C42.285D6
G = < a,b,c,d | a4=b4=c6=1, d2=a2b, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=a2c-1 >

Subgroups: 184 in 118 conjugacy classes, 79 normal (41 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C22, C6, C6, C8, C2×C4, C2×C4, C2×C4, C23, C12, C12, C2×C6, C2×C6, C2×C6, C42, C2×C8, C22×C4, C3⋊C8, C2×C12, C2×C12, C2×C12, C22×C6, C4×C8, C22⋊C8, C4⋊C8, C2×C42, C2×C3⋊C8, C4×C12, C22×C12, C42.12C4, C4×C3⋊C8, C12⋊C8, C12.55D4, C2×C4×C12, C42.285D6
Quotients: C1, C2, C4, C22, S3, C8, C2×C4, C23, Dic3, D6, C2×C8, M4(2), C22×C4, C4○D4, C3⋊C8, C2×Dic3, C22×S3, C42⋊C2, C22×C8, C2×M4(2), C2×C3⋊C8, C4.Dic3, C4○D12, C22×Dic3, C42.12C4, C22×C3⋊C8, C2×C4.Dic3, C23.26D6, C42.285D6

Smallest permutation representation of C42.285D6
On 96 points
Generators in S96
```(1 54 79 9)(2 55 80 10)(3 56 73 11)(4 49 74 12)(5 50 75 13)(6 51 76 14)(7 52 77 15)(8 53 78 16)(17 42 31 58)(18 43 32 59)(19 44 25 60)(20 45 26 61)(21 46 27 62)(22 47 28 63)(23 48 29 64)(24 41 30 57)(33 82 67 92)(34 83 68 93)(35 84 69 94)(36 85 70 95)(37 86 71 96)(38 87 72 89)(39 88 65 90)(40 81 66 91)
(1 73 5 77)(2 74 6 78)(3 75 7 79)(4 76 8 80)(9 56 13 52)(10 49 14 53)(11 50 15 54)(12 51 16 55)(17 25 21 29)(18 26 22 30)(19 27 23 31)(20 28 24 32)(33 69 37 65)(34 70 38 66)(35 71 39 67)(36 72 40 68)(41 59 45 63)(42 60 46 64)(43 61 47 57)(44 62 48 58)(81 93 85 89)(82 94 86 90)(83 95 87 91)(84 96 88 92)
(1 95 62)(2 47 96 80 63 86)(3 89 64)(4 41 90 74 57 88)(5 91 58)(6 43 92 76 59 82)(7 93 60)(8 45 94 78 61 84)(9 70 27)(10 22 71 55 28 37)(11 72 29)(12 24 65 49 30 39)(13 66 31)(14 18 67 51 32 33)(15 68 25)(16 20 69 53 26 35)(17 50 40)(19 52 34)(21 54 36)(23 56 38)(42 75 81)(44 77 83)(46 79 85)(48 73 87)
(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)```

`G:=sub<Sym(96)| (1,54,79,9)(2,55,80,10)(3,56,73,11)(4,49,74,12)(5,50,75,13)(6,51,76,14)(7,52,77,15)(8,53,78,16)(17,42,31,58)(18,43,32,59)(19,44,25,60)(20,45,26,61)(21,46,27,62)(22,47,28,63)(23,48,29,64)(24,41,30,57)(33,82,67,92)(34,83,68,93)(35,84,69,94)(36,85,70,95)(37,86,71,96)(38,87,72,89)(39,88,65,90)(40,81,66,91), (1,73,5,77)(2,74,6,78)(3,75,7,79)(4,76,8,80)(9,56,13,52)(10,49,14,53)(11,50,15,54)(12,51,16,55)(17,25,21,29)(18,26,22,30)(19,27,23,31)(20,28,24,32)(33,69,37,65)(34,70,38,66)(35,71,39,67)(36,72,40,68)(41,59,45,63)(42,60,46,64)(43,61,47,57)(44,62,48,58)(81,93,85,89)(82,94,86,90)(83,95,87,91)(84,96,88,92), (1,95,62)(2,47,96,80,63,86)(3,89,64)(4,41,90,74,57,88)(5,91,58)(6,43,92,76,59,82)(7,93,60)(8,45,94,78,61,84)(9,70,27)(10,22,71,55,28,37)(11,72,29)(12,24,65,49,30,39)(13,66,31)(14,18,67,51,32,33)(15,68,25)(16,20,69,53,26,35)(17,50,40)(19,52,34)(21,54,36)(23,56,38)(42,75,81)(44,77,83)(46,79,85)(48,73,87), (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)>;`

