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## G = C24.23D4order 192 = 26·3

### 23rd non-split extension by C24 of D4 acting via D4/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C12 — C24.23D4
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C4○D12 — C8○D12 — C24.23D4
 Lower central C3 — C6 — C2×C12 — C24.23D4
 Upper central C1 — C2 — C2×C4 — C2×D8

Generators and relations for C24.23D4
G = < a,b,c | a24=c2=1, b4=a12, bab-1=a-1, cac=a17, cbc=a12b3 >

Subgroups: 312 in 108 conjugacy classes, 37 normal (27 characteristic)
C1, C2, C2 [×4], C3, C4 [×2], C4, C22, C22 [×5], S3, C6, C6 [×3], C8 [×2], C8 [×3], C2×C4, C2×C4, D4 [×6], Q8, C23 [×2], Dic3, C12 [×2], D6, C2×C6, C2×C6 [×4], C2×C8, C2×C8, M4(2) [×4], D8 [×4], SD16 [×2], C2×D4 [×2], C4○D4, C3⋊C8 [×3], C24 [×2], Dic6, C4×S3, D12, C3⋊D4, C2×C12, C3×D4 [×4], C22×C6 [×2], C4.D4 [×2], C8.C4, C8○D4, C2×D8, C8⋊C22 [×2], S3×C8, C8⋊S3, C4.Dic3, C4.Dic3 [×2], D4⋊S3 [×2], D4.S3 [×2], C2×C24, C3×D8 [×2], C4○D12, C6×D4 [×2], D4.4D4, C24.C4, C12.D4 [×2], C8○D12, D126C22 [×2], C6×D8, C24.23D4
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×4], C23, D6 [×3], C2×D4 [×2], C4○D4, C3⋊D4 [×2], C22×S3, C4⋊D4, S3×D4, D42S3, C2×C3⋊D4, D4.4D4, D63D4, C24.23D4

