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G = C22×F9  order 288 = 25·32

Direct product of C22 and F9

direct product, metabelian, soluble, monomial, A-group

Aliases: C22×F9, C62⋊2C8, C32⋊(C22×C8), C32⋊C4.2C23, C3⋊S3⋊(C2×C8), (C3×C6)⋊(C2×C8), (C2×C3⋊S3)⋊2C8, (C2×C32⋊C4).7C4, C32⋊C4.6(C2×C4), C3⋊S3.3(C22×C4), (C22×C3⋊S3).5C4, (C22×C32⋊C4).9C2, (C2×C32⋊C4).26C22, (C2×C3⋊S3).4(C2×C4), SmallGroup(288,1030)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C32 — C22×F9
Chief series
C1 — C32 — C3⋊S3 — C32⋊C4 — F9 — C2×F9 — C22×F9
Lower central
C32 — C22×F9

Subgroups: 492 in 92 conjugacy classes, 43 normal (9 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×4], C22, C22 [×6], S3 [×4], C6 [×3], C8 [×4], C2×C4 [×6], C23, C32, D6 [×6], C2×C6, C2×C8 [×6], C22×C4, C3⋊S3, C3⋊S3 [×3], C3×C6 [×3], C22×S3, C22×C8, C32⋊C4, C32⋊C4 [×3], C2×C3⋊S3 [×6], C62, F9 [×4], C2×C32⋊C4 [×6], C22×C3⋊S3, C2×F9 [×6], C22×C32⋊C4, C22×F9

Quotients:
C1, C2 [×7], C4 [×4], C22 [×7], C8 [×4], C2×C4 [×6], C23, C2×C8 [×6], C22×C4, C22×C8, F9, C2×F9 [×3], C22×F9

Generators and relations
 G = < a,b,c,d,e | a2=b2=c3=d3=e8=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, ece-1=cd=dc, ede-1=c >

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

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

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

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

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

Matrix representation ►G ⊆ GL10(𝔽73)

72000000000
07200000000
0010000000
0001000000
0000100000
0000010000
0000001000
0000000100
0000000010
0000000001
,
72000000000
0100000000
0010000000
0001000000
0000100000
0000010000
0000001000
0000000100
0000000010
0000000001
,
1000000000
0100000000
00000007210
00000007201
00000007200
00100007200
00010007200
00001007200
00000107200
00000017200
,
1000000000
0100000000
00072000000
00172000000
00072001000
00072100000
00072010000
00072000001
00072000100
00072000010
,
51000000000
05100000000
0000100000
0000000100
0000010000
0000000010
0010000000
0000000001
0001000000
0000001000

G:=sub<GL(10,GF(73))| [72,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[72,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,72,72,72,72,72,72,72,72,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0],[1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,72,72,72,72,72,72,72,72,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0],[51,0,0,0,0,0,0,0,0,0,0,51,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0] >;
 

36 conjugacy classes

class 1 2A2B2C2D2E2F2G 3 4A···4H6A6B6C8A···8P
order1222222234···46668···8
size1111999989···98889···9

36 irreducible representations

dim111111188
type+++++
imageC1C2C2C4C4C8C8F9C2×F9
kernelC22×F9C2×F9C22×C32⋊C4C2×C32⋊C4C22×C3⋊S3C2×C3⋊S3C62C22C2
# reps1616212413

In GAP, Magma, Sage, TeX 

C_2^2\times F_9
 
% in TeX
 
G:=Group("C2^2xF9");
 
// GroupNames label
 
G:=SmallGroup(288,1030);
 
// by ID
 
G=gap.SmallGroup(288,1030);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,56,80,4037,1202,201,10982,1595,622]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^2=b^2=c^3=d^3=e^8=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,e*c*e^-1=c*d=d*c,e*d*e^-1=c>;
 
// generators/relations
 

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