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## G = C24.3D6order 192 = 26·3

### 2nd non-split extension by C24 of D6 acting via D6/C2=S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C2×A4 — C24.3D6
 Chief series C1 — C22 — A4 — C2×A4 — C22×A4 — C2×A4⋊C4 — C24.3D6
 Lower central A4 — C2×A4 — C24.3D6
 Upper central C1 — C22 — C2×C4

Generators and relations for C24.3D6
G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e6=f2=a, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, fcf-1=ede-1=cd=dc, ece-1=d, df=fd, fef-1=be5 >

Subgroups: 394 in 109 conjugacy classes, 23 normal (21 characteristic)
C1, C2, C2, C3, C4, C22, C22, C6, C2×C4, C2×C4, C23, C23, Dic3, C12, A4, C2×C6, C22⋊C4, C4⋊C4, C22×C4, C24, C2×Dic3, C2×C12, C2×A4, C2.C42, C2×C22⋊C4, C2×C4⋊C4, C23×C4, Dic3⋊C4, A4⋊C4, A4⋊C4, C4×A4, C22×A4, C23.8Q8, C2×A4⋊C4, C2×C4×A4, C24.3D6
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Q8, D6, C4⋊C4, Dic6, C4×S3, C3⋊D4, S4, Dic3⋊C4, C2×S4, A4⋊Q8, C4×S4, A4⋊D4, C24.3D6

Character table of C24.3D6

 class 1 2A 2B 2C 2D 2E 2F 2G 3 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 6A 6B 6C 12A 12B 12C 12D size 1 1 1 1 3 3 3 3 8 2 2 6 6 12 12 12 12 12 12 12 12 8 8 8 8 8 8 8 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 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 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 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 linear of order 2 ρ5 1 1 -1 -1 -1 1 1 -1 1 -i i -i i i 1 -1 -1 -i -i i 1 -1 -1 1 -i i i -i linear of order 4 ρ6 1 1 -1 -1 -1 1 1 -1 1 -i i -i i -i -1 1 1 i i -i -1 -1 -1 1 -i i i -i linear of order 4 ρ7 1 1 -1 -1 -1 1 1 -1 1 i -i i -i -i 1 -1 -1 i i -i 1 -1 -1 1 i -i -i i linear of order 4 ρ8 1 1 -1 -1 -1 1 1 -1 1 i -i i -i i -1 1 1 -i -i i -1 -1 -1 1 i -i -i i linear of order 4 ρ9 2 -2 2 -2 2 -2 2 -2 2 0 0 0 0 0 0 0 0 0 0 0 0 2 -2 -2 0 0 0 0 orthogonal lifted from D4 ρ10 2 2 2 2 2 2 2 2 -1 -2 -2 -2 -2 0 0 0 0 0 0 0 0 -1 -1 -1 1 1 1 1 orthogonal lifted from D6 ρ11 2 2 2 2 2 2 2 2 -1 2 2 2 2 0 0 0 0 0 0 0 0 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ12 2 -2 -2 2 -2 -2 2 2 2 0 0 0 0 0 0 0 0 0 0 0 0 -2 2 -2 0 0 0 0 symplectic lifted from Q8, Schur index 2 ρ13 2 -2 -2 2 -2 -2 2 2 -1 0 0 0 0 0 0 0 0 0 0 0 0 1 -1 1 √3 √3 -√3 -√3 symplectic lifted from Dic6, Schur index 2 ρ14 2 -2 -2 2 -2 -2 2 2 -1 0 0 0 0 0 0 0 0 0 0 0 0 1 -1 1 -√3 -√3 √3 √3 symplectic lifted from Dic6, Schur index 2 ρ15 2 2 -2 -2 -2 2 2 -2 -1 -2i 2i -2i 2i 0 0 0 0 0 0 0 0 1 1 -1 i -i -i i complex lifted from C4×S3 ρ16 2 2 -2 -2 -2 2 2 -2 -1 2i -2i 2i -2i 0 0 0 0 0 0 0 0 1 1 -1 -i i i -i complex lifted from C4×S3 ρ17 2 -2 2 -2 2 -2 2 -2 -1 0 0 0 0 0 0 0 0 0 0 0 0 -1 1 1 -√-3 √-3 -√-3 √-3 complex lifted from C3⋊D4 ρ18 2 -2 2 -2 2 -2 2 -2 -1 0 0 0 0 0 0 0 0 0 0 0 0 -1 1 1 √-3 -√-3 √-3 -√-3 complex lifted from C3⋊D4 ρ19 3 3 3 3 -1 -1 -1 -1 0 -3 -3 1 1 1 -1 1 -1 -1 1 -1 1 0 0 0 0 0 0 0 orthogonal lifted from C2×S4 ρ20 3 3 3 3 -1 -1 -1 -1 0 3 3 -1 -1 1 1 -1 1 -1 1 -1 -1 0 0 0 0 0 0 0 orthogonal lifted from S4 ρ21 3 3 3 3 -1 -1 -1 -1 0 -3 -3 1 1 -1 1 -1 1 1 -1 1 -1 0 0 0 0 0 0 0 orthogonal lifted from C2×S4 ρ22 3 3 3 3 -1 -1 -1 -1 0 3 3 -1 -1 -1 -1 1 -1 1 -1 1 1 0 0 0 0 0 0 0 orthogonal lifted from S4 ρ23 3 3 -3 -3 1 -1 -1 1 0 3i -3i -i i -i 1 1 -1 -i i i -1 0 0 0 0 0 0 0 complex lifted from C4×S4 ρ24 3 3 -3 -3 1 -1 -1 1 0 3i -3i -i i i -1 -1 1 i -i -i 1 0 0 0 0 0 0 0 complex lifted from C4×S4 ρ25 3 3 -3 -3 1 -1 -1 1 0 -3i 3i i -i -i -1 -1 1 -i i i 1 0 0 0 0 0 0 0 complex lifted from C4×S4 ρ26 3 3 -3 -3 1 -1 -1 1 0 -3i 3i i -i i 1 1 -1 i -i -i -1 0 0 0 0 0 0 0 complex lifted from C4×S4 ρ27 6 -6 6 -6 -2 2 -2 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from A4⋊D4 ρ28 6 -6 -6 6 2 2 -2 -2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from A4⋊Q8, Schur index 2

