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## G = C2×D12order 48 = 24·3

### Direct product of C2 and D12

Aliases: C2×D12, C42D6, C61D4, C122C22, D61C22, C6.3C23, C22.10D6, C31(C2×D4), (C2×C4)⋊2S3, (C2×C12)⋊3C2, (C22×S3)⋊1C2, C2.4(C22×S3), (C2×C6).10C22, SmallGroup(48,36)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C2×D12
 Chief series C1 — C3 — C6 — D6 — C22×S3 — C2×D12
 Lower central C3 — C6 — C2×D12
 Upper central C1 — C22 — C2×C4

Generators and relations for C2×D12
G = < a,b,c | a2=b12=c2=1, ab=ba, ac=ca, cbc=b-1 >

Subgroups: 124 in 54 conjugacy classes, 27 normal (9 characteristic)
C1, C2, C2 [×2], C2 [×4], C3, C4 [×2], C22, C22 [×8], S3 [×4], C6, C6 [×2], C2×C4, D4 [×4], C23 [×2], C12 [×2], D6 [×4], D6 [×4], C2×C6, C2×D4, D12 [×4], C2×C12, C22×S3 [×2], C2×D12
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D6 [×3], C2×D4, D12 [×2], C22×S3, C2×D12

Character table of C2×D12

 class 1 2A 2B 2C 2D 2E 2F 2G 3 4A 4B 6A 6B 6C 12A 12B 12C 12D size 1 1 1 1 6 6 6 6 2 2 2 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 -1 -2 -2 -1 -1 -1 1 1 1 1 orthogonal lifted from D6 ρ10 2 2 2 2 0 0 0 0 -1 2 2 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ11 2 2 -2 -2 0 0 0 0 2 0 0 2 -2 -2 0 0 0 0 orthogonal lifted from D4 ρ12 2 -2 -2 2 0 0 0 0 -1 2 -2 1 1 -1 1 -1 -1 1 orthogonal lifted from D6 ρ13 2 -2 -2 2 0 0 0 0 -1 -2 2 1 1 -1 -1 1 1 -1 orthogonal lifted from D6 ρ14 2 -2 2 -2 0 0 0 0 2 0 0 -2 2 -2 0 0 0 0 orthogonal lifted from D4 ρ15 2 2 -2 -2 0 0 0 0 -1 0 0 -1 1 1 √3 -√3 √3 -√3 orthogonal lifted from D12 ρ16 2 2 -2 -2 0 0 0 0 -1 0 0 -1 1 1 -√3 √3 -√3 √3 orthogonal lifted from D12 ρ17 2 -2 2 -2 0 0 0 0 -1 0 0 1 -1 1 √3 √3 -√3 -√3 orthogonal lifted from D12 ρ18 2 -2 2 -2 0 0 0 0 -1 0 0 1 -1 1 -√3 -√3 √3 √3 orthogonal lifted from D12

Permutation representations of C2×D12
On 24 points - transitive group 24T29
Generators in S24
(1 16)(2 17)(3 18)(4 19)(5 20)(6 21)(7 22)(8 23)(9 24)(10 13)(11 14)(12 15)
(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 21)(14 20)(15 19)(16 18)(22 24)

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

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

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

G:=TransitiveGroup(24,29);

C2×D12 is a maximal subgroup of
C6.D8  C2.D24  C12.46D4  C4⋊D12  C427S3  D6⋊D4  Dic3⋊D4  Dic35D4  D6.D4  C12⋊D4  C8⋊D6  C127D4  C123D4  C12.23D4  D4⋊D6  C2×S3×D4  D4○D12  Q8⋊D12
C2×D12 is a maximal quotient of
C122Q8  C4⋊D12  C427S3  D6⋊D4  C23.21D6  C12⋊D4  C4.D12  C4○D24  C8⋊D6  C8.D6  C127D4

Matrix representation of C2×D12 in GL3(𝔽13) generated by

 12 0 0 0 1 0 0 0 1
,
 1 0 0 0 3 10 0 3 6
,
 12 0 0 0 3 10 0 7 10
G:=sub<GL(3,GF(13))| [12,0,0,0,1,0,0,0,1],[1,0,0,0,3,3,0,10,6],[12,0,0,0,3,7,0,10,10] >;

C2×D12 in GAP, Magma, Sage, TeX

C_2\times D_{12}
% in TeX

G:=Group("C2xD12");
// GroupNames label

G:=SmallGroup(48,36);
// by ID

G=gap.SmallGroup(48,36);
# by ID

G:=PCGroup([5,-2,-2,-2,-2,-3,182,42,804]);
// Polycyclic

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

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