Copied to
clipboard

G = (C2×C6)⋊4S4order 288 = 25·32

2nd semidirect product of C2×C6 and S4 acting via S4/A4=C2

non-abelian, soluble, monomial

Aliases: (C2×C6)⋊4S4, (C3×A4)⋊8D4, C6.38(C2×S4), (C23×C6)⋊4S3, A43(C3⋊D4), C6.7S44C2, C223(C3⋊S4), C242(C3⋊S3), (C22×A4)⋊4S3, (C2×A4).12D6, C33(A4⋊D4), C22⋊(C327D4), (C22×C6).23D6, (C6×A4).17C22, (A4×C2×C6)⋊3C2, (C2×C3⋊S4)⋊4C2, C2.11(C2×C3⋊S4), (C2×C6)⋊4(C3⋊D4), C23.5(C2×C3⋊S3), SmallGroup(288,917)

Series: Derived Chief Lower central Upper central

C1C22C6×A4 — (C2×C6)⋊4S4
C1C22C2×C6C3×A4C6×A4C2×C3⋊S4 — (C2×C6)⋊4S4
C3×A4C6×A4 — (C2×C6)⋊4S4
C1C2C22

Generators and relations for (C2×C6)⋊4S4
 G = < a,b,c,d,e,f | a2=b6=c2=d2=e3=f2=1, ab=ba, ac=ca, ad=da, ae=ea, faf=ab3, bc=cb, bd=db, be=eb, fbf=b-1, ece-1=fcf=cd=dc, ede-1=c, df=fd, fef=e-1 >

Subgroups: 952 in 156 conjugacy classes, 27 normal (18 characteristic)
C1, C2, C2 [×5], C3, C3 [×3], C4 [×3], C22 [×2], C22 [×11], S3 [×4], C6, C6 [×10], C2×C4 [×3], D4 [×6], C23, C23 [×5], C32, Dic3 [×6], A4 [×3], D6 [×6], C2×C6 [×2], C2×C6 [×11], C22⋊C4 [×3], C2×D4 [×3], C24, C3⋊S3, C3×C6 [×2], C2×Dic3 [×3], C3⋊D4 [×9], S4 [×3], C2×A4 [×3], C2×A4 [×3], C22×S3, C22×C6, C22×C6 [×4], C22≀C2, C3⋊Dic3, C3×A4, C2×C3⋊S3, C62, C6.D4 [×3], A4⋊C4 [×3], C2×C3⋊D4 [×3], C2×S4 [×3], C22×A4 [×3], C23×C6, C327D4, C3⋊S4, C6×A4, C6×A4, C244S3, A4⋊D4 [×3], C6.7S4, C2×C3⋊S4, A4×C2×C6, (C2×C6)⋊4S4
Quotients: C1, C2 [×3], C22, S3 [×4], D4, D6 [×4], C3⋊S3, C3⋊D4 [×4], S4, C2×C3⋊S3, C2×S4, C327D4, C3⋊S4, A4⋊D4, C2×C3⋊S4, (C2×C6)⋊4S4

