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## G = C2×C3⋊D4order 48 = 24·3

### Direct product of C2 and C3⋊D4

Aliases: C2×C3⋊D4, C62D4, C232S3, C223D6, D63C22, C6.10C23, Dic32C22, C33(C2×D4), (C2×C6)⋊3C22, (C22×C6)⋊2C2, (C22×S3)⋊3C2, (C2×Dic3)⋊4C2, C2.10(C22×S3), SmallGroup(48,43)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C2×C3⋊D4
 Chief series C1 — C3 — C6 — D6 — C22×S3 — C2×C3⋊D4
 Lower central C3 — C6 — C2×C3⋊D4
 Upper central C1 — C22 — C23

Generators and relations for C2×C3⋊D4
G = < a,b,c,d | a2=b3=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c-1 >

Subgroups: 108 in 54 conjugacy classes, 27 normal (11 characteristic)
C1, C2, C2 [×2], C2 [×4], C3, C4 [×2], C22, C22 [×2], C22 [×6], S3 [×2], C6, C6 [×2], C6 [×2], C2×C4, D4 [×4], C23, C23, Dic3 [×2], D6 [×2], D6 [×2], C2×C6, C2×C6 [×2], C2×C6 [×2], C2×D4, C2×Dic3, C3⋊D4 [×4], C22×S3, C22×C6, C2×C3⋊D4
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D6 [×3], C2×D4, C3⋊D4 [×2], C22×S3, C2×C3⋊D4

Character table of C2×C3⋊D4

 class 1 2A 2B 2C 2D 2E 2F 2G 3 4A 4B 6A 6B 6C 6D 6E 6F 6G size 1 1 1 1 2 2 6 6 2 6 6 2 2 2 2 2 2 2 ρ1 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 linear of order 2 ρ3 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 linear of order 2 ρ5 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 linear of order 2 ρ7 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 linear of order 2 ρ9 2 -2 2 -2 0 0 0 0 2 0 0 0 0 0 2 -2 -2 0 orthogonal lifted from D4 ρ10 2 -2 -2 2 2 -2 0 0 -1 0 0 1 -1 1 1 -1 1 -1 orthogonal lifted from D6 ρ11 2 2 2 2 -2 -2 0 0 -1 0 0 1 1 1 -1 -1 -1 1 orthogonal lifted from D6 ρ12 2 -2 -2 2 -2 2 0 0 -1 0 0 -1 1 -1 1 -1 1 1 orthogonal lifted from D6 ρ13 2 2 2 2 2 2 0 0 -1 0 0 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ14 2 2 -2 -2 0 0 0 0 2 0 0 0 0 0 -2 -2 2 0 orthogonal lifted from D4 ρ15 2 2 -2 -2 0 0 0 0 -1 0 0 -√-3 -√-3 √-3 1 1 -1 √-3 complex lifted from C3⋊D4 ρ16 2 -2 2 -2 0 0 0 0 -1 0 0 -√-3 √-3 √-3 -1 1 1 -√-3 complex lifted from C3⋊D4 ρ17 2 -2 2 -2 0 0 0 0 -1 0 0 √-3 -√-3 -√-3 -1 1 1 √-3 complex lifted from C3⋊D4 ρ18 2 2 -2 -2 0 0 0 0 -1 0 0 √-3 √-3 -√-3 1 1 -1 -√-3 complex lifted from C3⋊D4

Permutation representations of C2×C3⋊D4
On 24 points - transitive group 24T25
Generators in S24
(1 5)(2 6)(3 7)(4 8)(9 23)(10 24)(11 21)(12 22)(13 18)(14 19)(15 20)(16 17)
(1 20 21)(2 22 17)(3 18 23)(4 24 19)(5 15 11)(6 12 16)(7 13 9)(8 10 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 7)(2 6)(3 5)(4 8)(9 20)(10 19)(11 18)(12 17)(13 21)(14 24)(15 23)(16 22)

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

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

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

G:=TransitiveGroup(24,25);

On 24 points - transitive group 24T45
Generators in S24
(1 19)(2 20)(3 17)(4 18)(5 13)(6 14)(7 15)(8 16)(9 23)(10 24)(11 21)(12 22)
(1 9 5)(2 6 10)(3 11 7)(4 8 12)(13 19 23)(14 24 20)(15 17 21)(16 22 18)
(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 4)(2 3)(5 8)(6 7)(9 12)(10 11)(13 16)(14 15)(17 20)(18 19)(21 24)(22 23)

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

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

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

G:=TransitiveGroup(24,45);

Matrix representation of C2×C3⋊D4 in GL3(𝔽13) generated by

 12 0 0 0 1 0 0 0 1
,
 1 0 0 0 12 12 0 1 0
,
 1 0 0 0 11 9 0 11 2
,
 12 0 0 0 1 0 0 12 12
G:=sub<GL(3,GF(13))| [12,0,0,0,1,0,0,0,1],[1,0,0,0,12,1,0,12,0],[1,0,0,0,11,11,0,9,2],[12,0,0,0,1,12,0,0,12] >;

C2×C3⋊D4 in GAP, Magma, Sage, TeX

C_2\times C_3\rtimes D_4
% in TeX

G:=Group("C2xC3:D4");
// GroupNames label

G:=SmallGroup(48,43);
// by ID

G=gap.SmallGroup(48,43);
# by ID

G:=PCGroup([5,-2,-2,-2,-2,-3,182,804]);
// Polycyclic

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

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