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## G = C42.72D6order 192 = 26·3

### 72nd non-split extension by C42 of D6 acting via D6/C3=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C12 — C42.72D6
 Chief series C1 — C3 — C6 — C2×C6 — C2×C12 — C2×C3⋊C8 — C42.S3 — C42.72D6
 Lower central C3 — C6 — C2×C12 — C42.72D6
 Upper central C1 — C22 — C42 — C4⋊1D4

Generators and relations for C42.72D6
G = < a,b,c,d | a4=b4=c6=1, d2=a2, ab=ba, cac-1=a-1, dad-1=a-1b2, cbc-1=dbd-1=b-1, dcd-1=a2bc-1 >

Subgroups: 304 in 110 conjugacy classes, 39 normal (15 characteristic)
C1, C2, C2 [×2], C2 [×2], C3, C4 [×2], C4 [×4], C22, C22 [×6], C6, C6 [×2], C6 [×2], C8 [×2], C2×C4, C2×C4 [×2], C2×C4 [×2], D4 [×8], C23 [×2], Dic3 [×2], C12 [×2], C12 [×2], C2×C6, C2×C6 [×6], C42, C4⋊C4 [×4], C2×C8 [×2], C2×D4 [×2], C2×D4 [×2], C3⋊C8 [×2], C2×Dic3 [×2], C2×C12, C2×C12 [×2], C3×D4 [×8], C22×C6 [×2], C8⋊C4, D4⋊C4 [×4], C42.C2, C41D4, C2×C3⋊C8 [×2], Dic3⋊C4 [×2], C4⋊Dic3 [×2], C4×C12, C6×D4 [×2], C6×D4 [×2], C42.29C22, C42.S3, D4⋊Dic3 [×4], C12.6Q8, C3×C41D4, C42.72D6
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D6 [×3], C2×D4, C4○D4 [×2], C3⋊D4 [×2], C22×S3, C4.4D4, C8⋊C22 [×2], D42S3 [×2], C2×C3⋊D4, C42.29C22, D126C22 [×2], C23.12D6, C42.72D6