Character table of (C2×C6)⋊4S4

 class 12A2B2C2D2E2F3A3B3C3D4A4B4C6A6B6C6D6E6F6G6H6I6J6K6L6M6N6O6P
 size 1123363628883636362226666888888888
ρ1111111111111111111111111111111    trivial
ρ211-111-1-1111111-1-11-11-1-111-1-1-1-1-111-1    linear of order 2
ρ3111111-11111-1-1-11111111111111111    linear of order 2
ρ411-111-111111-1-11-11-11-1-111-1-1-1-1-111-1    linear of order 2
ρ522-222-20-1-12-10001-11-111-1-1111-21-12-2    orthogonal lifted from D6
ρ62222220-12-1-1000-1-1-1-1-1-1-1-1-12-1-122-1-1    orthogonal lifted from S3
ρ722-222-202-1-1-1000-22-22-2-22-111111-1-11    orthogonal lifted from D6
ρ822-222-20-12-1-10001-11-111-1-11-211-22-11    orthogonal lifted from D6
ρ922222202-1-1-10002222222-1-1-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ102-20-220022220000-20200-2-200000-2-20    orthogonal lifted from D4
ρ112222220-1-1-12000-1-1-1-1-1-1-122-12-1-1-1-1-1    orthogonal lifted from S3
ρ122222220-1-12-1000-1-1-1-1-1-1-1-1-1-1-12-1-122    orthogonal lifted from S3
ρ1322-222-20-1-1-120001-11-111-12-21-211-1-11    orthogonal lifted from D6
ρ142-20-2200-1-12-1000--31-3-1-3--311-3-3--30--31-20    complex lifted from C3⋊D4
ρ152-20-2200-12-1-1000-31--3-1--3-311-30--3-30-21--3    complex lifted from C3⋊D4
ρ162-20-22002-1-1-10000-20200-21--3-3-3-3--311--3    complex lifted from C3⋊D4
ρ172-20-2200-12-1-1000--31-3-1-3--311--30-3--30-21-3    complex lifted from C3⋊D4
ρ182-20-2200-1-1-12000-31--3-1--3-31-20-30--3--311-3    complex lifted from C3⋊D4
ρ192-20-2200-1-1-12000--31-3-1-3--31-20--30-3-311--3    complex lifted from C3⋊D4
ρ202-20-2200-1-12-1000-31--3-1--3-311--3--3-30-31-20    complex lifted from C3⋊D4
ρ212-20-22002-1-1-10000-20200-21-3--3--3--3-311-3    complex lifted from C3⋊D4
ρ22333-1-1-113000-11-1333-1-1-1-1000000000    orthogonal lifted from S4
ρ2333-3-1-11130001-1-1-33-3-111-1000000000    orthogonal lifted from C2×S4
ρ24333-1-1-1-130001-11333-1-1-1-1000000000    orthogonal lifted from S4
ρ2533-3-1-11-13000-111-33-3-111-1000000000    orthogonal lifted from C2×S4
ρ26666-2-2-20-3000000-3-3-31111000000000    orthogonal lifted from C3⋊S4
ρ2766-6-2-220-30000003-331-1-11000000000    orthogonal lifted from C2×C3⋊S4
ρ286-602-20060000000-60-2002000000000    orthogonal lifted from A4⋊D4
ρ296-602-200-30000003-33-3-31-3--3-1000000000    complex faithful
ρ306-602-200-3000000-3-333-31--3-3-1000000000    complex faithful

Smallest permutation representation of (C2×C6)⋊4S4
On 36 points
Generators in S36
(1 18)(2 13)(3 14)(4 15)(5 16)(6 17)(7 36)(8 31)(9 32)(10 33)(11 34)(12 35)(19 25)(20 26)(21 27)(22 28)(23 29)(24 30)
(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)
(2 6)(3 5)(7 28)(8 27)(9 26)(10 25)(11 30)(12 29)(13 14)(15 18)(16 17)(19 36)(20 35)(21 34)(22 33)(23 32)(24 31)

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

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

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

Matrix representation of (C2×C6)⋊4S4 in GL5(𝔽13)

10000
012000
001200
000120
000012
,
40000
010000
00100
00010
00001
,
10000
01000
001200
000120
00001
,
10000
01000
00100
000120
000012
,
10000
01000
00001
00100
00010
,
01000
10000
00100
00001
00010

G:=sub<GL(5,GF(13))| [1,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12],[4,0,0,0,0,0,10,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,12,0,0,0,0,0,12],[1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0],[0,1,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,1,0] >;

(C2×C6)⋊4S4 in GAP, Magma, Sage, TeX

(C_2\times C_6)\rtimes_4S_4
% in TeX

G:=Group("(C2xC6):4S4");
// GroupNames label

G:=SmallGroup(288,917);
// by ID

G=gap.SmallGroup(288,917);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-3,-2,2,85,451,1684,6053,782,3534,1350]);
// Polycyclic

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

Export

Character table of (C2×C6)⋊4S4 in TeX

׿
×
𝔽