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## G = C24.22D4order 192 = 26·3

### 22nd non-split extension by C24 of D4 acting via D4/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C12 — C24.22D4
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C4×Dic3 — C23.12D6 — C24.22D4
 Lower central C3 — C6 — C2×C12 — C24.22D4
 Upper central C1 — C22 — C2×C4 — C2×D8

Generators and relations for C24.22D4
G = < a,b,c | a24=b4=1, c2=a12, bab-1=a17, cac-1=a-1, cbc-1=a12b-1 >

Subgroups: 376 in 130 conjugacy classes, 43 normal (21 characteristic)
C1, C2, C2 [×2], C2 [×2], C3, C4 [×2], C4 [×4], C22, C22 [×6], C6, C6 [×2], C6 [×2], C8 [×2], C8 [×2], C2×C4, C2×C4 [×4], D4 [×4], Q8 [×4], C23 [×2], Dic3 [×4], C12 [×2], C2×C6, C2×C6 [×6], C42, C22⋊C4 [×4], C2×C8, C2×C8, D8 [×2], SD16 [×4], Q16 [×2], C2×D4 [×2], C2×Q8 [×2], C3⋊C8 [×2], C24 [×2], Dic6 [×4], C2×Dic3 [×2], C2×Dic3 [×2], C2×C12, C3×D4 [×4], C22×C6 [×2], C4×C8, C4.4D4 [×2], C2×D8, C2×SD16 [×2], C2×Q16, Dic12 [×2], C2×C3⋊C8, C4×Dic3, D4.S3 [×4], C6.D4 [×4], C2×C24, C3×D8 [×2], C2×Dic6 [×2], C6×D4 [×2], C8.12D4, C8×Dic3, C2×Dic12, C2×D4.S3 [×2], C23.12D6 [×2], C6×D8, C24.22D4
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×6], C23, D6 [×3], C2×D4 [×3], C3⋊D4 [×2], C22×S3, C41D4, C4○D8 [×2], S3×D4 [×2], C2×C3⋊D4, C8.12D4, D83S3 [×2], C123D4, C24.22D4

Smallest permutation representation of C24.22D4
On 96 points
Generators in S96
```(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 38 85 53)(2 31 86 70)(3 48 87 63)(4 41 88 56)(5 34 89 49)(6 27 90 66)(7 44 91 59)(8 37 92 52)(9 30 93 69)(10 47 94 62)(11 40 95 55)(12 33 96 72)(13 26 73 65)(14 43 74 58)(15 36 75 51)(16 29 76 68)(17 46 77 61)(18 39 78 54)(19 32 79 71)(20 25 80 64)(21 42 81 57)(22 35 82 50)(23 28 83 67)(24 45 84 60)
(1 65 13 53)(2 64 14 52)(3 63 15 51)(4 62 16 50)(5 61 17 49)(6 60 18 72)(7 59 19 71)(8 58 20 70)(9 57 21 69)(10 56 22 68)(11 55 23 67)(12 54 24 66)(25 74 37 86)(26 73 38 85)(27 96 39 84)(28 95 40 83)(29 94 41 82)(30 93 42 81)(31 92 43 80)(32 91 44 79)(33 90 45 78)(34 89 46 77)(35 88 47 76)(36 87 48 75)```

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

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

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

36 conjugacy classes

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

36 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 S3 D4 D4 D4 D6 D6 C3⋊D4 C4○D8 S3×D4 S3×D4 D8⋊3S3 kernel C24.22D4 C8×Dic3 C2×Dic12 C2×D4.S3 C23.12D6 C6×D8 C2×D8 C3⋊C8 C24 C2×Dic3 C2×C8 C2×D4 C8 C6 C4 C22 C2 # reps 1 1 1 2 2 1 1 2 2 2 1 2 4 8 1 1 4

Matrix representation of C24.22D4 in GL4(𝔽73) generated by

 64 0 0 0 8 8 0 0 0 0 32 32 0 0 57 0
,
 72 7 0 0 31 1 0 0 0 0 46 0 0 0 0 46
,
 72 7 0 0 0 1 0 0 0 0 46 0 0 0 27 27
`G:=sub<GL(4,GF(73))| [64,8,0,0,0,8,0,0,0,0,32,57,0,0,32,0],[72,31,0,0,7,1,0,0,0,0,46,0,0,0,0,46],[72,0,0,0,7,1,0,0,0,0,46,27,0,0,0,27] >;`

C24.22D4 in GAP, Magma, Sage, TeX

`C_{24}._{22}D_4`
`% in TeX`

`G:=Group("C24.22D4");`
`// GroupNames label`

`G:=SmallGroup(192,714);`
`// by ID`

`G=gap.SmallGroup(192,714);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,701,1094,135,570,297,136,6278]);`
`// Polycyclic`

`G:=Group<a,b,c|a^24=b^4=1,c^2=a^12,b*a*b^-1=a^17,c*a*c^-1=a^-1,c*b*c^-1=a^12*b^-1>;`
`// generators/relations`

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