Character table of C42.72D6

 class 1 2A 2B 2C 2D 2E 3 4A 4B 4C 4D 4E 4F 6A 6B 6C 6D 6E 6F 6G 8A 8B 8C 8D 12A 12B 12C 12D 12E 12F size 1 1 1 1 8 8 2 2 2 4 4 24 24 2 2 2 8 8 8 8 12 12 12 12 4 4 4 4 4 4 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 1 1 1 1 1 1 1 1 1 -1 -1 1 1 1 1 1 1 1 -1 -1 -1 -1 1 1 1 1 1 1 linear of order 2 ρ3 1 1 1 1 -1 -1 1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 1 1 1 1 1 1 linear of order 2 ρ4 1 1 1 1 -1 -1 1 1 1 1 1 -1 -1 1 1 1 -1 -1 -1 -1 1 1 1 1 1 1 1 1 1 1 linear of order 2 ρ5 1 1 1 1 1 -1 1 1 1 -1 -1 1 -1 1 1 1 1 -1 1 -1 -1 -1 1 1 -1 1 -1 -1 1 -1 linear of order 2 ρ6 1 1 1 1 1 -1 1 1 1 -1 -1 -1 1 1 1 1 1 -1 1 -1 1 1 -1 -1 -1 1 -1 -1 1 -1 linear of order 2 ρ7 1 1 1 1 -1 1 1 1 1 -1 -1 1 -1 1 1 1 -1 1 -1 1 1 1 -1 -1 -1 1 -1 -1 1 -1 linear of order 2 ρ8 1 1 1 1 -1 1 1 1 1 -1 -1 -1 1 1 1 1 -1 1 -1 1 -1 -1 1 1 -1 1 -1 -1 1 -1 linear of order 2 ρ9 2 2 2 2 -2 -2 -1 2 2 2 2 0 0 -1 -1 -1 1 1 1 1 0 0 0 0 -1 -1 -1 -1 -1 -1 orthogonal lifted from D6 ρ10 2 2 2 2 2 -2 -1 2 2 -2 -2 0 0 -1 -1 -1 -1 1 -1 1 0 0 0 0 1 -1 1 1 -1 1 orthogonal lifted from D6 ρ11 2 2 2 2 0 0 2 -2 -2 -2 2 0 0 2 2 2 0 0 0 0 0 0 0 0 2 -2 -2 -2 -2 2 orthogonal lifted from D4 ρ12 2 2 2 2 0 0 2 -2 -2 2 -2 0 0 2 2 2 0 0 0 0 0 0 0 0 -2 -2 2 2 -2 -2 orthogonal lifted from D4 ρ13 2 2 2 2 -2 2 -1 2 2 -2 -2 0 0 -1 -1 -1 1 -1 1 -1 0 0 0 0 1 -1 1 1 -1 1 orthogonal lifted from D6 ρ14 2 2 2 2 2 2 -1 2 2 2 2 0 0 -1 -1 -1 -1 -1 -1 -1 0 0 0 0 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ15 2 2 2 2 0 0 -1 -2 -2 2 -2 0 0 -1 -1 -1 -√-3 -√-3 √-3 √-3 0 0 0 0 1 1 -1 -1 1 1 complex lifted from C3⋊D4 ρ16 2 2 2 2 0 0 -1 -2 -2 -2 2 0 0 -1 -1 -1 -√-3 √-3 √-3 -√-3 0 0 0 0 -1 1 1 1 1 -1 complex lifted from C3⋊D4 ρ17 2 2 2 2 0 0 -1 -2 -2 -2 2 0 0 -1 -1 -1 √-3 -√-3 -√-3 √-3 0 0 0 0 -1 1 1 1 1 -1 complex lifted from C3⋊D4 ρ18 2 2 2 2 0 0 -1 -2 -2 2 -2 0 0 -1 -1 -1 √-3 √-3 -√-3 -√-3 0 0 0 0 1 1 -1 -1 1 1 complex lifted from C3⋊D4 ρ19 2 -2 -2 2 0 0 2 -2 2 0 0 0 0 -2 2 -2 0 0 0 0 0 0 2i -2i 0 -2 0 0 2 0 complex lifted from C4○D4 ρ20 2 -2 -2 2 0 0 2 2 -2 0 0 0 0 -2 2 -2 0 0 0 0 -2i 2i 0 0 0 2 0 0 -2 0 complex lifted from C4○D4 ρ21 2 -2 -2 2 0 0 2 -2 2 0 0 0 0 -2 2 -2 0 0 0 0 0 0 -2i 2i 0 -2 0 0 2 0 complex lifted from C4○D4 ρ22 2 -2 -2 2 0 0 2 2 -2 0 0 0 0 -2 2 -2 0 0 0 0 2i -2i 0 0 0 2 0 0 -2 0 complex lifted from C4○D4 ρ23 4 -4 4 -4 0 0 4 0 0 0 0 0 0 4 -4 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C8⋊C22 ρ24 4 4 -4 -4 0 0 4 0 0 0 0 0 0 -4 -4 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C8⋊C22 ρ25 4 -4 -4 4 0 0 -2 4 -4 0 0 0 0 2 -2 2 0 0 0 0 0 0 0 0 0 -2 0 0 2 0 symplectic lifted from D4⋊2S3, Schur index 2 ρ26 4 -4 -4 4 0 0 -2 -4 4 0 0 0 0 2 -2 2 0 0 0 0 0 0 0 0 0 2 0 0 -2 0 symplectic lifted from D4⋊2S3, Schur index 2 ρ27 4 4 -4 -4 0 0 -2 0 0 0 0 0 0 2 2 -2 0 0 0 0 0 0 0 0 0 0 -2√-3 2√-3 0 0 complex lifted from D12⋊6C22 ρ28 4 4 -4 -4 0 0 -2 0 0 0 0 0 0 2 2 -2 0 0 0 0 0 0 0 0 0 0 2√-3 -2√-3 0 0 complex lifted from D12⋊6C22 ρ29 4 -4 4 -4 0 0 -2 0 0 0 0 0 0 -2 2 2 0 0 0 0 0 0 0 0 2√-3 0 0 0 0 -2√-3 complex lifted from D12⋊6C22 ρ30 4 -4 4 -4 0 0 -2 0 0 0 0 0 0 -2 2 2 0 0 0 0 0 0 0 0 -2√-3 0 0 0 0 2√-3 complex lifted from D12⋊6C22

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

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

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

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

Matrix representation of C42.72D6 in GL6(𝔽73)

 0 72 0 0 0 0 1 0 0 0 0 0 0 0 0 0 30 13 0 0 0 0 60 43 0 0 43 60 0 0 0 0 13 30 0 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 72 0 0 0 0 0 0 72 0 0
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 60 43 0 0 0 0 30 30 0 0 60 43 0 0 0 0 30 30 0 0
,
 46 0 0 0 0 0 0 27 0 0 0 0 0 0 8 30 65 43 0 0 38 65 35 8 0 0 65 43 65 43 0 0 35 8 35 8

`G:=sub<GL(6,GF(73))| [0,1,0,0,0,0,72,0,0,0,0,0,0,0,0,0,43,13,0,0,0,0,60,30,0,0,30,60,0,0,0,0,13,43,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,1,0,0,0,0,0,0,1,0,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,60,30,0,0,0,0,43,30,0,0,60,30,0,0,0,0,43,30,0,0],[46,0,0,0,0,0,0,27,0,0,0,0,0,0,8,38,65,35,0,0,30,65,43,8,0,0,65,35,65,35,0,0,43,8,43,8] >;`

C42.72D6 in GAP, Magma, Sage, TeX

`C_4^2._{72}D_6`
`% in TeX`

`G:=Group("C4^2.72D6");`
`// GroupNames label`

`G:=SmallGroup(192,630);`
`// by ID`

`G=gap.SmallGroup(192,630);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,477,64,590,135,438,102,6278]);`
`// 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=a^-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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