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## G = C23⋊D12order 192 = 26·3

### The semidirect product of C23 and D12 acting via D12/C3=D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22×C6 — C23⋊D12
 Chief series C1 — C3 — C6 — C2×C6 — C22×C6 — C2×C3⋊D4 — D4⋊6D6 — C23⋊D12
 Lower central C3 — C6 — C22×C6 — C23⋊D12
 Upper central C1 — C2 — C23 — C23⋊C4

Generators and relations for C23⋊D12
G = < a,b,c,d,e | a2=b2=c2=d12=e2=1, ab=ba, ac=ca, dad-1=eae=abc, dbd-1=bc=cb, be=eb, cd=dc, ce=ec, ede=d-1 >

Subgroups: 800 in 198 conjugacy classes, 39 normal (21 characteristic)
C1, C2, C2 [×8], C3, C4 [×6], C22, C22 [×2], C22 [×18], S3 [×4], C6, C6 [×4], C2×C4, C2×C4 [×8], D4 [×15], Q8, C23 [×2], C23 [×8], Dic3 [×3], C12 [×3], D6 [×15], C2×C6, C2×C6 [×2], C2×C6 [×3], C22⋊C4 [×2], C22⋊C4 [×4], C2×D4, C2×D4 [×8], C4○D4 [×3], C24, Dic6, C4×S3 [×2], D12 [×5], C2×Dic3 [×2], C2×Dic3 [×2], C3⋊D4 [×8], C2×C12, C2×C12 [×2], C3×D4 [×2], C22×S3 [×2], C22×S3 [×6], C22×C6 [×2], C23⋊C4, C23⋊C4 [×2], C22≀C2 [×3], 2+ 1+4, D6⋊C4 [×3], C6.D4, C3×C22⋊C4 [×2], C2×D12 [×2], C4○D12, S3×D4 [×2], D42S3 [×2], C2×C3⋊D4 [×2], C2×C3⋊D4 [×2], C6×D4, S3×C23, C2≀C22, C23.6D6 [×2], C3×C23⋊C4, D6⋊D4 [×2], C232D6, D46D6, C23⋊D12
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×6], C23, D6 [×3], C2×D4 [×3], D12 [×2], C22×S3, C22≀C2, C2×D12, S3×D4 [×2], C2≀C22, D6⋊D4, C23⋊D12

Character table of C23⋊D12

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 3 4A 4B 4C 4D 4E 4F 6A 6B 6C 6D 6E 12A 12B 12C 12D 12E size 1 1 2 2 2 4 12 12 12 12 2 4 8 8 12 12 24 2 4 4 4 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 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 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 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 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 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 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 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 linear of order 2 ρ9 2 2 -2 2 -2 0 0 2 0 0 2 0 0 0 0 -2 0 2 2 -2 -2 0 0 0 0 0 0 orthogonal lifted from D4 ρ10 2 2 -2 -2 2 0 -2 0 0 0 2 0 0 0 2 0 0 2 -2 -2 2 0 0 0 0 0 0 orthogonal lifted from D4 ρ11 2 2 2 2 2 -2 0 0 0 0 -1 -2 -2 2 0 0 0 -1 -1 -1 -1 1 1 -1 1 1 -1 orthogonal lifted from D6 ρ12 2 2 2 2 2 -2 0 0 0 0 -1 -2 2 -2 0 0 0 -1 -1 -1 -1 1 -1 1 -1 1 1 orthogonal lifted from D6 ρ13 2 2 -2 -2 2 0 2 0 0 0 2 0 0 0 -2 0 0 2 -2 -2 2 0 0 0 0 0 0 orthogonal lifted from D4 ρ14 2 2 -2 2 -2 0 0 -2 0 0 2 0 0 0 0 2 0 2 2 -2 -2 0 0 0 0 0 0 orthogonal lifted from D4 ρ15 2 2 2 2 2 2 0 0 0 0 -1 2 -2 -2 0 0 0 -1 -1 -1 -1 -1 1 1 1 -1 1 orthogonal lifted from D6 ρ16 2 2 2 2 2 2 0 0 0 0 -1 2 2 2 0 0 0 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ17 2 2 2 -2 -2 2 0 0 0 0 2 -2 0 0 0 0 0 2 -2 2 -2 2 0 0 0 -2 0 orthogonal lifted from D4 ρ18 2 2 2 -2 -2 -2 0 0 0 0 2 2 0 0 0 0 0 2 -2 2 -2 -2 0 0 0 2 0 orthogonal lifted from D4 ρ19 2 2 2 -2 -2 2 0 0 0 0 -1 -2 0 0 0 0 0 -1 1 -1 1 -1 √3 √3 -√3 1 -√3 orthogonal lifted from D12 ρ20 2 2 2 -2 -2 -2 0 0 0 0 -1 2 0 0 0 0 0 -1 1 -1 1 1 √3 -√3 -√3 -1 √3 orthogonal lifted from D12 ρ21 2 2 2 -2 -2 -2 0 0 0 0 -1 2 0 0 0 0 0 -1 1 -1 1 1 -√3 √3 √3 -1 -√3 orthogonal lifted from D12 ρ22 2 2 2 -2 -2 2 0 0 0 0 -1 -2 0 0 0 0 0 -1 1 -1 1 -1 -√3 -√3 √3 1 √3 orthogonal lifted from D12 ρ23 4 -4 0 0 0 0 0 0 2 -2 4 0 0 0 0 0 0 -4 0 0 0 0 0 0 0 0 0 orthogonal lifted from C2≀C22 ρ24 4 4 -4 -4 4 0 0 0 0 0 -2 0 0 0 0 0 0 -2 2 2 -2 0 0 0 0 0 0 orthogonal lifted from S3×D4 ρ25 4 4 -4 4 -4 0 0 0 0 0 -2 0 0 0 0 0 0 -2 -2 2 2 0 0 0 0 0 0 orthogonal lifted from S3×D4 ρ26 4 -4 0 0 0 0 0 0 -2 2 4 0 0 0 0 0 0 -4 0 0 0 0 0 0 0 0 0 orthogonal lifted from C2≀C22 ρ27 8 -8 0 0 0 0 0 0 0 0 -4 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 orthogonal faithful

