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## G = C42.240D6order 192 = 26·3

### 60th non-split extension by C42 of D6 acting via D6/S3=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C42.240D6
 Chief series C1 — C3 — C6 — C2×C6 — C22×S3 — S3×C2×C4 — S3×C42 — C42.240D6
 Lower central C3 — C2×C6 — C42.240D6
 Upper central C1 — C22 — C4⋊Q8

Generators and relations for C42.240D6
G = < a,b,c,d | a4=b4=d2=1, c6=b2, ab=ba, cac-1=dad=a-1, cbc-1=dbd=b-1, dcd=b2c5 >

Subgroups: 848 in 310 conjugacy classes, 111 normal (19 characteristic)
C1, C2, C2 [×2], C2 [×6], C3, C4 [×6], C4 [×8], C22, C22 [×16], S3 [×6], C6, C6 [×2], C2×C4, C2×C4 [×6], C2×C4 [×19], D4 [×20], Q8 [×4], C23 [×5], Dic3 [×2], Dic3 [×2], C12 [×6], C12 [×4], D6 [×2], D6 [×14], C2×C6, C42, C42 [×3], C22⋊C4 [×8], C4⋊C4 [×4], C22×C4 [×7], C2×D4 [×10], C2×Q8 [×2], C4○D4 [×8], C4×S3 [×4], C4×S3 [×12], D12 [×20], C2×Dic3, C2×Dic3 [×2], C2×C12, C2×C12 [×6], C3×Q8 [×4], C22×S3, C22×S3 [×4], C2×C42, C4×D4 [×4], C4⋊D4 [×4], C4.4D4 [×2], C41D4, C4⋊Q8, C2×C4○D4 [×2], C4×Dic3, C4×Dic3 [×2], D6⋊C4 [×8], C4×C12, C3×C4⋊C4 [×4], S3×C2×C4, S3×C2×C4 [×6], C2×D12 [×10], Q83S3 [×8], C6×Q8 [×2], C22.26C24, S3×C42, C4⋊D12, Dic35D4 [×4], C12⋊D4 [×4], C12.23D4 [×2], C3×C4⋊Q8, C2×Q83S3 [×2], C42.240D6
Quotients: C1, C2 [×15], C22 [×35], S3, D4 [×4], C23 [×15], D6 [×7], C2×D4 [×6], C4○D4 [×4], C24, C22×S3 [×7], C22×D4, C2×C4○D4 [×2], S3×D4 [×2], Q83S3 [×4], S3×C23, C22.26C24, C2×S3×D4, C2×Q83S3 [×2], C42.240D6

Smallest permutation representation of C42.240D6
On 96 points
Generators in S96
```(1 78 18 63)(2 64 19 79)(3 80 20 65)(4 66 21 81)(5 82 22 67)(6 68 23 83)(7 84 24 69)(8 70 13 73)(9 74 14 71)(10 72 15 75)(11 76 16 61)(12 62 17 77)(25 89 60 43)(26 44 49 90)(27 91 50 45)(28 46 51 92)(29 93 52 47)(30 48 53 94)(31 95 54 37)(32 38 55 96)(33 85 56 39)(34 40 57 86)(35 87 58 41)(36 42 59 88)
(1 30 7 36)(2 25 8 31)(3 32 9 26)(4 27 10 33)(5 34 11 28)(6 29 12 35)(13 54 19 60)(14 49 20 55)(15 56 21 50)(16 51 22 57)(17 58 23 52)(18 53 24 59)(37 79 43 73)(38 74 44 80)(39 81 45 75)(40 76 46 82)(41 83 47 77)(42 78 48 84)(61 92 67 86)(62 87 68 93)(63 94 69 88)(64 89 70 95)(65 96 71 90)(66 91 72 85)
(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)
(1 9)(2 8)(3 7)(4 6)(10 12)(13 19)(14 18)(15 17)(20 24)(21 23)(26 36)(27 35)(28 34)(29 33)(30 32)(37 95)(38 94)(39 93)(40 92)(41 91)(42 90)(43 89)(44 88)(45 87)(46 86)(47 85)(48 96)(49 59)(50 58)(51 57)(52 56)(53 55)(61 76)(62 75)(63 74)(64 73)(65 84)(66 83)(67 82)(68 81)(69 80)(70 79)(71 78)(72 77)```

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

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

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

42 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 3 4A ··· 4F 4G 4H 4I 4J 4K 4L 4M 4N 4O 4P 4Q 4R 6A 6B 6C 12A ··· 12F 12G 12H 12I 12J order 1 2 2 2 2 2 2 2 2 2 3 4 ··· 4 4 4 4 4 4 4 4 4 4 4 4 4 6 6 6 12 ··· 12 12 12 12 12 size 1 1 1 1 6 6 12 12 12 12 2 2 ··· 2 3 3 3 3 4 4 4 4 6 6 6 6 2 2 2 4 ··· 4 8 8 8 8

42 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 S3 D4 D6 D6 D6 C4○D4 S3×D4 Q8⋊3S3 kernel C42.240D6 S3×C42 C4⋊D12 Dic3⋊5D4 C12⋊D4 C12.23D4 C3×C4⋊Q8 C2×Q8⋊3S3 C4⋊Q8 C4×S3 C42 C4⋊C4 C2×Q8 C12 C4 C4 # reps 1 1 1 4 4 2 1 2 1 4 1 4 2 8 2 4

Matrix representation of C42.240D6 in GL6(𝔽13)

 5 0 0 0 0 0 3 8 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 8 0 0 0 0 0 10 5 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 0 8 0 0 0 0 8 0
,
 1 1 0 0 0 0 11 12 0 0 0 0 0 0 0 12 0 0 0 0 1 12 0 0 0 0 0 0 0 1 0 0 0 0 12 0
,
 12 12 0 0 0 0 0 1 0 0 0 0 0 0 1 12 0 0 0 0 0 12 0 0 0 0 0 0 1 0 0 0 0 0 0 12

`G:=sub<GL(6,GF(13))| [5,3,0,0,0,0,0,8,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[8,10,0,0,0,0,0,5,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,8,0,0,0,0,8,0],[1,11,0,0,0,0,1,12,0,0,0,0,0,0,0,1,0,0,0,0,12,12,0,0,0,0,0,0,0,12,0,0,0,0,1,0],[12,0,0,0,0,0,12,1,0,0,0,0,0,0,1,0,0,0,0,0,12,12,0,0,0,0,0,0,1,0,0,0,0,0,0,12] >;`

C42.240D6 in GAP, Magma, Sage, TeX

`C_4^2._{240}D_6`
`% in TeX`

`G:=Group("C4^2.240D6");`
`// GroupNames label`

`G:=SmallGroup(192,1284);`
`// by ID`

`G=gap.SmallGroup(192,1284);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,100,675,570,185,80,6278]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^4=b^4=d^2=1,c^6=b^2,a*b=b*a,c*a*c^-1=d*a*d=a^-1,c*b*c^-1=d*b*d=b^-1,d*c*d=b^2*c^5>;`
`// generators/relations`

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