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## G = C24.C23order 192 = 26·3

### 6th non-split extension by C24 of C23 acting faithfully

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — C24.C23
 Chief series C1 — C3 — C6 — C12 — C4×S3 — C4○D12 — Q8.15D6 — C24.C23
 Lower central C3 — C6 — C12 — C24.C23
 Upper central C1 — C2 — C2×C4 — C8.C22

Generators and relations for C24.C23
G = < a,b,c,d | a24=b2=1, c2=d2=a12, bab=a5, cac-1=a7, dad-1=a19, bc=cb, dbd-1=a12b, cd=dc >

Subgroups: 736 in 258 conjugacy classes, 99 normal (45 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, Q8, C23, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C2×C6, C2×C8, M4(2), M4(2), D8, SD16, SD16, Q16, Q16, C2×D4, C2×Q8, C2×Q8, C4○D4, C4○D4, C3⋊C8, C24, Dic6, Dic6, C4×S3, C4×S3, D12, D12, D12, C3⋊D4, C3⋊D4, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C3×Q8, C3×Q8, C22×S3, C8○D4, C2×SD16, C4○D8, C8⋊C22, C8.C22, C8.C22, 2+ 1+4, 2- 1+4, S3×C8, C8⋊S3, C24⋊C2, D24, C2×C3⋊C8, D4⋊S3, D4.S3, Q82S3, Q82S3, C3⋊Q16, C3×M4(2), C3×SD16, C3×Q16, C2×D12, C2×D12, C4○D12, C4○D12, S3×D4, S3×D4, S3×Q8, S3×Q8, Q83S3, Q83S3, C6×Q8, C3×C4○D4, D4○SD16, D12.C4, C8⋊D6, S3×SD16, Q83D6, Q16⋊S3, D24⋊C2, C2×Q82S3, Q8.13D6, C3×C8.C22, Q8.15D6, D4○D12, C24.C23
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C24, C22×S3, C22×D4, S3×D4, S3×C23, D4○SD16, C2×S3×D4, C24.C23

