Copied to
clipboard

## G = C3×C22⋊C4order 48 = 24·3

### Direct product of C3 and C22⋊C4

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

Aliases: C3×C22⋊C4, C6.12D4, C222C12, C23.2C6, (C2×C4)⋊1C6, (C2×C6)⋊1C4, (C2×C12)⋊2C2, C2.1(C3×D4), C6.10(C2×C4), C2.1(C2×C12), C22.2(C2×C6), (C22×C6).1C2, (C2×C6).13C22, SmallGroup(48,21)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C3×C22⋊C4
 Chief series C1 — C2 — C22 — C2×C6 — C2×C12 — C3×C22⋊C4
 Lower central C1 — C2 — C3×C22⋊C4
 Upper central C1 — C2×C6 — C3×C22⋊C4

Generators and relations for C3×C22⋊C4
G = < a,b,c,d | a3=b2=c2=d4=1, ab=ba, ac=ca, ad=da, dbd-1=bc=cb, cd=dc >

Character table of C3×C22⋊C4

 class 1 2A 2B 2C 2D 2E 3A 3B 4A 4B 4C 4D 6A 6B 6C 6D 6E 6F 6G 6H 6I 6J 12A 12B 12C 12D 12E 12F 12G 12H size 1 1 1 1 2 2 1 1 2 2 2 2 1 1 1 1 1 1 2 2 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 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 ζ32 ζ3 -1 -1 -1 -1 ζ32 ζ32 ζ32 ζ3 ζ3 ζ3 ζ32 ζ3 ζ3 ζ32 ζ65 ζ65 ζ6 ζ6 ζ6 ζ6 ζ65 ζ65 linear of order 6 ρ6 1 1 1 1 -1 -1 ζ32 ζ3 1 -1 -1 1 ζ32 ζ32 ζ32 ζ3 ζ3 ζ3 ζ6 ζ65 ζ65 ζ6 ζ65 ζ65 ζ6 ζ6 ζ32 ζ32 ζ3 ζ3 linear of order 6 ρ7 1 1 1 1 -1 -1 ζ3 ζ32 1 -1 -1 1 ζ3 ζ3 ζ3 ζ32 ζ32 ζ32 ζ65 ζ6 ζ6 ζ65 ζ6 ζ6 ζ65 ζ65 ζ3 ζ3 ζ32 ζ32 linear of order 6 ρ8 1 1 1 1 -1 -1 ζ3 ζ32 -1 1 1 -1 ζ3 ζ3 ζ3 ζ32 ζ32 ζ32 ζ65 ζ6 ζ6 ζ65 ζ32 ζ32 ζ3 ζ3 ζ65 ζ65 ζ6 ζ6 linear of order 6 ρ9 1 1 1 1 -1 -1 ζ32 ζ3 -1 1 1 -1 ζ32 ζ32 ζ32 ζ3 ζ3 ζ3 ζ6 ζ65 ζ65 ζ6 ζ3 ζ3 ζ32 ζ32 ζ6 ζ6 ζ65 ζ65 linear of order 6 ρ10 1 1 1 1 1 1 ζ32 ζ3 1 1 1 1 ζ32 ζ32 ζ32 ζ3 ζ3 ζ3 ζ32 ζ3 ζ3 ζ32 ζ3 ζ3 ζ32 ζ32 ζ32 ζ32 ζ3 ζ3 linear of order 3 ρ11 1 1 1 1 1 1 ζ3 ζ32 -1 -1 -1 -1 ζ3 ζ3 ζ3 ζ32 ζ32 ζ32 ζ3 ζ32 ζ32 ζ3 ζ6 ζ6 ζ65 ζ65 ζ65 ζ65 ζ6 ζ6 linear of order 6 ρ12 1 1 1 1 1 1 ζ3 ζ32 1 1 1 1 ζ3 ζ3 ζ3 ζ32 ζ32 ζ32 ζ3 ζ32 ζ32 ζ3 ζ32 ζ32 ζ3 ζ3 ζ3 ζ3 ζ32 ζ32 linear of order 3 ρ13 1 -1 1 -1 1 -1 1 1 i i -i -i 1 -1 -1 -1 1 -1 1 -1 1 -1 i -i i -i -i i -i i linear of order 4 ρ14 1 -1 1 -1 -1 1 1 1 -i i -i i 1 -1 -1 -1 1 -1 -1 1 -1 1 i -i i -i i -i i -i linear of order 4 ρ15 1 -1 1 -1 1 -1 1 1 -i -i i i 1 -1 -1 -1 1 -1 1 -1 1 -1 -i i -i i i -i i -i linear of order 4 ρ16 1 -1 1 -1 -1 1 1 1 i -i i -i 1 -1 -1 -1 1 -1 -1 1 -1 1 -i i -i i -i i -i i linear of order 4 ρ17 1 -1 1 -1 1 -1 ζ32 ζ3 i i -i -i ζ32 ζ6 ζ6 ζ65 ζ3 ζ65 ζ32 ζ65 ζ3 ζ6 ζ4ζ3 ζ43ζ3 ζ4ζ32 ζ43ζ32 ζ43ζ32 ζ4ζ32 ζ43ζ3 ζ4ζ3 linear of order 12 ρ18 1 -1 1 -1 -1 1 ζ3 ζ32 i -i i -i ζ3 ζ65 ζ65 ζ6 ζ32 ζ6 ζ65 ζ32 ζ6 ζ3 ζ43ζ32 ζ4ζ32 ζ43ζ3 ζ4ζ3 ζ43ζ3 ζ4ζ3 ζ43ζ32 ζ4ζ32 linear of order 12 ρ19 1 -1 1 -1 1 -1 ζ3 ζ32 -i -i i i ζ3 ζ65 ζ65 ζ6 ζ32 ζ6 ζ3 ζ6 ζ32 ζ65 ζ43ζ32 ζ4ζ32 ζ43ζ3 ζ4ζ3 ζ4ζ3 ζ43ζ3 ζ4ζ32 ζ43ζ32 linear of order 12 ρ20 1 -1 1 -1 1 -1 ζ3 ζ32 i i -i -i ζ3 ζ65 ζ65 ζ6 ζ32 ζ6 ζ3 ζ6 ζ32 ζ65 ζ4ζ32 ζ43ζ32 ζ4ζ3 ζ43ζ3 ζ43ζ3 ζ4ζ3 ζ43ζ32 ζ4ζ32 linear of order 12 ρ21 1 -1 1 -1 -1 1 ζ3 ζ32 -i i -i i ζ3 ζ65 ζ65 ζ6 ζ32 ζ6 ζ65 ζ32 ζ6 ζ3 ζ4ζ32 ζ43ζ32 ζ4ζ3 ζ43ζ3 ζ4ζ3 ζ43ζ3 ζ4ζ32 ζ43ζ32 linear of order 12 ρ22 1 -1 1 -1 -1 1 ζ32 ζ3 -i i -i i ζ32 ζ6 ζ6 ζ65 ζ3 ζ65 ζ6 ζ3 ζ65 ζ32 ζ4ζ3 ζ43ζ3 ζ4ζ32 ζ43ζ32 ζ4ζ32 ζ43ζ32 ζ4ζ3 ζ43ζ3 linear of order 12 ρ23 1 -1 1 -1 1 -1 ζ32 ζ3 -i -i i i ζ32 ζ6 ζ6 ζ65 ζ3 ζ65 ζ32 ζ65 ζ3 ζ6 ζ43ζ3 ζ4ζ3 ζ43ζ32 ζ4ζ32 ζ4ζ32 ζ43ζ32 ζ4ζ3 ζ43ζ3 linear of order 12 ρ24 1 -1 1 -1 -1 1 ζ32 ζ3 i -i i -i ζ32 ζ6 ζ6 ζ65 ζ3 ζ65 ζ6 ζ3 ζ65 ζ32 ζ43ζ3 ζ4ζ3 ζ43ζ32 ζ4ζ32 ζ43ζ32 ζ4ζ32 ζ43ζ3 ζ4ζ3 linear of order 12 ρ25 2 -2 -2 2 0 0 2 2 0 0 0 0 -2 2 -2 -2 -2 2 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ26 2 2 -2 -2 0 0 2 2 0 0 0 0 -2 -2 2 2 -2 -2 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ27 2 2 -2 -2 0 0 -1-√-3 -1+√-3 0 0 0 0 1+√-3 1+√-3 -1-√-3 -1+√-3 1-√-3 1-√-3 0 0 0 0 0 0 0 0 0 0 0 0 complex lifted from C3×D4 ρ28 2 -2 -2 2 0 0 -1-√-3 -1+√-3 0 0 0 0 1+√-3 -1-√-3 1+√-3 1-√-3 1-√-3 -1+√-3 0 0 0 0 0 0 0 0 0 0 0 0 complex lifted from C3×D4 ρ29 2 -2 -2 2 0 0 -1+√-3 -1-√-3 0 0 0 0 1-√-3 -1+√-3 1-√-3 1+√-3 1+√-3 -1-√-3 0 0 0 0 0 0 0 0 0 0 0 0 complex lifted from C3×D4 ρ30 2 2 -2 -2 0 0 -1+√-3 -1-√-3 0 0 0 0 1-√-3 1-√-3 -1+√-3 -1-√-3 1+√-3 1+√-3 0 0 0 0 0 0 0 0 0 0 0 0 complex lifted from C3×D4

