C2^2xC3.S4 - GroupNames

G = C₂²×C₃.S₄ order 288 = 2⁵·3²

Direct product of C₂² and C₃.S₄

direct product, non-abelian, soluble, monomial

Aliases: C₂²×C₃.S₄, C₂³⋊D₁₈, C₂⁴⋊₂D₉, 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, C₂, C₃, C₄, C₂², C₂², S₃, C₆, C₆, C₂×C₄, D₄, C₂³, C₂³, C₉, Dic₃, D₆, C₂×C₆, C₂×C₆, C₂²×C₄, C₂×D₄, C₂⁴, C₂⁴, D₉, C₁₈, C₂×Dic₃, C₃⋊D₄, C₂²×S₃, C₂²×C₆, C₂²×C₆, C₂²×D₄, C₃.A₄, D₁₈, C₂×C₁₈, C₂²×Dic₃, C₂×C₃⋊D₄, S₃×C₂³, C₂³×C₆, C₃.S₄, C₂×C₃.A₄, C₂²×D₉, C₂²×C₃⋊D₄, C₂×C₃.S₄, C₂²×C₃.A₄, C₂²×C₃.S₄
Quotients: C₁, C₂, C₂², S₃, C₂³, D₆, D₉, S₄, C₂²×S₃, D₁₈, C₂×S₄, C₃.S₄, C₂²×D₉, C₂²×S₄, C₂×C₃.S₄, 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 34)(20 35)(21 36)(22 28)(23 29)(24 30)(25 31)(26 32)(27 33)

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

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

G = C22×C3.S4 order 288 = 25·32

Direct product of C22 and C3.S4

G = C₂²×C₃.S₄ order 288 = 2⁵·3²

Direct product of C₂² and C₃.S₄