Copied to
clipboard

## G = C42.19D6order 192 = 26·3

### 19th non-split extension by C42 of D6 acting via D6/C3=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C12 — C42.19D6
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C2×D12 — C4⋊D12 — C42.19D6
 Lower central C3 — C6 — C2×C12 — C42.19D6
 Upper central C1 — C22 — C42 — C8⋊C4

Generators and relations for C42.19D6
G = < a,b,c,d | a4=b4=1, c6=a2b-1, d2=a2b2, ab=ba, cac-1=ab2, dad-1=a-1b2, bc=cb, dbd-1=b-1, dcd-1=b-1c5 >

Subgroups: 440 in 110 conjugacy classes, 39 normal (15 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, C2×C4, D4, C23, Dic3, C12, C12, D6, C2×C6, C42, C4⋊C4, C2×C8, C2×D4, C24, D12, C2×Dic3, C2×C12, C2×C12, C22×S3, C8⋊C4, D4⋊C4, C42.C2, C41D4, Dic3⋊C4, C4⋊Dic3, C4×C12, C2×C24, C2×D12, C2×D12, C42.29C22, C2.D24, C3×C8⋊C4, C12.6Q8, C4⋊D12, C42.19D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, D12, C22×S3, C4.4D4, C8⋊C22, C2×D12, C4○D12, C42.29C22, C427S3, C8⋊D6, C42.19D6

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

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

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

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

36 conjugacy classes

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

36 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + image C1 C2 C2 C2 C2 S3 D4 D6 D6 C4○D4 D12 C4○D12 C8⋊C22 C8⋊D6 kernel C42.19D6 C2.D24 C3×C8⋊C4 C12.6Q8 C4⋊D12 C8⋊C4 C2×C12 C42 C2×C8 C12 C2×C4 C4 C6 C2 # reps 1 4 1 1 1 1 2 1 2 4 4 8 2 4

Matrix representation of C42.19D6 in GL6(𝔽73)

 1 55 0 0 0 0 65 72 0 0 0 0 0 0 9 18 58 65 0 0 55 64 8 66 0 0 66 8 64 55 0 0 65 58 18 9
,
 72 0 0 0 0 0 0 72 0 0 0 0 0 0 66 59 0 0 0 0 14 7 0 0 0 0 0 0 66 59 0 0 0 0 14 7
,
 46 48 0 0 0 0 70 27 0 0 0 0 0 0 0 0 72 72 0 0 0 0 1 0 0 0 7 66 0 0 0 0 7 14 0 0
,
 27 25 0 0 0 0 0 46 0 0 0 0 0 0 0 0 66 7 0 0 0 0 14 7 0 0 66 7 0 0 0 0 14 7 0 0

`G:=sub<GL(6,GF(73))| [1,65,0,0,0,0,55,72,0,0,0,0,0,0,9,55,66,65,0,0,18,64,8,58,0,0,58,8,64,18,0,0,65,66,55,9],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,66,14,0,0,0,0,59,7,0,0,0,0,0,0,66,14,0,0,0,0,59,7],[46,70,0,0,0,0,48,27,0,0,0,0,0,0,0,0,7,7,0,0,0,0,66,14,0,0,72,1,0,0,0,0,72,0,0,0],[27,0,0,0,0,0,25,46,0,0,0,0,0,0,0,0,66,14,0,0,0,0,7,7,0,0,66,14,0,0,0,0,7,7,0,0] >;`

C42.19D6 in GAP, Magma, Sage, TeX

`C_4^2._{19}D_6`
`% in TeX`

`G:=Group("C4^2.19D6");`
`// GroupNames label`

`G:=SmallGroup(192,272);`
`// by ID`

`G=gap.SmallGroup(192,272);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,120,254,387,142,1123,136,6278]);`
`// Polycyclic`

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

׿
×
𝔽