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G = C2×C3⋊D4order 48 = 24·3

Direct product of C2 and C3⋊D4

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

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

C1C6 — C2×C3⋊D4
C1C3C6D6C22×S3 — C2×C3⋊D4
C3C6 — C2×C3⋊D4
C1C22C23

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 12A2B2C2D2E2F2G34A4B6A6B6C6D6E6F6G
 size 111122662662222222
ρ1111111111111111111    trivial
ρ2111111-1-11-1-11111111    linear of order 2
ρ31111-1-1111-1-1-1-1-1111-1    linear of order 2
ρ41-1-111-1-111-11-11-1-11-11    linear of order 2
ρ51-1-11-111-11-111-11-11-1-1    linear of order 2
ρ61111-1-1-1-1111-1-1-1111-1    linear of order 2
ρ71-1-111-11-111-1-11-1-11-11    linear of order 2
ρ81-1-11-11-1111-11-11-11-1-1    linear of order 2
ρ92-22-200002000002-2-20    orthogonal lifted from D4
ρ102-2-222-200-1001-111-11-1    orthogonal lifted from D6
ρ112222-2-200-100111-1-1-11    orthogonal lifted from D6
ρ122-2-22-2200-100-11-11-111    orthogonal lifted from D6
ρ1322222200-100-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ1422-2-20000200000-2-220    orthogonal lifted from D4
ρ1522-2-20000-100--3--3-311-1-3    complex lifted from C3⋊D4
ρ162-22-20000-100--3-3-3-111--3    complex lifted from C3⋊D4
ρ172-22-20000-100-3--3--3-111-3    complex lifted from C3⋊D4
ρ1822-2-20000-100-3-3--311-1--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

1200
010
001
,
100
01212
010
,
100
0119
0112
,
1200
010
01212
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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