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## G = F5×SL2(𝔽3)  order 480 = 25·3·5

### Direct product of F5 and SL2(𝔽3)

Aliases: F5×SL2(𝔽3), (Q8×F5)⋊C3, Q8⋊(C3×F5), (C5×Q8)⋊C12, (C2×F5).A4, (Q8×D5).C6, C2.3(A4×F5), C10.2(C4×A4), C5⋊(C4×SL2(𝔽3)), D5.(C4.A4), D10.3(C2×A4), D5.(C2×SL2(𝔽3)), (C5×SL2(𝔽3))⋊3C4, (D5×SL2(𝔽3)).3C2, SmallGroup(480,965)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C5×Q8 — F5×SL2(𝔽3)
 Chief series C1 — C2 — C10 — C5×Q8 — Q8×D5 — D5×SL2(𝔽3) — F5×SL2(𝔽3)
 Lower central C5×Q8 — F5×SL2(𝔽3)
 Upper central C1 — C2

Generators and relations for F5×SL2(𝔽3)
G = < a,b,c,d,e | a5=b4=c4=e3=1, d2=c2, bab-1=a3, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd-1=c-1, ece-1=d, ede-1=cd >

Subgroups: 378 in 60 conjugacy classes, 18 normal (16 characteristic)
C1, C2, C2, C3, C4, C22, C5, C6, C2×C4, Q8, Q8, D5, C10, C12, C2×C6, C15, C42, C4⋊C4, C2×Q8, Dic5, C20, F5, F5, D10, SL2(𝔽3), C2×C12, C3×D5, C30, C4×Q8, Dic10, C4×D5, C5×Q8, C2×F5, C2×F5, C2×SL2(𝔽3), C3×F5, C6×D5, C4×F5, C4⋊F5, Q8×D5, C4×SL2(𝔽3), C5×SL2(𝔽3), C6×F5, Q8×F5, D5×SL2(𝔽3), F5×SL2(𝔽3)
Quotients: C1, C2, C3, C4, C6, C12, A4, F5, SL2(𝔽3), C2×A4, C4×A4, C2×SL2(𝔽3), C4.A4, C3×F5, C4×SL2(𝔽3), A4×F5, F5×SL2(𝔽3)

Smallest permutation representation of F5×SL2(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 13)(2 15 5 11)(3 12 4 14)(6 39 10 37)(7 36 9 40)(8 38)(16 24 20 22)(17 21 19 25)(18 23)(26 34 30 32)(27 31 29 35)(28 33)
(1 23 13 18)(2 24 14 19)(3 25 15 20)(4 21 11 16)(5 22 12 17)(6 26 36 31)(7 27 37 32)(8 28 38 33)(9 29 39 34)(10 30 40 35)
(1 8 13 38)(2 9 14 39)(3 10 15 40)(4 6 11 36)(5 7 12 37)(16 26 21 31)(17 27 22 32)(18 28 23 33)(19 29 24 34)(20 30 25 35)
(6 21 31)(7 22 32)(8 23 33)(9 24 34)(10 25 35)(16 26 36)(17 27 37)(18 28 38)(19 29 39)(20 30 40)

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

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

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

35 conjugacy classes

 class 1 2A 2B 2C 3A 3B 4A 4B 4C 4D 4E 4F 4G 4H 5 6A 6B 6C 6D 6E 6F 10 12A ··· 12H 15A 15B 20 30A 30B order 1 2 2 2 3 3 4 4 4 4 4 4 4 4 5 6 6 6 6 6 6 10 12 ··· 12 15 15 20 30 30 size 1 1 5 5 4 4 5 5 5 5 6 30 30 30 4 4 4 20 20 20 20 4 20 ··· 20 16 16 24 16 16

35 irreducible representations

 dim 1 1 1 1 1 1 12 2 2 2 3 3 3 4 4 8 8 type + + + - + + + - image C1 C2 C3 C4 C6 C12 A4×F5 SL2(𝔽3) SL2(𝔽3) C4.A4 A4 C2×A4 C4×A4 F5 C3×F5 F5×SL2(𝔽3) F5×SL2(𝔽3) kernel F5×SL2(𝔽3) D5×SL2(𝔽3) Q8×F5 C5×SL2(𝔽3) Q8×D5 C5×Q8 C2 F5 F5 D5 C2×F5 D10 C10 SL2(𝔽3) Q8 C1 C1 # reps 1 1 2 2 2 4 1 2 4 6 1 1 2 1 2 1 2

Matrix representation of F5×SL2(𝔽3) in GL6(𝔽61)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 60 0 0 1 0 0 60 0 0 0 1 0 60 0 0 0 0 1 60
,
 11 0 0 0 0 0 0 11 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0
,
 0 1 0 0 0 0 60 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 48 14 0 0 0 0 14 13 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 14 0 0 0 0 0 13 0 0 0 0 0 0 47 0 0 0 0 0 0 47 0 0 0 0 0 0 47 0 0 0 0 0 0 47

G:=sub<GL(6,GF(61))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,60,60,60,60],[11,0,0,0,0,0,0,11,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,1,0],[0,60,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[48,14,0,0,0,0,14,13,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,14,13,0,0,0,0,0,0,47,0,0,0,0,0,0,47,0,0,0,0,0,0,47,0,0,0,0,0,0,47] >;

F5×SL2(𝔽3) in GAP, Magma, Sage, TeX

F_5\times {\rm SL}_2({\mathbb F}_3)
% in TeX

G:=Group("F5xSL(2,3)");
// GroupNames label

G:=SmallGroup(480,965);
// by ID

G=gap.SmallGroup(480,965);
# by ID

G:=PCGroup([7,-2,-3,-2,-2,2,-5,-2,42,514,584,221,795,382,4037,1363]);
// Polycyclic

G:=Group<a,b,c,d,e|a^5=b^4=c^4=e^3=1,d^2=c^2,b*a*b^-1=a^3,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,d*c*d^-1=c^-1,e*c*e^-1=d,e*d*e^-1=c*d>;
// generators/relations

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