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G = C22×C3.S4order 288 = 25·32

Direct product of C22 and C3.S4

direct product, non-abelian, soluble, monomial

Aliases: C22×C3.S4, C23⋊D18, C242D9, C3.A4⋊C23, C3.(C22×S4), C6.27(C2×S4), (C2×C6).13S4, C22⋊(C22×D9), (C23×C6).4S3, (C22×C6).18D6, (C2×C3.A4)⋊C22, (C2×C6).(C22×S3), (C22×C3.A4)⋊3C2, SmallGroup(288,835)

Series: Derived Chief Lower central Upper central

C1C22C3.A4 — C22×C3.S4
C1C22C2×C6C3.A4C3.S4C2×C3.S4 — C22×C3.S4
C3.A4 — C22×C3.S4
C1C22

Generators and relations for C22×C3.S4
 G = < a,b,c,d,e,f,g | a2=b2=c3=d2=e2=g2=1, f3=c, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, bc=cb, bd=db, be=eb, bf=fb, bg=gb, cd=dc, ce=ec, cf=fc, gcg=c-1, fdf-1=gdg=de=ed, fef-1=d, eg=ge, gfg=c-1f2 >

Subgroups: 1306 in 230 conjugacy classes, 36 normal (11 characteristic)
C1, C2 [×3], C2 [×8], C3, C4 [×4], C22 [×2], C22 [×26], S3 [×4], C6 [×3], C6 [×4], C2×C4 [×6], D4 [×16], C23 [×3], C23 [×14], C9, Dic3 [×4], D6 [×16], C2×C6 [×2], C2×C6 [×10], C22×C4, C2×D4 [×12], C24, C24, D9 [×4], C18 [×3], C2×Dic3 [×6], C3⋊D4 [×16], C22×S3 [×10], C22×C6 [×3], C22×C6 [×4], C22×D4, C3.A4, D18 [×6], C2×C18, C22×Dic3, C2×C3⋊D4 [×12], S3×C23, C23×C6, C3.S4 [×4], C2×C3.A4 [×3], C22×D9, C22×C3⋊D4, C2×C3.S4 [×6], C22×C3.A4, C22×C3.S4
Quotients: C1, C2 [×7], C22 [×7], S3, C23, D6 [×3], D9, S4, C22×S3, D18 [×3], C2×S4 [×3], C3.S4, C22×D9, C22×S4, C2×C3.S4 [×3], C22×C3.S4

Smallest permutation representation of C22×C3.S4
On 36 points
Generators in S36
(1 15)(2 16)(3 17)(4 18)(5 10)(6 11)(7 12)(8 13)(9 14)(19 30)(20 31)(21 32)(22 33)(23 34)(24 35)(25 36)(26 28)(27 29)
(1 26)(2 27)(3 19)(4 20)(5 21)(6 22)(7 23)(8 24)(9 25)(10 32)(11 33)(12 34)(13 35)(14 36)(15 28)(16 29)(17 30)(18 31)
(1 4 7)(2 5 8)(3 6 9)(10 13 16)(11 14 17)(12 15 18)(19 22 25)(20 23 26)(21 24 27)(28 31 34)(29 32 35)(30 33 36)
(1 26)(2 27)(4 20)(5 21)(7 23)(8 24)(10 32)(12 34)(13 35)(15 28)(16 29)(18 31)
(2 27)(3 19)(5 21)(6 22)(8 24)(9 25)(10 32)(11 33)(13 35)(14 36)(16 29)(17 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 26)(2 25)(3 24)(4 23)(5 22)(6 21)(7 20)(8 19)(9 27)(10 33)(11 32)(12 31)(13 30)(14 29)(15 28)(16 36)(17 35)(18 34)

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

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

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

36 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I2J2K 3 4A4B4C4D6A6B6C6D6E6F6G9A9B9C18A···18I
order12222222222234444666666699918···18
size111133331818181821818181822266668888···8

36 irreducible representations

dim11122223366
type+++++++++++
imageC1C2C2S3D6D9D18S4C2×S4C3.S4C2×C3.S4
kernelC22×C3.S4C2×C3.S4C22×C3.A4C23×C6C22×C6C24C23C2×C6C6C22C2
# reps16113392613

Matrix representation of C22×C3.S4 in GL7(𝔽37)

1000000
0100000
00360000
00036000
0000100
0000010
0000001
,
36000000
03600000
00360000
00036000
0000100
0000010
0000001
,
363600000
1000000
0010000
0001000
0000100
0000010
0000001
,
1000000
0100000
0010000
0001000
00003600
0000010
00000036
,
1000000
0100000
0010000
0001000
00003600
00000360
0000001
,
312000000
171100000
00361000
00360000
0000001
00003600
00000360
,
36000000
1100000
00036000
00360000
0000010
0000100
00000036

G:=sub<GL(7,GF(37))| [1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[36,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[36,1,0,0,0,0,0,36,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,36],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,1],[31,17,0,0,0,0,0,20,11,0,0,0,0,0,0,0,36,36,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,36,0,0,0,0,1,0,0],[36,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,36,0,0,0,0,0,36,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,36] >;

C22×C3.S4 in GAP, Magma, Sage, TeX

C_2^2\times C_3.S_4
% in TeX

G:=Group("C2^2xC3.S4");
// GroupNames label

G:=SmallGroup(288,835);
// by ID

G=gap.SmallGroup(288,835);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-3,-2,2,1123,192,1684,6053,782,3534,1350]);
// Polycyclic

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

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