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## G = C6×GL2(𝔽3)  order 288 = 25·32

### Direct product of C6 and GL2(𝔽3)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — Q8 — SL2(𝔽3) — C6×GL2(𝔽3)
 Chief series C1 — C2 — Q8 — SL2(𝔽3) — C3×SL2(𝔽3) — C3×GL2(𝔽3) — C6×GL2(𝔽3)
 Lower central SL2(𝔽3) — C6×GL2(𝔽3)
 Upper central C1 — C2×C6

Generators and relations for C6×GL2(𝔽3)
G = < a,b,c,d,e | a6=b4=d3=e2=1, c2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=ece=b-1, dbd-1=bc, ebe=b2c, dcd-1=b, ede=d-1 >

Subgroups: 430 in 109 conjugacy classes, 24 normal (16 characteristic)
C1, C2, C2 [×2], C2 [×2], C3, C3 [×2], C4 [×2], C22, C22 [×4], S3 [×4], C6, C6 [×2], C6 [×8], C8 [×2], C2×C4, D4 [×3], Q8, Q8, C23, C32, C12 [×2], D6 [×6], C2×C6, C2×C6 [×6], C2×C8, SD16 [×4], C2×D4, C2×Q8, C3×S3 [×4], C3×C6 [×3], C24 [×2], SL2(𝔽3), SL2(𝔽3), C2×C12, C3×D4 [×3], C3×Q8, C3×Q8, C22×S3, C22×C6, C2×SD16, S3×C6 [×6], C62, C2×C24, C3×SD16 [×4], GL2(𝔽3) [×2], C2×SL2(𝔽3), C2×SL2(𝔽3), C6×D4, C6×Q8, C3×SL2(𝔽3), S3×C2×C6, C6×SD16, C2×GL2(𝔽3), C3×GL2(𝔽3) [×2], C6×SL2(𝔽3), C6×GL2(𝔽3)
Quotients: C1, C2 [×3], C3, C22, S3, C6 [×3], D6, C2×C6, C3×S3, S4, S3×C6, GL2(𝔽3) [×2], C2×S4, C3×S4, C2×GL2(𝔽3), C3×GL2(𝔽3) [×2], C6×S4, C6×GL2(𝔽3)

Smallest permutation representation of C6×GL2(𝔽3)
On 48 points
Generators in S48
(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)(37 38 39 40 41 42)(43 44 45 46 47 48)
(1 24 46 12)(2 19 47 7)(3 20 48 8)(4 21 43 9)(5 22 44 10)(6 23 45 11)(13 37 30 32)(14 38 25 33)(15 39 26 34)(16 40 27 35)(17 41 28 36)(18 42 29 31)
(1 16 46 27)(2 17 47 28)(3 18 48 29)(4 13 43 30)(5 14 44 25)(6 15 45 26)(7 41 19 36)(8 42 20 31)(9 37 21 32)(10 38 22 33)(11 39 23 34)(12 40 24 35)
(7 28 41)(8 29 42)(9 30 37)(10 25 38)(11 26 39)(12 27 40)(13 32 21)(14 33 22)(15 34 23)(16 35 24)(17 36 19)(18 31 20)
(7 17)(8 18)(9 13)(10 14)(11 15)(12 16)(19 28)(20 29)(21 30)(22 25)(23 26)(24 27)(31 42)(32 37)(33 38)(34 39)(35 40)(36 41)

G:=sub<Sym(48)| (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)(37,38,39,40,41,42)(43,44,45,46,47,48), (1,24,46,12)(2,19,47,7)(3,20,48,8)(4,21,43,9)(5,22,44,10)(6,23,45,11)(13,37,30,32)(14,38,25,33)(15,39,26,34)(16,40,27,35)(17,41,28,36)(18,42,29,31), (1,16,46,27)(2,17,47,28)(3,18,48,29)(4,13,43,30)(5,14,44,25)(6,15,45,26)(7,41,19,36)(8,42,20,31)(9,37,21,32)(10,38,22,33)(11,39,23,34)(12,40,24,35), (7,28,41)(8,29,42)(9,30,37)(10,25,38)(11,26,39)(12,27,40)(13,32,21)(14,33,22)(15,34,23)(16,35,24)(17,36,19)(18,31,20), (7,17)(8,18)(9,13)(10,14)(11,15)(12,16)(19,28)(20,29)(21,30)(22,25)(23,26)(24,27)(31,42)(32,37)(33,38)(34,39)(35,40)(36,41)>;

