Copied to
clipboard

## G = C4×C3⋊S3order 72 = 23·32

### Direct product of C4 and C3⋊S3

Aliases: C4×C3⋊S3, C122S3, C6.13D6, C32(C4×S3), (C3×C12)⋊4C2, C325(C2×C4), C4(C3⋊Dic3), C3⋊Dic34C2, (C3×C6).12C22, C2.1(C2×C3⋊S3), (C2×C3⋊S3).3C2, SmallGroup(72,32)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C4×C3⋊S3
 Chief series C1 — C3 — C32 — C3×C6 — C2×C3⋊S3 — C4×C3⋊S3
 Lower central C32 — C4×C3⋊S3
 Upper central C1 — C4

Generators and relations for C4×C3⋊S3
G = < a,b,c,d | a4=b3=c3=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b-1, dcd=c-1 >

Character table of C4×C3⋊S3

 class 1 2A 2B 2C 3A 3B 3C 3D 4A 4B 4C 4D 6A 6B 6C 6D 12A 12B 12C 12D 12E 12F 12G 12H size 1 1 9 9 2 2 2 2 1 1 9 9 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 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 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 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 linear of order 2 ρ5 1 -1 -1 1 1 1 1 1 -i i i -i -1 -1 -1 -1 -i i i i -i -i i -i linear of order 4 ρ6 1 -1 1 -1 1 1 1 1 -i i -i i -1 -1 -1 -1 -i i i i -i -i i -i linear of order 4 ρ7 1 -1 1 -1 1 1 1 1 i -i i -i -1 -1 -1 -1 i -i -i -i i i -i i linear of order 4 ρ8 1 -1 -1 1 1 1 1 1 i -i -i i -1 -1 -1 -1 i -i -i -i i i -i i linear of order 4 ρ9 2 2 0 0 -1 -1 2 -1 2 2 0 0 -1 -1 2 -1 -1 -1 2 -1 -1 2 -1 -1 orthogonal lifted from S3 ρ10 2 2 0 0 -1 -1 2 -1 -2 -2 0 0 -1 -1 2 -1 1 1 -2 1 1 -2 1 1 orthogonal lifted from D6 ρ11 2 2 0 0 2 -1 -1 -1 2 2 0 0 -1 2 -1 -1 2 2 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ12 2 2 0 0 -1 -1 -1 2 -2 -2 0 0 -1 -1 -1 2 1 1 1 -2 -2 1 1 1 orthogonal lifted from D6 ρ13 2 2 0 0 -1 -1 -1 2 2 2 0 0 -1 -1 -1 2 -1 -1 -1 2 2 -1 -1 -1 orthogonal lifted from S3 ρ14 2 2 0 0 -1 2 -1 -1 -2 -2 0 0 2 -1 -1 -1 1 1 1 1 1 1 -2 -2 orthogonal lifted from D6 ρ15 2 2 0 0 2 -1 -1 -1 -2 -2 0 0 -1 2 -1 -1 -2 -2 1 1 1 1 1 1 orthogonal lifted from D6 ρ16 2 2 0 0 -1 2 -1 -1 2 2 0 0 2 -1 -1 -1 -1 -1 -1 -1 -1 -1 2 2 orthogonal lifted from S3 ρ17 2 -2 0 0 -1 -1 2 -1 -2i 2i 0 0 1 1 -2 1 i -i 2i -i i -2i -i i complex lifted from C4×S3 ρ18 2 -2 0 0 -1 -1 2 -1 2i -2i 0 0 1 1 -2 1 -i i -2i i -i 2i i -i complex lifted from C4×S3 ρ19 2 -2 0 0 2 -1 -1 -1 2i -2i 0 0 1 -2 1 1 2i -2i i i -i -i i -i complex lifted from C4×S3 ρ20 2 -2 0 0 -1 -1 -1 2 -2i 2i 0 0 1 1 1 -2 i -i -i 2i -2i i -i i complex lifted from C4×S3 ρ21 2 -2 0 0 -1 -1 -1 2 2i -2i 0 0 1 1 1 -2 -i i i -2i 2i -i i -i complex lifted from C4×S3 ρ22 2 -2 0 0 -1 2 -1 -1 -2i 2i 0 0 -2 1 1 1 i -i -i -i i i 2i -2i complex lifted from C4×S3 ρ23 2 -2 0 0 -1 2 -1 -1 2i -2i 0 0 -2 1 1 1 -i i i i -i -i -2i 2i complex lifted from C4×S3 ρ24 2 -2 0 0 2 -1 -1 -1 -2i 2i 0 0 1 -2 1 1 -2i 2i -i -i i i -i i complex lifted from C4×S3

