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## G = C24.43D4order 192 = 26·3

### 43rd non-split extension by C24 of D4 acting via D4/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C12 — C24.43D4
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C4×Dic3 — C23.12D6 — C24.43D4
 Lower central C3 — C6 — C2×C12 — C24.43D4
 Upper central C1 — C22 — C2×C4 — C2×SD16

Generators and relations for C24.43D4
G = < a,b,c | a24=b4=1, c2=a12, bab-1=a17, cac-1=a11, cbc-1=a12b-1 >

Subgroups: 408 in 130 conjugacy classes, 43 normal (31 characteristic)
C1, C2, C2 [×2], C2 [×2], C3, C4 [×2], C4 [×4], C22, C22 [×6], S3, C6, C6 [×2], C6, C8 [×2], C8 [×2], C2×C4, C2×C4 [×4], D4 [×4], Q8 [×4], C23 [×2], Dic3 [×3], C12 [×2], C12, D6 [×3], C2×C6, C2×C6 [×3], C42, C22⋊C4 [×4], C2×C8, C2×C8, D8 [×2], SD16 [×4], Q16 [×2], C2×D4, C2×D4, C2×Q8, C2×Q8, C3⋊C8 [×2], C24 [×2], Dic6 [×2], D12 [×2], C2×Dic3 [×2], C2×Dic3, C2×C12, C2×C12, C3×D4 [×2], C3×Q8 [×2], C22×S3, C22×C6, C4×C8, C4.4D4 [×2], C2×D8, C2×SD16, C2×SD16, C2×Q16, C24⋊C2 [×2], C2×C3⋊C8, C4×Dic3, D6⋊C4 [×2], D4⋊S3 [×2], C3⋊Q16 [×2], C6.D4 [×2], C2×C24, C3×SD16 [×2], C2×Dic6, C2×D12, C6×D4, C6×Q8, C8.12D4, C8×Dic3, C2×C24⋊C2, C2×D4⋊S3, C23.12D6, C2×C3⋊Q16, C12.23D4, C6×SD16, C24.43D4
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, Q8.7D6 [×2], C123D4, C24.43D4

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

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

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

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

36 conjugacy classes

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

36 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 S3 D4 D4 D4 D6 D6 D6 C3⋊D4 C4○D8 S3×D4 S3×D4 Q8.7D6 kernel C24.43D4 C8×Dic3 C2×C24⋊C2 C2×D4⋊S3 C23.12D6 C2×C3⋊Q16 C12.23D4 C6×SD16 C2×SD16 C3⋊C8 C24 C2×Dic3 C2×C8 C2×D4 C2×Q8 C8 C6 C4 C22 C2 # reps 1 1 1 1 1 1 1 1 1 2 2 2 1 1 1 4 8 1 1 4

Matrix representation of C24.43D4 in GL6(𝔽73)

 67 6 0 0 0 0 67 67 0 0 0 0 0 0 0 1 0 0 0 0 72 1 0 0 0 0 0 0 53 71 0 0 0 0 18 20
,
 0 46 0 0 0 0 27 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 20 2 0 0 0 0 55 53
,
 6 67 0 0 0 0 67 67 0 0 0 0 0 0 0 72 0 0 0 0 72 0 0 0 0 0 0 0 53 71 0 0 0 0 17 20

`G:=sub<GL(6,GF(73))| [67,67,0,0,0,0,6,67,0,0,0,0,0,0,0,72,0,0,0,0,1,1,0,0,0,0,0,0,53,18,0,0,0,0,71,20],[0,27,0,0,0,0,46,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,20,55,0,0,0,0,2,53],[6,67,0,0,0,0,67,67,0,0,0,0,0,0,0,72,0,0,0,0,72,0,0,0,0,0,0,0,53,17,0,0,0,0,71,20] >;`

C24.43D4 in GAP, Magma, Sage, TeX

`C_{24}._{43}D_4`
`% in TeX`

`G:=Group("C24.43D4");`
`// GroupNames label`

`G:=SmallGroup(192,727);`
`// by ID`

`G=gap.SmallGroup(192,727);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,701,1094,135,184,570,297,136,6278]);`
`// 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^11,c*b*c^-1=a^12*b^-1>;`
`// generators/relations`

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