Smallest permutation representation of C24.3D6
On 48 points
Generators in S48
```(1 7)(2 8)(3 9)(4 10)(5 11)(6 12)(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)(25 31)(26 32)(27 33)(28 34)(29 35)(30 36)(37 43)(38 44)(39 45)(40 46)(41 47)(42 48)
(1 47)(2 48)(3 37)(4 38)(5 39)(6 40)(7 41)(8 42)(9 43)(10 44)(11 45)(12 46)(13 36)(14 25)(15 26)(16 27)(17 28)(18 29)(19 30)(20 31)(21 32)(22 33)(23 34)(24 35)
(1 18)(2 13)(3 9)(4 21)(5 16)(6 12)(7 24)(8 19)(10 15)(11 22)(14 20)(17 23)(25 31)(26 44)(27 39)(28 34)(29 47)(30 42)(32 38)(33 45)(35 41)(36 48)(37 43)(40 46)
(1 24)(2 8)(3 20)(4 15)(5 11)(6 23)(7 18)(9 14)(10 21)(12 17)(13 19)(16 22)(25 43)(26 38)(27 33)(28 46)(29 41)(30 36)(31 37)(32 44)(34 40)(35 47)(39 45)(42 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 14 7 20)(2 30 8 36)(3 24 9 18)(4 28 10 34)(5 22 11 16)(6 26 12 32)(13 48 19 42)(15 46 21 40)(17 44 23 38)(25 41 31 47)(27 39 33 45)(29 37 35 43)```

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

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

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

Matrix representation of C24.3D6 in GL5(𝔽13)

 12 0 0 0 0 0 12 0 0 0 0 0 12 0 0 0 0 0 12 0 0 0 0 0 12
,
 12 0 0 0 0 0 12 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 12 0 0 0 0 0 12
,
 1 0 0 0 0 0 1 0 0 0 0 0 12 0 0 0 0 0 1 0 0 0 0 0 12
,
 10 3 0 0 0 10 7 0 0 0 0 0 0 12 0 0 0 0 0 1 0 0 8 0 0
,
 5 5 0 0 0 0 8 0 0 0 0 0 0 0 8 0 0 0 8 0 0 0 8 0 0

`G:=sub<GL(5,GF(13))| [12,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12],[12,0,0,0,0,0,12,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,12,0,0,0,0,0,12],[1,0,0,0,0,0,1,0,0,0,0,0,12,0,0,0,0,0,1,0,0,0,0,0,12],[10,10,0,0,0,3,7,0,0,0,0,0,0,0,8,0,0,12,0,0,0,0,0,1,0],[5,0,0,0,0,5,8,0,0,0,0,0,0,0,8,0,0,0,8,0,0,0,8,0,0] >;`

C24.3D6 in GAP, Magma, Sage, TeX

`C_2^4._3D_6`
`% in TeX`

`G:=Group("C2^4.3D6");`
`// GroupNames label`

`G:=SmallGroup(192,970);`
`// by ID`

`G=gap.SmallGroup(192,970);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-3,-2,2,56,141,36,1124,4037,285,2358,475]);`
`// Polycyclic`

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

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