Smallest permutation representation of C4×C3⋊S3
On 36 points
Generators in S36
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)(25 26 27 28)(29 30 31 32)(33 34 35 36)
(1 21 29)(2 22 30)(3 23 31)(4 24 32)(5 35 11)(6 36 12)(7 33 9)(8 34 10)(13 28 19)(14 25 20)(15 26 17)(16 27 18)
(1 7 16)(2 8 13)(3 5 14)(4 6 15)(9 18 29)(10 19 30)(11 20 31)(12 17 32)(21 33 27)(22 34 28)(23 35 25)(24 36 26)
(5 14)(6 15)(7 16)(8 13)(9 27)(10 28)(11 25)(12 26)(17 36)(18 33)(19 34)(20 35)(21 29)(22 30)(23 31)(24 32)

G:=sub<Sym(36)| (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24)(25,26,27,28)(29,30,31,32)(33,34,35,36), (1,21,29)(2,22,30)(3,23,31)(4,24,32)(5,35,11)(6,36,12)(7,33,9)(8,34,10)(13,28,19)(14,25,20)(15,26,17)(16,27,18), (1,7,16)(2,8,13)(3,5,14)(4,6,15)(9,18,29)(10,19,30)(11,20,31)(12,17,32)(21,33,27)(22,34,28)(23,35,25)(24,36,26), (5,14)(6,15)(7,16)(8,13)(9,27)(10,28)(11,25)(12,26)(17,36)(18,33)(19,34)(20,35)(21,29)(22,30)(23,31)(24,32)>;

G:=Group( (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24)(25,26,27,28)(29,30,31,32)(33,34,35,36), (1,21,29)(2,22,30)(3,23,31)(4,24,32)(5,35,11)(6,36,12)(7,33,9)(8,34,10)(13,28,19)(14,25,20)(15,26,17)(16,27,18), (1,7,16)(2,8,13)(3,5,14)(4,6,15)(9,18,29)(10,19,30)(11,20,31)(12,17,32)(21,33,27)(22,34,28)(23,35,25)(24,36,26), (5,14)(6,15)(7,16)(8,13)(9,27)(10,28)(11,25)(12,26)(17,36)(18,33)(19,34)(20,35)(21,29)(22,30)(23,31)(24,32) );

G=PermutationGroup([[(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16),(17,18,19,20),(21,22,23,24),(25,26,27,28),(29,30,31,32),(33,34,35,36)], [(1,21,29),(2,22,30),(3,23,31),(4,24,32),(5,35,11),(6,36,12),(7,33,9),(8,34,10),(13,28,19),(14,25,20),(15,26,17),(16,27,18)], [(1,7,16),(2,8,13),(3,5,14),(4,6,15),(9,18,29),(10,19,30),(11,20,31),(12,17,32),(21,33,27),(22,34,28),(23,35,25),(24,36,26)], [(5,14),(6,15),(7,16),(8,13),(9,27),(10,28),(11,25),(12,26),(17,36),(18,33),(19,34),(20,35),(21,29),(22,30),(23,31),(24,32)]])

C4×C3⋊S3 is a maximal subgroup of
C12.29D6  C12.31D6  C24⋊S3  C3⋊S33C8  C32⋊M4(2)  C4⋊(C32⋊C4)  D12⋊S3  Dic3.D6  D6.D6  C4×S32  D6⋊D6  C12.59D6  C12.D6  C12.26D6  C338(C2×C4)  C12.14S4  C30.D6
C4×C3⋊S3 is a maximal quotient of
C24⋊S3  C6.Dic6  C6.11D12  C338(C2×C4)  C30.D6

Matrix representation of C4×C3⋊S3 in GL4(𝔽13) generated by

 5 0 0 0 0 5 0 0 0 0 8 0 0 0 0 8
,
 0 1 0 0 12 12 0 0 0 0 0 1 0 0 12 12
,
 1 0 0 0 0 1 0 0 0 0 12 12 0 0 1 0
,
 1 0 0 0 12 12 0 0 0 0 1 1 0 0 0 12
G:=sub<GL(4,GF(13))| [5,0,0,0,0,5,0,0,0,0,8,0,0,0,0,8],[0,12,0,0,1,12,0,0,0,0,0,12,0,0,1,12],[1,0,0,0,0,1,0,0,0,0,12,1,0,0,12,0],[1,12,0,0,0,12,0,0,0,0,1,0,0,0,1,12] >;

C4×C3⋊S3 in GAP, Magma, Sage, TeX

C_4\times C_3\rtimes S_3
% in TeX

G:=Group("C4xC3:S3");
// GroupNames label

G:=SmallGroup(72,32);
// by ID

G=gap.SmallGroup(72,32);
# by ID

G:=PCGroup([5,-2,-2,-2,-3,-3,26,323,1204]);
// Polycyclic

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

Export

׿
×
𝔽