Permutation representations of C3×C22⋊C4
On 24 points - transitive group 24T39
Generators in S24
(1 6 19)(2 7 20)(3 8 17)(4 5 18)(9 15 21)(10 16 22)(11 13 23)(12 14 24)
(2 12)(4 10)(5 16)(7 14)(18 22)(20 24)
(1 11)(2 12)(3 9)(4 10)(5 16)(6 13)(7 14)(8 15)(17 21)(18 22)(19 23)(20 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)

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

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

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

G:=TransitiveGroup(24,39);

C3×C22⋊C4 is a maximal subgroup of
C23.6D6  C23.16D6  Dic3.D4  C23.8D6  Dic34D4  D6⋊D4  C23.9D6  Dic3⋊D4  C23.11D6  C23.21D6  D4×C12  (C2×Q8)⋊C12  SL2(𝔽3)⋊5D4  D14⋊C12  C23.2F7  C22⋊(He3⋊C4)
C3×C22⋊C4 is a maximal quotient of
D14⋊C12  C23.2F7

Matrix representation of C3×C22⋊C4 in GL3(𝔽13) generated by

 3 0 0 0 9 0 0 0 9
,
 1 0 0 0 12 0 0 3 1
,
 1 0 0 0 12 0 0 0 12
,
 5 0 0 0 8 1 0 0 5
G:=sub<GL(3,GF(13))| [3,0,0,0,9,0,0,0,9],[1,0,0,0,12,3,0,0,1],[1,0,0,0,12,0,0,0,12],[5,0,0,0,8,0,0,1,5] >;

C3×C22⋊C4 in GAP, Magma, Sage, TeX

C_3\times C_2^2\rtimes C_4
% in TeX

G:=Group("C3xC2^2:C4");
// GroupNames label

G:=SmallGroup(48,21);
// by ID

G=gap.SmallGroup(48,21);
# by ID

G:=PCGroup([5,-2,-2,-3,-2,-2,120,141]);
// Polycyclic

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

Export

׿
×
𝔽