Permutation representations of C23⋊D12
On 24 points - transitive group 24T336
Generators in S24
(1 16)(2 17)(3 24)(4 13)(5 20)(6 21)(7 19)(8 14)(9 15)(10 22)(11 23)(12 18)
(1 10)(3 12)(5 8)(14 20)(16 22)(18 24)
(1 10)(2 11)(3 12)(4 7)(5 8)(6 9)(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)
(1 2 3 4 5 6)(7 8 9 10 11 12)(13 14 15 16 17 18 19 20 21 22 23 24)
(1 3)(4 6)(7 9)(10 12)(13 15)(16 24)(17 23)(18 22)(19 21)

G:=sub<Sym(24)| (1,16)(2,17)(3,24)(4,13)(5,20)(6,21)(7,19)(8,14)(9,15)(10,22)(11,23)(12,18), (1,10)(3,12)(5,8)(14,20)(16,22)(18,24), (1,10)(2,11)(3,12)(4,7)(5,8)(6,9)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24), (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24), (1,3)(4,6)(7,9)(10,12)(13,15)(16,24)(17,23)(18,22)(19,21)>;

G:=Group( (1,16)(2,17)(3,24)(4,13)(5,20)(6,21)(7,19)(8,14)(9,15)(10,22)(11,23)(12,18), (1,10)(3,12)(5,8)(14,20)(16,22)(18,24), (1,10)(2,11)(3,12)(4,7)(5,8)(6,9)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24), (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24), (1,3)(4,6)(7,9)(10,12)(13,15)(16,24)(17,23)(18,22)(19,21) );

G=PermutationGroup([(1,16),(2,17),(3,24),(4,13),(5,20),(6,21),(7,19),(8,14),(9,15),(10,22),(11,23),(12,18)], [(1,10),(3,12),(5,8),(14,20),(16,22),(18,24)], [(1,10),(2,11),(3,12),(4,7),(5,8),(6,9),(13,19),(14,20),(15,21),(16,22),(17,23),(18,24)], [(1,2,3,4,5,6),(7,8,9,10,11,12),(13,14,15,16,17,18,19,20,21,22,23,24)], [(1,3),(4,6),(7,9),(10,12),(13,15),(16,24),(17,23),(18,22),(19,21)])

G:=TransitiveGroup(24,336);

On 24 points - transitive group 24T345
Generators in S24
(2 16)(3 17)(6 20)(7 21)(10 24)(11 13)
(2 16)(4 18)(6 20)(8 22)(10 24)(12 14)
(1 15)(2 16)(3 17)(4 18)(5 19)(6 20)(7 21)(8 22)(9 23)(10 24)(11 13)(12 14)
(1 2 3 4 5 6 7 8 9 10 11 12)(13 14 15 16 17 18 19 20 21 22 23 24)
(1 3)(4 12)(5 11)(6 10)(7 9)(13 19)(14 18)(15 17)(20 24)(21 23)

G:=sub<Sym(24)| (2,16)(3,17)(6,20)(7,21)(10,24)(11,13), (2,16)(4,18)(6,20)(8,22)(10,24)(12,14), (1,15)(2,16)(3,17)(4,18)(5,19)(6,20)(7,21)(8,22)(9,23)(10,24)(11,13)(12,14), (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24), (1,3)(4,12)(5,11)(6,10)(7,9)(13,19)(14,18)(15,17)(20,24)(21,23)>;

