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## G = D5×GL2(𝔽3)  order 480 = 25·3·5

### Direct product of D5 and GL2(𝔽3)

Aliases: D5×GL2(𝔽3), D10.6S4, SL2(𝔽3)⋊2D10, (C5×Q8)⋊D6, Q8⋊(S3×D5), (Q8×D5)⋊3S3, C10.8(C2×S4), Q8⋊D156C2, C2.11(D5×S4), C51(C2×GL2(𝔽3)), (D5×SL2(𝔽3))⋊3C2, (C5×GL2(𝔽3))⋊4C2, (C5×SL2(𝔽3))⋊2C22, SmallGroup(480,974)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — Q8 — C5×SL2(𝔽3) — D5×GL2(𝔽3)
 Chief series C1 — C2 — Q8 — C5×Q8 — C5×SL2(𝔽3) — D5×SL2(𝔽3) — D5×GL2(𝔽3)
 Lower central C5×SL2(𝔽3) — D5×GL2(𝔽3)
 Upper central C1 — C2

Generators and relations for D5×GL2(𝔽3)
G = < a,b,c,d,e,f | a5=b2=c4=e3=f2=1, d2=c2, bab=a-1, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, dcd-1=fdf=c-1, ece-1=cd, fcf=c2d, ede-1=c, fef=e-1 >

Subgroups: 938 in 102 conjugacy classes, 17 normal (15 characteristic)
C1, C2, C2, C3, C4, C22, C5, S3, C6, C8, C2×C4, D4, Q8, Q8, C23, D5, D5, C10, C10, D6, C2×C6, C15, C2×C8, SD16, C2×D4, C2×Q8, Dic5, C20, D10, D10, C2×C10, SL2(𝔽3), C22×S3, C5×S3, C3×D5, D15, C30, C2×SD16, C52C8, C40, Dic10, C4×D5, D20, C5⋊D4, C5×D4, C5×Q8, C22×D5, GL2(𝔽3), GL2(𝔽3), C2×SL2(𝔽3), S3×D5, C6×D5, S3×C10, D30, C8×D5, C40⋊C2, D4.D5, Q8⋊D5, C5×SD16, D4×D5, Q8×D5, C2×GL2(𝔽3), C5×SL2(𝔽3), C2×S3×D5, D5×SD16, C5×GL2(𝔽3), Q8⋊D15, D5×SL2(𝔽3), D5×GL2(𝔽3)
Quotients: C1, C2, C22, S3, D5, D6, D10, S4, GL2(𝔽3), C2×S4, S3×D5, C2×GL2(𝔽3), D5×S4, D5×GL2(𝔽3)

Smallest permutation representation of D5×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 11)(2 15)(3 14)(4 13)(5 12)(6 19)(7 18)(8 17)(9 16)(10 20)(21 39)(22 38)(23 37)(24 36)(25 40)(26 33)(27 32)(28 31)(29 35)(30 34)
(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), (1,11)(2,15)(3,14)(4,13)(5,12)(6,19)(7,18)(8,17)(9,16)(10,20)(21,39)(22,38)(23,37)(24,36)(25,40)(26,33)(27,32)(28,31)(29,35)(30,34), (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), (1,11)(2,15)(3,14)(4,13)(5,12)(6,19)(7,18)(8,17)(9,16)(10,20)(21,39)(22,38)(23,37)(24,36)(25,40)(26,33)(27,32)(28,31)(29,35)(30,34), (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)], [(1,11),(2,15),(3,14),(4,13),(5,12),(6,19),(7,18),(8,17),(9,16),(10,20),(21,39),(22,38),(23,37),(24,36),(25,40),(26,33),(27,32),(28,31),(29,35),(30,34)], [(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)]])

32 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3 4A 4B 5A 5B 6A 6B 6C 8A 8B 8C 8D 10A 10B 10C 10D 15A 15B 20A 20B 30A 30B 40A 40B 40C 40D order 1 2 2 2 2 2 3 4 4 5 5 6 6 6 8 8 8 8 10 10 10 10 15 15 20 20 30 30 40 40 40 40 size 1 1 5 5 12 60 8 6 30 2 2 8 40 40 6 6 30 30 2 2 24 24 16 16 12 12 16 16 12 12 12 12

32 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 3 3 4 4 4 6 8 type + + + + + + + + + + + + + + image C1 C2 C2 C2 S3 D5 D6 D10 GL2(𝔽3) S4 C2×S4 GL2(𝔽3) S3×D5 D5×GL2(𝔽3) D5×S4 D5×GL2(𝔽3) kernel D5×GL2(𝔽3) C5×GL2(𝔽3) Q8⋊D15 D5×SL2(𝔽3) Q8×D5 GL2(𝔽3) C5×Q8 SL2(𝔽3) D5 D10 C10 D5 Q8 C1 C2 C1 # reps 1 1 1 1 1 2 1 2 4 2 2 2 2 4 4 2

Matrix representation of D5×GL2(𝔽3) in GL4(𝔽241) generated by

 1 2 0 0 214 188 0 0 0 0 1 0 0 0 0 1
,
 1 2 0 0 0 240 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 1 0 0 0 0 25 12 0 0 229 216
,
 1 0 0 0 0 1 0 0 0 0 228 12 0 0 26 13
,
 1 0 0 0 0 1 0 0 0 0 12 25 0 0 13 228
,
 1 0 0 0 0 1 0 0 0 0 240 0 0 0 1 1
G:=sub<GL(4,GF(241))| [1,214,0,0,2,188,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,2,240,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,25,229,0,0,12,216],[1,0,0,0,0,1,0,0,0,0,228,26,0,0,12,13],[1,0,0,0,0,1,0,0,0,0,12,13,0,0,25,228],[1,0,0,0,0,1,0,0,0,0,240,1,0,0,0,1] >;

D5×GL2(𝔽3) in GAP, Magma, Sage, TeX

D_5\times {\rm GL}_2({\mathbb F}_3)
% in TeX

G:=Group("D5xGL(2,3)");
// GroupNames label

G:=SmallGroup(480,974);
// by ID

G=gap.SmallGroup(480,974);
# by ID

G:=PCGroup([7,-2,-2,-3,-5,-2,2,-2,93,1347,2111,3168,172,1272,1909,285,124]);
// Polycyclic

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

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