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## G = C22×C3.S4order 288 = 25·32

### Direct product of C22 and C3.S4

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

 Derived series C1 — C22 — C3.A4 — C22×C3.S4
 Chief series C1 — C22 — C2×C6 — C3.A4 — C3.S4 — C2×C3.S4 — C22×C3.S4
 Lower central C3.A4 — C22×C3.S4
 Upper central C1 — C22

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 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 2K 3 4A 4B 4C 4D 6A 6B 6C 6D 6E 6F 6G 9A 9B 9C 18A ··· 18I order 1 2 2 2 2 2 2 2 2 2 2 2 3 4 4 4 4 6 6 6 6 6 6 6 9 9 9 18 ··· 18 size 1 1 1 1 3 3 3 3 18 18 18 18 2 18 18 18 18 2 2 2 6 6 6 6 8 8 8 8 ··· 8

36 irreducible representations

 dim 1 1 1 2 2 2 2 3 3 6 6 type + + + + + + + + + + + image C1 C2 C2 S3 D6 D9 D18 S4 C2×S4 C3.S4 C2×C3.S4 kernel C22×C3.S4 C2×C3.S4 C22×C3.A4 C23×C6 C22×C6 C24 C23 C2×C6 C6 C22 C2 # reps 1 6 1 1 3 3 9 2 6 1 3

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

 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 36 0 0 0 0 0 1 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 20 0 0 0 0 0 17 11 0 0 0 0 0 0 0 36 1 0 0 0 0 0 36 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 36 0 0 0 0 0 0 0 36 0
,
 36 0 0 0 0 0 0 1 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

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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