G:=Group( (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)(37,38,39,40,41,42)(43,44,45,46,47,48), (1,24,46,12)(2,19,47,7)(3,20,48,8)(4,21,43,9)(5,22,44,10)(6,23,45,11)(13,37,30,32)(14,38,25,33)(15,39,26,34)(16,40,27,35)(17,41,28,36)(18,42,29,31), (1,16,46,27)(2,17,47,28)(3,18,48,29)(4,13,43,30)(5,14,44,25)(6,15,45,26)(7,41,19,36)(8,42,20,31)(9,37,21,32)(10,38,22,33)(11,39,23,34)(12,40,24,35), (7,28,41)(8,29,42)(9,30,37)(10,25,38)(11,26,39)(12,27,40)(13,32,21)(14,33,22)(15,34,23)(16,35,24)(17,36,19)(18,31,20), (7,17)(8,18)(9,13)(10,14)(11,15)(12,16)(19,28)(20,29)(21,30)(22,25)(23,26)(24,27)(31,42)(32,37)(33,38)(34,39)(35,40)(36,41) );

G=PermutationGroup([(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),(37,38,39,40,41,42),(43,44,45,46,47,48)], [(1,24,46,12),(2,19,47,7),(3,20,48,8),(4,21,43,9),(5,22,44,10),(6,23,45,11),(13,37,30,32),(14,38,25,33),(15,39,26,34),(16,40,27,35),(17,41,28,36),(18,42,29,31)], [(1,16,46,27),(2,17,47,28),(3,18,48,29),(4,13,43,30),(5,14,44,25),(6,15,45,26),(7,41,19,36),(8,42,20,31),(9,37,21,32),(10,38,22,33),(11,39,23,34),(12,40,24,35)], [(7,28,41),(8,29,42),(9,30,37),(10,25,38),(11,26,39),(12,27,40),(13,32,21),(14,33,22),(15,34,23),(16,35,24),(17,36,19),(18,31,20)], [(7,17),(8,18),(9,13),(10,14),(11,15),(12,16),(19,28),(20,29),(21,30),(22,25),(23,26),(24,27),(31,42),(32,37),(33,38),(34,39),(35,40),(36,41)])

48 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3A 3B 3C 3D 3E 4A 4B 6A ··· 6F 6G ··· 6O 6P 6Q 6R 6S 8A 8B 8C 8D 12A 12B 12C 12D 24A ··· 24H order 1 2 2 2 2 2 3 3 3 3 3 4 4 6 ··· 6 6 ··· 6 6 6 6 6 8 8 8 8 12 12 12 12 24 ··· 24 size 1 1 1 1 12 12 1 1 8 8 8 6 6 1 ··· 1 8 ··· 8 12 12 12 12 6 6 6 6 6 6 6 6 6 ··· 6

48 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 3 3 3 3 4 4 type + + + + + + + + image C1 C2 C2 C3 C6 C6 S3 D6 C3×S3 S3×C6 GL2(𝔽3) C3×GL2(𝔽3) S4 C2×S4 C3×S4 C6×S4 GL2(𝔽3) C3×GL2(𝔽3) kernel C6×GL2(𝔽3) C3×GL2(𝔽3) C6×SL2(𝔽3) C2×GL2(𝔽3) GL2(𝔽3) C2×SL2(𝔽3) C6×Q8 C3×Q8 C2×Q8 Q8 C6 C2 C2×C6 C6 C22 C2 C6 C2 # reps 1 2 1 2 4 2 1 1 2 2 4 8 2 2 4 4 2 4

Matrix representation of C6×GL2(𝔽3) in GL3(𝔽73) generated by

 72 0 0 0 65 0 0 0 65
,
 1 0 0 0 32 52 0 21 41
,
 1 0 0 0 20 52 0 33 53
,
 1 0 0 0 52 32 0 53 20
,
 72 0 0 0 1 0 0 72 72
G:=sub<GL(3,GF(73))| [72,0,0,0,65,0,0,0,65],[1,0,0,0,32,21,0,52,41],[1,0,0,0,20,33,0,52,53],[1,0,0,0,52,53,0,32,20],[72,0,0,0,1,72,0,0,72] >;

C6×GL2(𝔽3) in GAP, Magma, Sage, TeX

C_6\times {\rm GL}_2({\mathbb F}_3)
% in TeX

G:=Group("C6xGL(2,3)");
// GroupNames label

G:=SmallGroup(288,900);
// by ID

G=gap.SmallGroup(288,900);
# by ID

G:=PCGroup([7,-2,-2,-3,-3,-2,2,-2,675,2524,655,172,1517,404,285,124]);
// Polycyclic

G:=Group<a,b,c,d,e|a^6=b^4=d^3=e^2=1,c^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=e*c*e=b^-1,d*b*d^-1=b*c,e*b*e=b^2*c,d*c*d^-1=b,e*d*e=d^-1>;
// generators/relations

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