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## G = C42.213D6order 192 = 26·3

### 33rd non-split extension by C42 of D6 acting via D6/S3=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C12 — C42.213D6
 Chief series C1 — C3 — C6 — C2×C6 — C2×C12 — C2×C3⋊C8 — C4×C3⋊C8 — C42.213D6
 Lower central C3 — C6 — C2×C12 — C42.213D6
 Upper central C1 — C22 — C42 — C4.4D4

Generators and relations for C42.213D6
G = < a,b,c,d | a4=b4=c6=1, d2=a2, ab=ba, cac-1=dad-1=a-1b2, cbc-1=dbd-1=b-1, dcd-1=a2bc-1 >

Subgroups: 240 in 96 conjugacy classes, 39 normal (19 characteristic)
C1, C2, C2 [×2], C2, C3, C4 [×2], C4 [×5], C22, C22 [×3], C6, C6 [×2], C6, C8 [×2], C2×C4, C2×C4 [×2], C2×C4 [×3], D4 [×2], Q8 [×2], C23, Dic3 [×2], C12 [×2], C12 [×3], C2×C6, C2×C6 [×3], C42, C22⋊C4 [×2], C4⋊C4 [×4], C2×C8 [×2], C2×D4, C2×Q8, C3⋊C8 [×2], C2×Dic3 [×2], C2×C12, C2×C12 [×2], C2×C12, C3×D4 [×2], C3×Q8 [×2], C22×C6, C4×C8, D4⋊C4 [×2], Q8⋊C4 [×2], C4.4D4, C42.C2, C2×C3⋊C8 [×2], Dic3⋊C4 [×2], C4⋊Dic3 [×2], C4×C12, C3×C22⋊C4 [×2], C6×D4, C6×Q8, C42.78C22, C4×C3⋊C8, D4⋊Dic3 [×2], Q82Dic3 [×2], C12.6Q8, C3×C4.4D4, C42.213D6
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D6 [×3], C2×D4, C4○D4 [×2], C3⋊D4 [×2], C22×S3, C4.4D4, C4○D8 [×2], D42S3 [×2], C2×C3⋊D4, C42.78C22, C23.12D6, Q8.13D6 [×2], C42.213D6

Smallest permutation representation of C42.213D6
On 96 points
Generators in S96
```(1 39 6 34)(2 37 4 32)(3 41 5 36)(7 31 10 42)(8 35 11 40)(9 33 12 38)(13 63 77 69)(14 57 78 51)(15 65 73 71)(16 59 74 53)(17 61 75 67)(18 55 76 49)(19 29 87 26)(20 47 88 44)(21 25 89 28)(22 43 90 46)(23 27 85 30)(24 45 86 48)(50 81 56 94)(52 83 58 96)(54 79 60 92)(62 93 68 80)(64 95 70 82)(66 91 72 84)
(1 21 7 24)(2 19 8 22)(3 23 9 20)(4 87 11 90)(5 85 12 88)(6 89 10 86)(13 91 94 16)(14 17 95 92)(15 93 96 18)(25 31 45 39)(26 40 46 32)(27 33 47 41)(28 42 48 34)(29 35 43 37)(30 38 44 36)(49 71 62 58)(50 59 63 72)(51 67 64 60)(52 55 65 68)(53 69 66 56)(54 57 61 70)(73 80 83 76)(74 77 84 81)(75 82 79 78)
(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 58 6 52)(2 60 4 54)(3 56 5 50)(7 71 10 65)(8 67 11 61)(9 69 12 63)(13 41 77 36)(14 29 78 26)(15 39 73 34)(16 27 74 30)(17 37 75 32)(18 25 76 28)(19 64 87 70)(20 53 88 59)(21 62 89 68)(22 51 90 57)(23 66 85 72)(24 49 86 55)(31 83 42 96)(33 81 38 94)(35 79 40 92)(43 82 46 95)(44 91 47 84)(45 80 48 93)```

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

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

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

36 conjugacy classes

 class 1 2A 2B 2C 2D 3 4A ··· 4F 4G 4H 4I 6A 6B 6C 6D 6E 8A ··· 8H 12A ··· 12F 12G 12H order 1 2 2 2 2 3 4 ··· 4 4 4 4 6 6 6 6 6 8 ··· 8 12 ··· 12 12 12 size 1 1 1 1 8 2 2 ··· 2 8 24 24 2 2 2 8 8 6 ··· 6 4 ··· 4 8 8

36 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 D6 D6 D6 C4○D4 C3⋊D4 C4○D8 D4⋊2S3 Q8.13D6 kernel C42.213D6 C4×C3⋊C8 D4⋊Dic3 Q8⋊2Dic3 C12.6Q8 C3×C4.4D4 C4.4D4 C2×C12 C42 C2×D4 C2×Q8 C12 C2×C4 C6 C4 C2 # reps 1 1 2 2 1 1 1 2 1 1 1 4 4 8 2 4

Matrix representation of C42.213D6 in GL6(𝔽73)

 46 0 0 0 0 0 0 46 0 0 0 0 0 0 72 0 0 0 0 0 0 72 0 0 0 0 0 0 46 30 0 0 0 0 39 27
,
 0 1 0 0 0 0 72 0 0 0 0 0 0 0 72 0 0 0 0 0 0 72 0 0 0 0 0 0 72 66 0 0 0 0 42 1
,
 1 0 0 0 0 0 0 72 0 0 0 0 0 0 64 0 0 0 0 0 53 65 0 0 0 0 0 0 1 0 0 0 0 0 31 72
,
 67 6 0 0 0 0 6 6 0 0 0 0 0 0 69 44 0 0 0 0 71 4 0 0 0 0 0 0 0 34 0 0 0 0 58 0

`G:=sub<GL(6,GF(73))| [46,0,0,0,0,0,0,46,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,46,39,0,0,0,0,30,27],[0,72,0,0,0,0,1,0,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,42,0,0,0,0,66,1],[1,0,0,0,0,0,0,72,0,0,0,0,0,0,64,53,0,0,0,0,0,65,0,0,0,0,0,0,1,31,0,0,0,0,0,72],[67,6,0,0,0,0,6,6,0,0,0,0,0,0,69,71,0,0,0,0,44,4,0,0,0,0,0,0,0,58,0,0,0,0,34,0] >;`

C42.213D6 in GAP, Magma, Sage, TeX

`C_4^2._{213}D_6`
`% in TeX`

`G:=Group("C4^2.213D6");`
`// GroupNames label`

`G:=SmallGroup(192,615);`
`// by ID`

`G=gap.SmallGroup(192,615);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,64,590,471,438,102,6278]);`
`// Polycyclic`

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

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