Character table of C24.23D4

 class 1 2A 2B 2C 2D 2E 3 4A 4B 4C 6A 6B 6C 6D 6E 6F 6G 8A 8B 8C 8D 8E 8F 8G 12A 12B 24A 24B 24C 24D size 1 1 2 8 8 12 2 2 2 12 2 2 2 8 8 8 8 2 2 4 12 12 24 24 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 0 -1 2 2 0 -1 -1 -1 1 -1 1 -1 -2 -2 -2 0 0 0 0 -1 -1 1 1 1 1 orthogonal lifted from D6 ρ10 2 2 2 2 2 0 -1 2 2 0 -1 -1 -1 -1 -1 -1 -1 2 2 2 0 0 0 0 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ11 2 2 -2 0 0 0 2 2 -2 0 -2 -2 2 0 0 0 0 -2 -2 2 0 0 0 0 2 -2 -2 -2 2 2 orthogonal lifted from D4 ρ12 2 2 -2 0 0 2 2 -2 2 -2 -2 -2 2 0 0 0 0 0 0 0 0 0 0 0 -2 2 0 0 0 0 orthogonal lifted from D4 ρ13 2 2 -2 0 0 0 2 2 -2 0 -2 -2 2 0 0 0 0 2 2 -2 0 0 0 0 2 -2 2 2 -2 -2 orthogonal lifted from D4 ρ14 2 2 2 -2 -2 0 -1 2 2 0 -1 -1 -1 1 1 1 1 2 2 2 0 0 0 0 -1 -1 -1 -1 -1 -1 orthogonal lifted from D6 ρ15 2 2 -2 0 0 -2 2 -2 2 2 -2 -2 2 0 0 0 0 0 0 0 0 0 0 0 -2 2 0 0 0 0 orthogonal lifted from D4 ρ16 2 2 2 2 -2 0 -1 2 2 0 -1 -1 -1 -1 1 -1 1 -2 -2 -2 0 0 0 0 -1 -1 1 1 1 1 orthogonal lifted from D6 ρ17 2 2 -2 0 0 0 -1 2 -2 0 1 1 -1 √-3 -√-3 -√-3 √-3 -2 -2 2 0 0 0 0 -1 1 1 1 -1 -1 complex lifted from C3⋊D4 ρ18 2 2 -2 0 0 0 -1 2 -2 0 1 1 -1 √-3 √-3 -√-3 -√-3 2 2 -2 0 0 0 0 -1 1 -1 -1 1 1 complex lifted from C3⋊D4 ρ19 2 2 -2 0 0 0 -1 2 -2 0 1 1 -1 -√-3 -√-3 √-3 √-3 2 2 -2 0 0 0 0 -1 1 -1 -1 1 1 complex lifted from C3⋊D4 ρ20 2 2 -2 0 0 0 -1 2 -2 0 1 1 -1 -√-3 √-3 √-3 -√-3 -2 -2 2 0 0 0 0 -1 1 1 1 -1 -1 complex lifted from C3⋊D4 ρ21 2 2 2 0 0 0 2 -2 -2 0 2 2 2 0 0 0 0 0 0 0 -2i 2i 0 0 -2 -2 0 0 0 0 complex lifted from C4○D4 ρ22 2 2 2 0 0 0 2 -2 -2 0 2 2 2 0 0 0 0 0 0 0 2i -2i 0 0 -2 -2 0 0 0 0 complex lifted from C4○D4 ρ23 4 4 -4 0 0 0 -2 -4 4 0 2 2 -2 0 0 0 0 0 0 0 0 0 0 0 2 -2 0 0 0 0 orthogonal lifted from S3×D4 ρ24 4 -4 0 0 0 0 4 0 0 0 0 0 -4 0 0 0 0 -2√2 2√2 0 0 0 0 0 0 0 2√2 -2√2 0 0 orthogonal lifted from D4.4D4 ρ25 4 -4 0 0 0 0 4 0 0 0 0 0 -4 0 0 0 0 2√2 -2√2 0 0 0 0 0 0 0 -2√2 2√2 0 0 orthogonal lifted from D4.4D4 ρ26 4 4 4 0 0 0 -2 -4 -4 0 -2 -2 -2 0 0 0 0 0 0 0 0 0 0 0 2 2 0 0 0 0 symplectic lifted from D4⋊2S3, Schur index 2 ρ27 4 -4 0 0 0 0 -2 0 0 0 2√-3 -2√-3 2 0 0 0 0 2√2 -2√2 0 0 0 0 0 0 0 √2 -√2 √-6 -√-6 complex faithful ρ28 4 -4 0 0 0 0 -2 0 0 0 -2√-3 2√-3 2 0 0 0 0 2√2 -2√2 0 0 0 0 0 0 0 √2 -√2 -√-6 √-6 complex faithful ρ29 4 -4 0 0 0 0 -2 0 0 0 2√-3 -2√-3 2 0 0 0 0 -2√2 2√2 0 0 0 0 0 0 0 -√2 √2 -√-6 √-6 complex faithful ρ30 4 -4 0 0 0 0 -2 0 0 0 -2√-3 2√-3 2 0 0 0 0 -2√2 2√2 0 0 0 0 0 0 0 -√2 √2 √-6 -√-6 complex faithful

Smallest permutation representation of C24.23D4
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)
(1 26 19 32 13 38 7 44)(2 25 20 31 14 37 8 43)(3 48 21 30 15 36 9 42)(4 47 22 29 16 35 10 41)(5 46 23 28 17 34 11 40)(6 45 24 27 18 33 12 39)
(1 35)(2 28)(3 45)(4 38)(5 31)(6 48)(7 41)(8 34)(9 27)(10 44)(11 37)(12 30)(13 47)(14 40)(15 33)(16 26)(17 43)(18 36)(19 29)(20 46)(21 39)(22 32)(23 25)(24 42)```

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

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

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

Matrix representation of C24.23D4 in GL4(𝔽7) generated by

 2 1 2 1 6 1 3 1 6 4 3 3 2 2 0 4
,
 1 3 6 1 3 0 1 5 1 3 5 0 3 5 2 1
,
 5 1 0 0 4 2 0 0 6 3 5 1 5 1 4 2
`G:=sub<GL(4,GF(7))| [2,6,6,2,1,1,4,2,2,3,3,0,1,1,3,4],[1,3,1,3,3,0,3,5,6,1,5,2,1,5,0,1],[5,4,6,5,1,2,3,1,0,0,5,4,0,0,1,2] >;`

C24.23D4 in GAP, Magma, Sage, TeX

`C_{24}._{23}D_4`
`% in TeX`

`G:=Group("C24.23D4");`
`// GroupNames label`

`G:=SmallGroup(192,719);`
`// by ID`

`G=gap.SmallGroup(192,719);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,120,254,219,1123,297,136,438,102,6278]);`
`// Polycyclic`

`G:=Group<a,b,c|a^24=c^2=1,b^4=a^12,b*a*b^-1=a^-1,c*a*c=a^17,c*b*c=a^12*b^3>;`
`// generators/relations`

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