Smallest permutation representation of C24.C23
On 48 points
Generators in S48
```(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)
(2 6)(3 11)(4 16)(5 21)(8 12)(9 17)(10 22)(14 18)(15 23)(20 24)(25 41)(26 46)(28 32)(29 37)(30 42)(31 47)(34 38)(35 43)(36 48)(40 44)
(1 39 13 27)(2 46 14 34)(3 29 15 41)(4 36 16 48)(5 43 17 31)(6 26 18 38)(7 33 19 45)(8 40 20 28)(9 47 21 35)(10 30 22 42)(11 37 23 25)(12 44 24 32)
(1 4 13 16)(2 23 14 11)(3 18 15 6)(5 8 17 20)(7 22 19 10)(9 12 21 24)(25 34 37 46)(26 29 38 41)(27 48 39 36)(28 43 40 31)(30 33 42 45)(32 47 44 35)```

`G:=sub<Sym(48)| (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), (2,6)(3,11)(4,16)(5,21)(8,12)(9,17)(10,22)(14,18)(15,23)(20,24)(25,41)(26,46)(28,32)(29,37)(30,42)(31,47)(34,38)(35,43)(36,48)(40,44), (1,39,13,27)(2,46,14,34)(3,29,15,41)(4,36,16,48)(5,43,17,31)(6,26,18,38)(7,33,19,45)(8,40,20,28)(9,47,21,35)(10,30,22,42)(11,37,23,25)(12,44,24,32), (1,4,13,16)(2,23,14,11)(3,18,15,6)(5,8,17,20)(7,22,19,10)(9,12,21,24)(25,34,37,46)(26,29,38,41)(27,48,39,36)(28,43,40,31)(30,33,42,45)(32,47,44,35)>;`

`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), (2,6)(3,11)(4,16)(5,21)(8,12)(9,17)(10,22)(14,18)(15,23)(20,24)(25,41)(26,46)(28,32)(29,37)(30,42)(31,47)(34,38)(35,43)(36,48)(40,44), (1,39,13,27)(2,46,14,34)(3,29,15,41)(4,36,16,48)(5,43,17,31)(6,26,18,38)(7,33,19,45)(8,40,20,28)(9,47,21,35)(10,30,22,42)(11,37,23,25)(12,44,24,32), (1,4,13,16)(2,23,14,11)(3,18,15,6)(5,8,17,20)(7,22,19,10)(9,12,21,24)(25,34,37,46)(26,29,38,41)(27,48,39,36)(28,43,40,31)(30,33,42,45)(32,47,44,35) );`

`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)], [(2,6),(3,11),(4,16),(5,21),(8,12),(9,17),(10,22),(14,18),(15,23),(20,24),(25,41),(26,46),(28,32),(29,37),(30,42),(31,47),(34,38),(35,43),(36,48),(40,44)], [(1,39,13,27),(2,46,14,34),(3,29,15,41),(4,36,16,48),(5,43,17,31),(6,26,18,38),(7,33,19,45),(8,40,20,28),(9,47,21,35),(10,30,22,42),(11,37,23,25),(12,44,24,32)], [(1,4,13,16),(2,23,14,11),(3,18,15,6),(5,8,17,20),(7,22,19,10),(9,12,21,24),(25,34,37,46),(26,29,38,41),(27,48,39,36),(28,43,40,31),(30,33,42,45),(32,47,44,35)]])`

33 conjugacy classes

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

33 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 8 type + + + + + + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 S3 D4 D4 D4 D6 D6 D6 D6 D6 S3×D4 S3×D4 D4○SD16 C24.C23 kernel C24.C23 D12.C4 C8⋊D6 S3×SD16 Q8⋊3D6 Q16⋊S3 D24⋊C2 C2×Q8⋊2S3 Q8.13D6 C3×C8.C22 Q8.15D6 D4○D12 C8.C22 Dic6 D12 C3⋊D4 M4(2) SD16 Q16 C2×Q8 C4○D4 C4 C22 C3 C1 # reps 1 1 1 2 2 2 2 1 1 1 1 1 1 1 1 2 1 2 2 1 1 1 1 2 1

Matrix representation of C24.C23 in GL6(𝔽73)

 0 72 0 0 0 0 1 72 0 0 0 0 0 0 0 0 12 61 0 0 6 0 0 61 0 0 0 67 6 67 0 0 67 6 6 67
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 1 0 72 0 0 0 1 0 0 72
,
 72 0 0 0 0 0 0 72 0 0 0 0 0 0 0 61 0 0 0 0 67 0 0 0 0 0 0 67 67 6 0 0 67 67 6 6
,
 72 0 0 0 0 0 0 72 0 0 0 0 0 0 0 0 12 61 0 0 67 0 12 0 0 0 0 6 6 67 0 0 67 6 6 67

`G:=sub<GL(6,GF(73))| [0,1,0,0,0,0,72,72,0,0,0,0,0,0,0,6,0,67,0,0,0,0,67,6,0,0,12,0,6,6,0,0,61,61,67,67],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,1,1,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,67,0,67,0,0,61,0,67,67,0,0,0,0,67,6,0,0,0,0,6,6],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,67,0,67,0,0,0,0,6,6,0,0,12,12,6,6,0,0,61,0,67,67] >;`

C24.C23 in GAP, Magma, Sage, TeX

`C_{24}.C_2^3`
`% in TeX`

`G:=Group("C24.C2^3");`
`// GroupNames label`

`G:=SmallGroup(192,1337);`
`// by ID`

`G=gap.SmallGroup(192,1337);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,477,184,570,185,438,235,102,6278]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^24=b^2=1,c^2=d^2=a^12,b*a*b=a^5,c*a*c^-1=a^7,d*a*d^-1=a^19,b*c=c*b,d*b*d^-1=a^12*b,c*d=d*c>;`
`// generators/relations`

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