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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, C2, C3, C4, C22, C22, S3, C6, C8, C2×C4, C23, C32, D6, C2×C6, C2×C8, C22×C4, C3⋊S3, C3⋊S3, C3×C6, C22×S3, C22×C8, C32⋊C4, C32⋊C4, C2×C3⋊S3, C62, F9, C2×C32⋊C4, C22×C3⋊S3, C2×F9, C22×C32⋊C4, C22×F9
Quotients: C1, C2, C4, C22, C8, C2×C4, C23, C2×C8, C22×C4, C22×C8, F9, C2×F9, C22×F9

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