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G = C6.7S4order 144 = 24·32

7th non-split extension by C6 of S4 acting via S4/A4=C2

non-abelian, soluble, monomial

Aliases: C6.7S4, A4⋊Dic3, C3⋊(A4⋊C4), (C2×A4).S3, (C3×A4)⋊2C4, C2.1(C3⋊S4), (C6×A4).2C2, C23.(C3⋊S3), (C2×C6)⋊2Dic3, C22⋊(C3⋊Dic3), (C22×C6).4S3, SmallGroup(144,126)

Series: Derived Chief Lower central Upper central

C1C22C3×A4 — C6.7S4
C1C22C2×C6C3×A4C6×A4 — C6.7S4
C3×A4 — C6.7S4
C1C2

Generators and relations for C6.7S4
 G = < a,b,c,d,e | a6=b2=c2=d3=1, 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=d-1 >

3C2
3C2
4C3
4C3
4C3
3C22
3C22
18C4
18C4
3C6
3C6
4C6
4C6
4C6
4C32
9C2×C4
9C2×C4
3C2×C6
3C2×C6
6Dic3
6Dic3
12Dic3
12Dic3
12Dic3
4C3×C6
9C22⋊C4
3C2×Dic3
3C2×Dic3
4C3⋊Dic3
3A4⋊C4
3C6.D4
3A4⋊C4
3A4⋊C4

Character table of C6.7S4

 class 12A2B2C3A3B3C3D4A4B4C4D6A6B6C6D6E6F
 size 1133288818181818266888
ρ1111111111111111111    trivial
ρ211111111-1-1-1-1111111    linear of order 2
ρ31-1-111111-ii-ii-11-1-1-1-1    linear of order 4
ρ41-1-111111i-ii-i-11-1-1-1-1    linear of order 4
ρ522222-1-1-10000222-1-1-1    orthogonal lifted from S3
ρ62222-12-1-10000-1-1-1-12-1    orthogonal lifted from S3
ρ72222-1-1-120000-1-1-1-1-12    orthogonal lifted from S3
ρ82222-1-12-10000-1-1-12-1-1    orthogonal lifted from S3
ρ92-2-22-1-1-1200001-1111-2    symplectic lifted from Dic3, Schur index 2
ρ102-2-22-12-1-100001-111-21    symplectic lifted from Dic3, Schur index 2
ρ112-2-222-1-1-10000-22-2111    symplectic lifted from Dic3, Schur index 2
ρ122-2-22-1-12-100001-11-211    symplectic lifted from Dic3, Schur index 2
ρ1333-1-1300011-1-13-1-1000    orthogonal lifted from S4
ρ1433-1-13000-1-1113-1-1000    orthogonal lifted from S4
ρ153-31-13000i-i-ii-3-11000    complex lifted from A4⋊C4
ρ163-31-13000-iii-i-3-11000    complex lifted from A4⋊C4
ρ1766-2-2-30000000-311000    orthogonal lifted from C3⋊S4
ρ186-62-2-3000000031-1000    symplectic faithful, Schur index 2

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

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

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

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

C6.7S4 is a maximal subgroup of
Dic3.S4  Dic3×S4  S3×A4⋊C4  D6⋊S4  A4⋊Dic6  C4×C3⋊S4  (C2×C6)⋊4S4  C625Dic3  A4⋊Dic9  C6210Dic3
C6.7S4 is a maximal quotient of
C12.12S4  C6.GL2(𝔽3)  C3⋊U2(𝔽3)  A4⋊Dic9  C62.10Dic3  C626Dic3  C6210Dic3

Matrix representation of C6.7S4 in GL5(𝔽13)

11000
120000
00100
00010
00001
,
10000
01000
001200
000120
001201
,
10000
01000
00100
008120
001012
,
10000
01000
001104
001103
00162
,
80000
55000
001200
00802
00470

G:=sub<GL(5,GF(13))| [1,12,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,12,0,12,0,0,0,12,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,1,8,1,0,0,0,12,0,0,0,0,0,12],[1,0,0,0,0,0,1,0,0,0,0,0,11,11,1,0,0,0,0,6,0,0,4,3,2],[8,5,0,0,0,0,5,0,0,0,0,0,12,8,4,0,0,0,0,7,0,0,0,2,0] >;

C6.7S4 in GAP, Magma, Sage, TeX

C_6._7S_4
% in TeX

G:=Group("C6.7S4");
// GroupNames label

G:=SmallGroup(144,126);
// by ID

G=gap.SmallGroup(144,126);
# by ID

G:=PCGroup([6,-2,-2,-3,-3,-2,2,12,146,579,2164,556,1301,989]);
// Polycyclic

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

Export

Subgroup lattice of C6.7S4 in TeX
Character table of C6.7S4 in TeX

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