`G:=Group( (1,54,79,9)(2,55,80,10)(3,56,73,11)(4,49,74,12)(5,50,75,13)(6,51,76,14)(7,52,77,15)(8,53,78,16)(17,42,31,58)(18,43,32,59)(19,44,25,60)(20,45,26,61)(21,46,27,62)(22,47,28,63)(23,48,29,64)(24,41,30,57)(33,82,67,92)(34,83,68,93)(35,84,69,94)(36,85,70,95)(37,86,71,96)(38,87,72,89)(39,88,65,90)(40,81,66,91), (1,73,5,77)(2,74,6,78)(3,75,7,79)(4,76,8,80)(9,56,13,52)(10,49,14,53)(11,50,15,54)(12,51,16,55)(17,25,21,29)(18,26,22,30)(19,27,23,31)(20,28,24,32)(33,69,37,65)(34,70,38,66)(35,71,39,67)(36,72,40,68)(41,59,45,63)(42,60,46,64)(43,61,47,57)(44,62,48,58)(81,93,85,89)(82,94,86,90)(83,95,87,91)(84,96,88,92), (1,95,62)(2,47,96,80,63,86)(3,89,64)(4,41,90,74,57,88)(5,91,58)(6,43,92,76,59,82)(7,93,60)(8,45,94,78,61,84)(9,70,27)(10,22,71,55,28,37)(11,72,29)(12,24,65,49,30,39)(13,66,31)(14,18,67,51,32,33)(15,68,25)(16,20,69,53,26,35)(17,50,40)(19,52,34)(21,54,36)(23,56,38)(42,75,81)(44,77,83)(46,79,85)(48,73,87), (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) );`

`G=PermutationGroup([[(1,54,79,9),(2,55,80,10),(3,56,73,11),(4,49,74,12),(5,50,75,13),(6,51,76,14),(7,52,77,15),(8,53,78,16),(17,42,31,58),(18,43,32,59),(19,44,25,60),(20,45,26,61),(21,46,27,62),(22,47,28,63),(23,48,29,64),(24,41,30,57),(33,82,67,92),(34,83,68,93),(35,84,69,94),(36,85,70,95),(37,86,71,96),(38,87,72,89),(39,88,65,90),(40,81,66,91)], [(1,73,5,77),(2,74,6,78),(3,75,7,79),(4,76,8,80),(9,56,13,52),(10,49,14,53),(11,50,15,54),(12,51,16,55),(17,25,21,29),(18,26,22,30),(19,27,23,31),(20,28,24,32),(33,69,37,65),(34,70,38,66),(35,71,39,67),(36,72,40,68),(41,59,45,63),(42,60,46,64),(43,61,47,57),(44,62,48,58),(81,93,85,89),(82,94,86,90),(83,95,87,91),(84,96,88,92)], [(1,95,62),(2,47,96,80,63,86),(3,89,64),(4,41,90,74,57,88),(5,91,58),(6,43,92,76,59,82),(7,93,60),(8,45,94,78,61,84),(9,70,27),(10,22,71,55,28,37),(11,72,29),(12,24,65,49,30,39),(13,66,31),(14,18,67,51,32,33),(15,68,25),(16,20,69,53,26,35),(17,50,40),(19,52,34),(21,54,36),(23,56,38),(42,75,81),(44,77,83),(46,79,85),(48,73,87)], [(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)]])`

72 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3 4A ··· 4L 4M ··· 4R 6A ··· 6G 8A ··· 8P 12A ··· 12X order 1 2 2 2 2 2 3 4 ··· 4 4 ··· 4 6 ··· 6 8 ··· 8 12 ··· 12 size 1 1 1 1 2 2 2 1 ··· 1 2 ··· 2 2 ··· 2 6 ··· 6 2 ··· 2

72 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 type + + + + + + - + - + image C1 C2 C2 C2 C2 C4 C4 C8 S3 Dic3 D6 Dic3 D6 M4(2) C4○D4 C3⋊C8 C4.Dic3 C4○D12 kernel C42.285D6 C4×C3⋊C8 C12⋊C8 C12.55D4 C2×C4×C12 C4×C12 C22×C12 C2×C12 C2×C42 C42 C42 C22×C4 C22×C4 C12 C12 C2×C4 C4 C4 # reps 1 2 2 2 1 4 4 16 1 2 2 2 1 4 4 8 8 8

Matrix representation of C42.285D6 in GL4(𝔽73) generated by

 1 0 0 0 0 1 0 0 0 0 46 0 0 0 0 46
,
 27 0 0 0 0 27 0 0 0 0 27 0 0 0 0 27
,
 0 72 0 0 1 72 0 0 0 0 8 0 0 0 58 9
,
 34 50 0 0 11 39 0 0 0 0 27 42 0 0 55 46
`G:=sub<GL(4,GF(73))| [1,0,0,0,0,1,0,0,0,0,46,0,0,0,0,46],[27,0,0,0,0,27,0,0,0,0,27,0,0,0,0,27],[0,1,0,0,72,72,0,0,0,0,8,58,0,0,0,9],[34,11,0,0,50,39,0,0,0,0,27,55,0,0,42,46] >;`

C42.285D6 in GAP, Magma, Sage, TeX

`C_4^2._{285}D_6`
`% in TeX`

`G:=Group("C4^2.285D6");`
`// GroupNames label`

`G:=SmallGroup(192,484);`
`// by ID`

`G=gap.SmallGroup(192,484);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,56,120,422,136,6278]);`
`// Polycyclic`

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

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