Copied to
clipboard

## G = C3×C4○D4order 48 = 24·3

### Direct product of C3 and C4○D4

direct product, metabelian, nilpotent (class 2), monomial, 2-elementary

Aliases: C3×C4○D4, D4C12, Q8C12, D42C6, Q83C6, C6.13C23, C12.21C22, C4(C3×D4), C4(C3×Q8), (C2×C4)⋊3C6, C12(C3×D4), C12(C3×Q8), (C2×C12)⋊7C2, (C3×D4)⋊5C2, C4.5(C2×C6), (C3×Q8)⋊5C2, C22.(C2×C6), C2.3(C22×C6), (C2×C6).2C22, SmallGroup(48,47)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C3×C4○D4
 Chief series C1 — C2 — C6 — C2×C6 — C3×D4 — C3×C4○D4
 Lower central C1 — C2 — C3×C4○D4
 Upper central C1 — C12 — C3×C4○D4

Generators and relations for C3×C4○D4
G = < a,b,c,d | a3=b4=d2=1, c2=b2, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=b2c >

Character table of C3×C4○D4

 class 1 2A 2B 2C 2D 3A 3B 4A 4B 4C 4D 4E 6A 6B 6C 6D 6E 6F 6G 6H 12A 12B 12C 12D 12E 12F 12G 12H 12I 12J size 1 1 2 2 2 1 1 1 1 2 2 2 1 1 2 2 2 2 2 2 1 1 1 1 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 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 -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 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 -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 -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 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 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 -1 -1 -1 linear of order 2 ρ9 1 1 -1 1 -1 ζ3 ζ32 1 1 1 -1 -1 ζ3 ζ32 ζ6 ζ3 ζ65 ζ6 ζ65 ζ32 ζ3 ζ3 ζ32 ζ32 ζ32 ζ65 ζ6 ζ6 ζ65 ζ3 linear of order 6 ρ10 1 1 1 1 1 ζ32 ζ3 1 1 1 1 1 ζ32 ζ3 ζ3 ζ32 ζ32 ζ3 ζ32 ζ3 ζ32 ζ32 ζ3 ζ3 ζ3 ζ32 ζ3 ζ3 ζ32 ζ32 linear of order 3 ρ11 1 1 -1 1 1 ζ3 ζ32 -1 -1 -1 1 -1 ζ3 ζ32 ζ6 ζ3 ζ3 ζ32 ζ65 ζ32 ζ65 ζ65 ζ6 ζ6 ζ6 ζ3 ζ32 ζ6 ζ65 ζ65 linear of order 6 ρ12 1 1 -1 -1 1 ζ32 ζ3 1 1 -1 -1 1 ζ32 ζ3 ζ65 ζ6 ζ32 ζ3 ζ6 ζ65 ζ32 ζ32 ζ3 ζ3 ζ65 ζ6 ζ65 ζ3 ζ32 ζ6 linear of order 6 ρ13 1 1 1 -1 1 ζ32 ζ3 -1 -1 1 -1 -1 ζ32 ζ3 ζ3 ζ6 ζ32 ζ3 ζ32 ζ65 ζ6 ζ6 ζ65 ζ65 ζ3 ζ6 ζ65 ζ65 ζ6 ζ32 linear of order 6 ρ14 1 1 1 -1 -1 ζ3 ζ32 1 1 -1 1 -1 ζ3 ζ32 ζ32 ζ65 ζ65 ζ6 ζ3 ζ6 ζ3 ζ3 ζ32 ζ32 ζ6 ζ3 ζ32 ζ6 ζ65 ζ65 linear of order 6 ρ15 1 1 -1 1 1 ζ32 ζ3 -1 -1 -1 1 -1 ζ32 ζ3 ζ65 ζ32 ζ32 ζ3 ζ6 ζ3 ζ6 ζ6 ζ65 ζ65 ζ65 ζ32 ζ3 ζ65 ζ6 ζ6 linear of order 6 ρ16 1 1 1 -1 1 ζ3 ζ32 -1 -1 1 -1 -1 ζ3 ζ32 ζ32 ζ65 ζ3 ζ32 ζ3 ζ6 ζ65 ζ65 ζ6 ζ6 ζ32 ζ65 ζ6 ζ6 ζ65 ζ3 linear of order 6 ρ17 1 1 1 1 1 ζ3 ζ32 1 1 1 1 1 ζ3 ζ32 ζ32 ζ3 ζ3 ζ32 ζ3 ζ32 ζ3 ζ3 ζ32 ζ32 ζ32 ζ3 ζ32 ζ32 ζ3 ζ3 linear of order 3 ρ18 1 1 1 -1 -1 ζ32 ζ3 1 1 -1 1 -1 ζ32 ζ3 ζ3 ζ6 ζ6 ζ65 ζ32 ζ65 ζ32 ζ32 ζ3 ζ3 ζ65 ζ32 ζ3 ζ65 ζ6 ζ6 linear of order 6 ρ19 1 1 -1 1 -1 ζ32 ζ3 1 1 1 -1 -1 ζ32 ζ3 ζ65 ζ32 ζ6 ζ65 ζ6 ζ3 ζ32 ζ32 ζ3 ζ3 ζ3 ζ6 ζ65 ζ65 ζ6 ζ32 linear of order 6 ρ20 1 1 1 1 -1 ζ3 ζ32 -1 -1 -1 -1 1 ζ3 ζ32 ζ32 ζ3 ζ65 ζ6 ζ3 ζ32 ζ65 ζ65 ζ6 ζ6 ζ6 ζ65 ζ6 ζ32 ζ3 ζ65 linear of order 6 ρ21 1 1 -1 -1 1 ζ3 ζ32 1 1 -1 -1 1 ζ3 ζ32 ζ6 ζ65 ζ3 ζ32 ζ65 ζ6 ζ3 ζ3 ζ32 ζ32 ζ6 ζ65 ζ6 ζ32 ζ3 ζ65 linear of order 6 ρ22 1 1 1 1 -1 ζ32 ζ3 -1 -1 -1 -1 1 ζ32 ζ3 ζ3 ζ32 ζ6 ζ65 ζ32 ζ3 ζ6 ζ6 ζ65 ζ65 ζ65 ζ6 ζ65 ζ3 ζ32 ζ6 linear of order 6 ρ23 1 1 -1 -1 -1 ζ3 ζ32 -1 -1 1 1 1 ζ3 ζ32 ζ6 ζ65 ζ65 ζ6 ζ65 ζ6 ζ65 ζ65 ζ6 ζ6 ζ32 ζ3 ζ32 ζ32 ζ3 ζ3 linear of order 6 ρ24 1 1 -1 -1 -1 ζ32 ζ3 -1 -1 1 1 1 ζ32 ζ3 ζ65 ζ6 ζ6 ζ65 ζ6 ζ65 ζ6 ζ6 ζ65 ζ65 ζ3 ζ32 ζ3 ζ3 ζ32 ζ32 linear of order 6 ρ25 2 -2 0 0 0 2 2 -2i 2i 0 0 0 -2 -2 0 0 0 0 0 0 2i -2i 2i -2i 0 0 0 0 0 0 complex lifted from C4○D4 ρ26 2 -2 0 0 0 2 2 2i -2i 0 0 0 -2 -2 0 0 0 0 0 0 -2i 2i -2i 2i 0 0 0 0 0 0 complex lifted from C4○D4 ρ27 2 -2 0 0 0 -1+√-3 -1-√-3 -2i 2i 0 0 0 1-√-3 1+√-3 0 0 0 0 0 0 2ζ4ζ3 2ζ43ζ3 2ζ4ζ32 2ζ43ζ32 0 0 0 0 0 0 complex faithful ρ28 2 -2 0 0 0 -1-√-3 -1+√-3 -2i 2i 0 0 0 1+√-3 1-√-3 0 0 0 0 0 0 2ζ4ζ32 2ζ43ζ32 2ζ4ζ3 2ζ43ζ3 0 0 0 0 0 0 complex faithful ρ29 2 -2 0 0 0 -1+√-3 -1-√-3 2i -2i 0 0 0 1-√-3 1+√-3 0 0 0 0 0 0 2ζ43ζ3 2ζ4ζ3 2ζ43ζ32 2ζ4ζ32 0 0 0 0 0 0 complex faithful ρ30 2 -2 0 0 0 -1-√-3 -1+√-3 2i -2i 0 0 0 1+√-3 1-√-3 0 0 0 0 0 0 2ζ43ζ32 2ζ4ζ32 2ζ43ζ3 2ζ4ζ3 0 0 0 0 0 0 complex faithful

