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G = C5×GL2(𝔽3)  order 240 = 24·3·5

Direct product of C5 and GL2(𝔽3)

direct product, non-abelian, soluble

Aliases: C5×GL2(𝔽3), C10.5S4, SL2(𝔽3)⋊C10, Q8⋊(C5×S3), C2.3(C5×S4), (C5×Q8)⋊2S3, (C5×SL2(𝔽3))⋊4C2, SmallGroup(240,103)

Series: Derived Chief Lower central Upper central

C1C2Q8SL2(𝔽3) — C5×GL2(𝔽3)
C1C2Q8SL2(𝔽3)C5×SL2(𝔽3) — C5×GL2(𝔽3)
SL2(𝔽3) — C5×GL2(𝔽3)
C1C10

Generators and relations for C5×GL2(𝔽3)
 G = < a,b,c,d,e | a5=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 >

12C2
4C3
3C4
6C22
4C6
4S3
4S3
12C10
4C15
3D4
3C8
4D6
3C20
6C2×C10
4C5×S3
4C5×S3
4C30
3SD16
3C5×D4
3C40
4S3×C10
3C5×SD16

Smallest permutation representation of C5×GL2(𝔽3)
On 40 points
Generators in S40
(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)
(1 20 12 6)(2 16 13 7)(3 17 14 8)(4 18 15 9)(5 19 11 10)(21 34 39 30)(22 35 40 26)(23 31 36 27)(24 32 37 28)(25 33 38 29)
(1 24 12 37)(2 25 13 38)(3 21 14 39)(4 22 15 40)(5 23 11 36)(6 32 20 28)(7 33 16 29)(8 34 17 30)(9 35 18 26)(10 31 19 27)
(6 37 32)(7 38 33)(8 39 34)(9 40 35)(10 36 31)(16 25 29)(17 21 30)(18 22 26)(19 23 27)(20 24 28)
(1 12)(2 13)(3 14)(4 15)(5 11)(6 37)(7 38)(8 39)(9 40)(10 36)(16 25)(17 21)(18 22)(19 23)(20 24)

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

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

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

C5×GL2(𝔽3) is a maximal subgroup of   Dic5.6S4  GL2(𝔽3)⋊D5  D10.1S4

40 conjugacy classes

class 1 2A2B 3  4 5A5B5C5D 6 8A8B10A10B10C10D10E10F10G10H15A15B15C15D20A20B20C20D30A30B30C30D40A···40H
order122345555688101010101010101015151515202020203030303040···40
size11128611118661111121212128888666688886···6

40 irreducible representations

dim111122223344
type+++++
imageC1C2C5C10S3C5×S3GL2(𝔽3)C5×GL2(𝔽3)S4C5×S4GL2(𝔽3)C5×GL2(𝔽3)
kernelC5×GL2(𝔽3)C5×SL2(𝔽3)GL2(𝔽3)SL2(𝔽3)C5×Q8Q8C5C1C10C2C5C1
# reps114414282814

Matrix representation of C5×GL2(𝔽3) in GL2(𝔽11) generated by

40
04
,
17
610
,
410
67
,
11
89
,
11
010
G:=sub<GL(2,GF(11))| [4,0,0,4],[1,6,7,10],[4,6,10,7],[1,8,1,9],[1,0,1,10] >;

C5×GL2(𝔽3) in GAP, Magma, Sage, TeX

C_5\times {\rm GL}_2({\mathbb F}_3)
% in TeX

G:=Group("C5xGL(2,3)");
// GroupNames label

G:=SmallGroup(240,103);
// by ID

G=gap.SmallGroup(240,103);
# by ID

G:=PCGroup([6,-2,-5,-3,-2,2,-2,362,1443,447,117,904,286,202,88]);
// Polycyclic

G:=Group<a,b,c,d,e|a^5=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

Export

Subgroup lattice of C5×GL2(𝔽3) in TeX

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