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## G = C22×F9order 288 = 25·32

### Direct product of C22 and F9

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
 Upper central C1 — C22

Generators and relations for C22×F9
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 >

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

Smallest permutation representation of C22×F9
On 36 points
Generators in S36
(1 2)(3 4)(5 20)(6 13)(7 14)(8 15)(9 16)(10 17)(11 18)(12 19)(21 31)(22 32)(23 33)(24 34)(25 35)(26 36)(27 29)(28 30)
(1 4)(2 3)(5 24)(6 25)(7 26)(8 27)(9 28)(10 21)(11 22)(12 23)(13 35)(14 36)(15 29)(16 30)(17 31)(18 32)(19 33)(20 34)
(1 5 9)(2 20 16)(3 34 30)(4 24 28)(6 12 18)(7 15 17)(8 10 14)(11 13 19)(21 36 27)(22 35 33)(23 32 25)(26 29 31)
(1 13 17)(2 6 10)(3 25 21)(4 35 31)(5 19 7)(8 16 18)(9 11 15)(12 14 20)(22 29 28)(23 36 34)(24 33 26)(27 30 32)
(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)(5,20)(6,13)(7,14)(8,15)(9,16)(10,17)(11,18)(12,19)(21,31)(22,32)(23,33)(24,34)(25,35)(26,36)(27,29)(28,30), (1,4)(2,3)(5,24)(6,25)(7,26)(8,27)(9,28)(10,21)(11,22)(12,23)(13,35)(14,36)(15,29)(16,30)(17,31)(18,32)(19,33)(20,34), (1,5,9)(2,20,16)(3,34,30)(4,24,28)(6,12,18)(7,15,17)(8,10,14)(11,13,19)(21,36,27)(22,35,33)(23,32,25)(26,29,31), (1,13,17)(2,6,10)(3,25,21)(4,35,31)(5,19,7)(8,16,18)(9,11,15)(12,14,20)(22,29,28)(23,36,34)(24,33,26)(27,30,32), (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)(5,20)(6,13)(7,14)(8,15)(9,16)(10,17)(11,18)(12,19)(21,31)(22,32)(23,33)(24,34)(25,35)(26,36)(27,29)(28,30), (1,4)(2,3)(5,24)(6,25)(7,26)(8,27)(9,28)(10,21)(11,22)(12,23)(13,35)(14,36)(15,29)(16,30)(17,31)(18,32)(19,33)(20,34), (1,5,9)(2,20,16)(3,34,30)(4,24,28)(6,12,18)(7,15,17)(8,10,14)(11,13,19)(21,36,27)(22,35,33)(23,32,25)(26,29,31), (1,13,17)(2,6,10)(3,25,21)(4,35,31)(5,19,7)(8,16,18)(9,11,15)(12,14,20)(22,29,28)(23,36,34)(24,33,26)(27,30,32), (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),(5,20),(6,13),(7,14),(8,15),(9,16),(10,17),(11,18),(12,19),(21,31),(22,32),(23,33),(24,34),(25,35),(26,36),(27,29),(28,30)], [(1,4),(2,3),(5,24),(6,25),(7,26),(8,27),(9,28),(10,21),(11,22),(12,23),(13,35),(14,36),(15,29),(16,30),(17,31),(18,32),(19,33),(20,34)], [(1,5,9),(2,20,16),(3,34,30),(4,24,28),(6,12,18),(7,15,17),(8,10,14),(11,13,19),(21,36,27),(22,35,33),(23,32,25),(26,29,31)], [(1,13,17),(2,6,10),(3,25,21),(4,35,31),(5,19,7),(8,16,18),(9,11,15),(12,14,20),(22,29,28),(23,36,34),(24,33,26),(27,30,32)], [(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)])

36 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3 4A ··· 4H 6A 6B 6C 8A ··· 8P order 1 2 2 2 2 2 2 2 3 4 ··· 4 6 6 6 8 ··· 8 size 1 1 1 1 9 9 9 9 8 9 ··· 9 8 8 8 9 ··· 9

36 irreducible representations

 dim 1 1 1 1 1 1 1 8 8 type + + + + + image C1 C2 C2 C4 C4 C8 C8 F9 C2×F9 kernel C22×F9 C2×F9 C22×C32⋊C4 C2×C32⋊C4 C22×C3⋊S3 C2×C3⋊S3 C62 C22 C2 # reps 1 6 1 6 2 12 4 1 3

Matrix representation of C22×F9 in GL10(𝔽73)

 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 0 0 72 1 0 0 0 0 0 0 0 0 72 0 1 0 0 0 0 0 0 0 72 0 0 0 0 1 0 0 0 0 72 0 0 0 0 0 1 0 0 0 72 0 0 0 0 0 0 1 0 0 72 0 0 0 0 0 0 0 1 0 72 0 0 0 0 0 0 0 0 1 72 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 72 0 0 0 0 0 0 0 0 1 72 0 0 0 0 0 0 0 0 0 72 0 0 1 0 0 0 0 0 0 72 1 0 0 0 0 0 0 0 0 72 0 1 0 0 0 0 0 0 0 72 0 0 0 0 0 1 0 0 0 72 0 0 0 1 0 0 0 0 0 72 0 0 0 0 1 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 1 0 0 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 1 0 0 0 1 0 0 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 0 1 0 0 0

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] >;

C22×F9 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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