G:=Group( (2,16)(3,17)(6,20)(7,21)(10,24)(11,13), (2,16)(4,18)(6,20)(8,22)(10,24)(12,14), (1,15)(2,16)(3,17)(4,18)(5,19)(6,20)(7,21)(8,22)(9,23)(10,24)(11,13)(12,14), (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24), (1,3)(4,12)(5,11)(6,10)(7,9)(13,19)(14,18)(15,17)(20,24)(21,23) );

G=PermutationGroup([(2,16),(3,17),(6,20),(7,21),(10,24),(11,13)], [(2,16),(4,18),(6,20),(8,22),(10,24),(12,14)], [(1,15),(2,16),(3,17),(4,18),(5,19),(6,20),(7,21),(8,22),(9,23),(10,24),(11,13),(12,14)], [(1,2,3,4,5,6,7,8,9,10,11,12),(13,14,15,16,17,18,19,20,21,22,23,24)], [(1,3),(4,12),(5,11),(6,10),(7,9),(13,19),(14,18),(15,17),(20,24),(21,23)])

G:=TransitiveGroup(24,345);

On 24 points - transitive group 24T367
Generators in S24
(1 19)(2 23)(3 15)(4 13)(5 17)(6 21)(7 22)(8 20)(9 18)(10 16)(11 14)(12 24)
(1 10)(2 8)(3 12)(4 7)(5 11)(6 9)(13 22)(14 17)(15 24)(16 19)(18 21)(20 23)
(1 4)(2 5)(3 6)(7 10)(8 11)(9 12)(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)
(1 2 3)(4 5 6)(7 8 9 10 11 12)(13 14 15 16 17 18 19 20 21 22 23 24)
(1 6)(2 5)(3 4)(7 12)(8 11)(9 10)(13 18)(14 17)(15 16)(19 24)(20 23)(21 22)

G:=sub<Sym(24)| (1,19)(2,23)(3,15)(4,13)(5,17)(6,21)(7,22)(8,20)(9,18)(10,16)(11,14)(12,24), (1,10)(2,8)(3,12)(4,7)(5,11)(6,9)(13,22)(14,17)(15,24)(16,19)(18,21)(20,23), (1,4)(2,5)(3,6)(7,10)(8,11)(9,12)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24), (1,2,3)(4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24), (1,6)(2,5)(3,4)(7,12)(8,11)(9,10)(13,18)(14,17)(15,16)(19,24)(20,23)(21,22)>;

G:=Group( (1,19)(2,23)(3,15)(4,13)(5,17)(6,21)(7,22)(8,20)(9,18)(10,16)(11,14)(12,24), (1,10)(2,8)(3,12)(4,7)(5,11)(6,9)(13,22)(14,17)(15,24)(16,19)(18,21)(20,23), (1,4)(2,5)(3,6)(7,10)(8,11)(9,12)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24), (1,2,3)(4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24), (1,6)(2,5)(3,4)(7,12)(8,11)(9,10)(13,18)(14,17)(15,16)(19,24)(20,23)(21,22) );

G=PermutationGroup([(1,19),(2,23),(3,15),(4,13),(5,17),(6,21),(7,22),(8,20),(9,18),(10,16),(11,14),(12,24)], [(1,10),(2,8),(3,12),(4,7),(5,11),(6,9),(13,22),(14,17),(15,24),(16,19),(18,21),(20,23)], [(1,4),(2,5),(3,6),(7,10),(8,11),(9,12),(13,19),(14,20),(15,21),(16,22),(17,23),(18,24)], [(1,2,3),(4,5,6),(7,8,9,10,11,12),(13,14,15,16,17,18,19,20,21,22,23,24)], [(1,6),(2,5),(3,4),(7,12),(8,11),(9,10),(13,18),(14,17),(15,16),(19,24),(20,23),(21,22)])

G:=TransitiveGroup(24,367);

Matrix representation of C23⋊D12 in GL6(𝔽13)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 1 0 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12
,
 8 7 0 0 0 0 12 9 0 0 0 0 0 0 1 0 0 0 0 0 0 12 0 0 0 0 0 0 0 12 0 0 0 0 1 0
,
 4 7 0 0 0 0 9 9 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 12 0 0 0 0 12 0

G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[8,12,0,0,0,0,7,9,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,12,0],[4,9,0,0,0,0,7,9,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,12,0] >;

C23⋊D12 in GAP, Magma, Sage, TeX

C_2^3\rtimes D_{12}
% in TeX

G:=Group("C2^3:D12");
// GroupNames label

G:=SmallGroup(192,300);
// by ID

G=gap.SmallGroup(192,300);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,254,219,58,570,1684,438,6278]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^12=e^2=1,a*b=b*a,a*c=c*a,d*a*d^-1=e*a*e=a*b*c,d*b*d^-1=b*c=c*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations

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