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## G = C6×SL2(𝔽3)  order 144 = 24·32

### Direct product of C6 and SL2(𝔽3)

Aliases: C6×SL2(𝔽3), Q8⋊(C3×C6), (C6×Q8)⋊C3, (C2×Q8)⋊C32, C6.9(C2×A4), C2.2(C6×A4), (C2×C6).6A4, (C3×Q8)⋊2C6, C22.2(C3×A4), SmallGroup(144,156)

Series: Derived Chief Lower central Upper central

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

Generators and relations for C6×SL2(𝔽3)
G = < a,b,c,d | a6=b4=d3=1, c2=b2, ab=ba, ac=ca, ad=da, cbc-1=b-1, dbd-1=c, dcd-1=bc >

Smallest permutation representation of C6×SL2(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 25 46 18)(2 26 47 13)(3 27 48 14)(4 28 43 15)(5 29 44 16)(6 30 45 17)(7 41 22 36)(8 42 23 31)(9 37 24 32)(10 38 19 33)(11 39 20 34)(12 40 21 35)
(1 12 46 21)(2 7 47 22)(3 8 48 23)(4 9 43 24)(5 10 44 19)(6 11 45 20)(13 41 26 36)(14 42 27 31)(15 37 28 32)(16 38 29 33)(17 39 30 34)(18 40 25 35)
(1 3 5)(2 4 6)(7 28 34)(8 29 35)(9 30 36)(10 25 31)(11 26 32)(12 27 33)(13 37 20)(14 38 21)(15 39 22)(16 40 23)(17 41 24)(18 42 19)(43 45 47)(44 46 48)

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

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

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

C6×SL2(𝔽3) is a maximal subgroup of   C6.GL2(𝔽3)  SL2(𝔽3).D6  SL2(𝔽3).11D6

42 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C ··· 3H 4A 4B 6A ··· 6F 6G ··· 6X 12A 12B 12C 12D order 1 2 2 2 3 3 3 ··· 3 4 4 6 ··· 6 6 ··· 6 12 12 12 12 size 1 1 1 1 1 1 4 ··· 4 6 6 1 ··· 1 4 ··· 4 6 6 6 6

42 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 3 3 3 3 type + + - + + image C1 C2 C3 C3 C6 C6 SL2(𝔽3) SL2(𝔽3) C3×SL2(𝔽3) A4 C2×A4 C3×A4 C6×A4 kernel C6×SL2(𝔽3) C3×SL2(𝔽3) C2×SL2(𝔽3) C6×Q8 SL2(𝔽3) C3×Q8 C6 C6 C2 C2×C6 C6 C22 C2 # reps 1 1 6 2 6 2 2 4 12 1 1 2 2

Matrix representation of C6×SL2(𝔽3) in GL3(𝔽13) generated by

 10 0 0 0 12 0 0 0 12
,
 1 0 0 0 0 12 0 1 0
,
 1 0 0 0 3 4 0 4 10
,
 1 0 0 0 1 0 0 10 9
G:=sub<GL(3,GF(13))| [10,0,0,0,12,0,0,0,12],[1,0,0,0,0,1,0,12,0],[1,0,0,0,3,4,0,4,10],[1,0,0,0,1,10,0,0,9] >;

C6×SL2(𝔽3) in GAP, Magma, Sage, TeX

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

G:=Group("C6xSL(2,3)");
// GroupNames label

G:=SmallGroup(144,156);
// by ID

G=gap.SmallGroup(144,156);
# by ID

G:=PCGroup([6,-2,-3,-3,-2,2,-2,441,117,820,202,88]);
// Polycyclic

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

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