Permutation representations of C3×C4○D4
On 24 points - transitive group 24T17
Generators in S24
(1 13 11)(2 14 12)(3 15 9)(4 16 10)(5 18 21)(6 19 22)(7 20 23)(8 17 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 21 3 23)(2 22 4 24)(5 15 7 13)(6 16 8 14)(9 20 11 18)(10 17 12 19)
(1 23)(2 24)(3 21)(4 22)(5 15)(6 16)(7 13)(8 14)(9 18)(10 19)(11 20)(12 17)

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

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

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

G:=TransitiveGroup(24,17);

C3×C4○D4 is a maximal subgroup of
Q83Dic3  D4.Dic3  D4⋊D6  Q8.13D6  Q8.14D6  D4○D12  Q8○D12  Q8.C18  2- 1+43C6  D286C6  D42F7  Q83F7
C3×C4○D4 is a maximal quotient of
D4×C12  Q8×C12  D286C6  D42F7  Q83F7

Matrix representation of C3×C4○D4 in GL2(𝔽13) generated by

 3 0 0 3
,
 5 0 0 5
,
 8 0 0 5
,
 0 5 8 0
G:=sub<GL(2,GF(13))| [3,0,0,3],[5,0,0,5],[8,0,0,5],[0,8,5,0] >;

C3×C4○D4 in GAP, Magma, Sage, TeX

C_3\times C_4\circ D_4
% in TeX

G:=Group("C3xC4oD4");
// GroupNames label

G:=SmallGroup(48,47);
// by ID

G=gap.SmallGroup(48,47);
# by ID

G:=PCGroup([5,-2,-2,-2,-3,-2,261,102]);
// Polycyclic

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

Export

׿
×
𝔽