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G = C6.S4order 144 = 24·32

3rd non-split extension by C6 of S4 acting via S4/A4=C2

non-abelian, soluble, monomial

Aliases: C6.3S4, C23.D9, C22⋊Dic9, C3.A4⋊C4, C3.(A4⋊C4), (C2×C6).Dic3, C2.1(C3.S4), (C22×C6).2S3, (C2×C3.A4).C2, SmallGroup(144,33)

Series: Derived Chief Lower central Upper central

C1C22C3.A4 — C6.S4
C1C22C2×C6C3.A4C2×C3.A4 — C6.S4
C3.A4 — C6.S4
C1C2

Generators and relations for C6.S4
 G = < a,b,c,d,e | a6=b2=c2=1, d3=a2, e2=a3, ab=ba, ac=ca, ad=da, eae-1=a-1, dbd-1=ebe-1=bc=cb, dcd-1=b, ce=ec, ede-1=a4d2 >

3C2
3C2
3C22
3C22
18C4
18C4
3C6
3C6
4C9
9C2×C4
9C2×C4
3C2×C6
3C2×C6
6Dic3
6Dic3
4C18
9C22⋊C4
3C2×Dic3
3C2×Dic3
4Dic9
3C6.D4

Character table of C6.S4

 class 12A2B2C34A4B4C4D6A6B6C9A9B9C18A18B18C
 size 1133218181818266888888
ρ1111111111111111111    trivial
ρ211111-1-1-1-1111111111    linear of order 2
ρ31-11-11-ii-ii-1-11111-1-1-1    linear of order 4
ρ41-11-11i-ii-i-1-11111-1-1-1    linear of order 4
ρ52222-10000-1-1-1ζ9792ζ9594ζ989ζ9594ζ989ζ9792    orthogonal lifted from D9
ρ6222220000222-1-1-1-1-1-1    orthogonal lifted from S3
ρ72222-10000-1-1-1ζ9594ζ989ζ9792ζ989ζ9792ζ9594    orthogonal lifted from D9
ρ82222-10000-1-1-1ζ989ζ9792ζ9594ζ9792ζ9594ζ989    orthogonal lifted from D9
ρ92-22-2-1000011-1ζ9792ζ9594ζ98995949899792    symplectic lifted from Dic9, Schur index 2
ρ102-22-220000-2-22-1-1-1111    symplectic lifted from Dic3, Schur index 2
ρ112-22-2-1000011-1ζ989ζ9792ζ959497929594989    symplectic lifted from Dic9, Schur index 2
ρ122-22-2-1000011-1ζ9594ζ989ζ979298997929594    symplectic lifted from Dic9, Schur index 2
ρ1333-1-13-1-1113-1-1000000    orthogonal lifted from S4
ρ1433-1-1311-1-13-1-1000000    orthogonal lifted from S4
ρ153-3-113-iii-i-31-1000000    complex lifted from A4⋊C4
ρ163-3-113i-i-ii-31-1000000    complex lifted from A4⋊C4
ρ1766-2-2-30000-311000000    orthogonal lifted from C3.S4
ρ186-6-22-300003-11000000    symplectic faithful, Schur index 2

Smallest permutation representation of C6.S4
On 36 points
Generators in S36
(1 14 4 17 7 11)(2 15 5 18 8 12)(3 16 6 10 9 13)(19 33 22 36 25 30)(20 34 23 28 26 31)(21 35 24 29 27 32)
(1 23)(2 32)(3 10)(4 26)(5 35)(6 13)(7 20)(8 29)(9 16)(11 34)(12 27)(14 28)(15 21)(17 31)(18 24)(19 36)(22 30)(25 33)
(1 17)(2 24)(3 33)(4 11)(5 27)(6 36)(7 14)(8 21)(9 30)(10 25)(12 35)(13 19)(15 29)(16 22)(18 32)(20 28)(23 31)(26 34)
(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 23 17 31)(2 22 18 30)(3 21 10 29)(4 20 11 28)(5 19 12 36)(6 27 13 35)(7 26 14 34)(8 25 15 33)(9 24 16 32)

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

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

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

C6.S4 is a maximal subgroup of   C12.1S4  C4×C3.S4  C23.D18  C62.Dic3  A4⋊Dic9  C62.10Dic3
C6.S4 is a maximal quotient of   C12.S4  Q8⋊Dic9  C12.9S4  C18.S4  C62.10Dic3

Matrix representation of C6.S4 in GL5(𝔽37)

01000
361000
00100
00010
00001
,
10000
01000
003600
000360
00001
,
10000
01000
00100
000360
000036
,
3117000
2011000
00001
00100
00010
,
031000
310000
003600
000036
000360

G:=sub<GL(5,GF(37))| [0,36,0,0,0,1,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,36,0,0,0,0,0,36,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,36,0,0,0,0,0,36],[31,20,0,0,0,17,11,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0],[0,31,0,0,0,31,0,0,0,0,0,0,36,0,0,0,0,0,0,36,0,0,0,36,0] >;

C6.S4 in GAP, Magma, Sage, TeX

C_6.S_4
% in TeX

G:=Group("C6.S4");
// GroupNames label

G:=SmallGroup(144,33);
// by ID

G=gap.SmallGroup(144,33);
# by ID

G:=PCGroup([6,-2,-2,-3,-3,-2,2,12,362,122,579,2164,556,1301,989]);
// Polycyclic

G:=Group<a,b,c,d,e|a^6=b^2=c^2=1,d^3=a^2,e^2=a^3,a*b=b*a,a*c=c*a,a*d=d*a,e*a*e^-1=a^-1,d*b*d^-1=e*b*e^-1=b*c=c*b,d*c*d^-1=b,c*e=e*c,e*d*e^-1=a^4*d^2>;
// generators/relations

Export

Subgroup lattice of C6.S4 in TeX
Character table of C6.S4 in TeX

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