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G = D10.S4order 480 = 25·3·5

3rd non-split extension by D10 of S4 acting via S4/A4=C2

non-abelian, soluble

Aliases: D10.3S4, D5.GL2(𝔽3), SL2(𝔽3)⋊2F5, D5.CSU2(𝔽3), Q8⋊(C3⋊F5), C5⋊(Q8⋊Dic3), (C5×Q8)⋊Dic3, (Q8×D5).2S3, C2.3(A4⋊F5), C10.2(A4⋊C4), (C5×SL2(𝔽3))⋊2C4, (D5×SL2(𝔽3)).2C2, SmallGroup(480,962)

Series: Derived Chief Lower central Upper central

C1C2Q8C5×SL2(𝔽3) — D10.S4
C1C2Q8C5×Q8C5×SL2(𝔽3)D5×SL2(𝔽3) — D10.S4
C5×SL2(𝔽3) — D10.S4
C1C2

Generators and relations for D10.S4
 G = < a,b,c,d,e,f | a10=b2=e3=1, c2=d2=a5, f2=a-1b, bab=a-1, ac=ca, ad=da, ae=ea, faf-1=a7, bc=cb, bd=db, be=eb, fbf-1=a6b, dcd-1=a5c, ece-1=a5cd, fcf-1=cd, ede-1=c, fdf-1=a5d, fef-1=e-1 >

5C2
5C2
4C3
3C4
5C22
15C4
60C4
4C6
20C6
20C6
4C15
15C2×C4
15Q8
30C8
30C2×C4
20Dic3
20C2×C6
20Dic3
3C20
3Dic5
12F5
4C30
4C3×D5
4C3×D5
5C2×Q8
15C2×C8
15C4⋊C4
20C2×Dic3
3C4×D5
3Dic10
6C5⋊C8
6C2×F5
4C3⋊F5
4C3⋊F5
4C6×D5
15Q8⋊C4
5C2×SL2(𝔽3)
3C4⋊F5
3D5⋊C8
4C2×C3⋊F5
5Q8⋊Dic3
3Q8⋊F5

Character table of D10.S4

 class 12A2B2C34A4B4C4D56A6B6C8A8B8C8D1015A15B2030A30B
 size 1155863060604840403030303041616241616
ρ111111111111111111111111    trivial
ρ21111111-1-11111-1-1-1-1111111    linear of order 2
ρ311-1-111-1-ii11-1-1-ii-ii111111    linear of order 4
ρ411-1-111-1i-i11-1-1i-ii-i111111    linear of order 4
ρ52222-122002-1-1-100002-1-12-1-1    orthogonal lifted from S3
ρ622-2-2-12-2002-11100002-1-12-1-1    symplectic lifted from Dic3, Schur index 2
ρ72-2-22-1000021-11-2-222-2-1-1011    symplectic lifted from CSU2(𝔽3), Schur index 2
ρ82-2-22-1000021-1122-2-2-2-1-1011    symplectic lifted from CSU2(𝔽3), Schur index 2
ρ92-22-2-10000211-1-2--2--2-2-2-1-1011    complex lifted from GL2(𝔽3)
ρ102-22-2-10000211-1--2-2-2--2-2-1-1011    complex lifted from GL2(𝔽3)
ρ1133330-1-1113000-1-1-1-1300-100    orthogonal lifted from S4
ρ1233330-1-1-1-130001111300-100    orthogonal lifted from S4
ρ1333-3-30-11i-i3000-ii-ii300-100    complex lifted from A4⋊C4
ρ1433-3-30-11-ii3000i-ii-i300-100    complex lifted from A4⋊C4
ρ154-44-4100004-1-110000-4110-1-1    orthogonal lifted from GL2(𝔽3)
ρ16440044000-14000000-1-1-1-1-1-1    orthogonal lifted from F5
ρ174-4-44100004-11-10000-4110-1-1    symplectic lifted from CSU2(𝔽3), Schur index 2
ρ184400-24000-1-2000000-11--15/21+-15/2-11+-15/21--15/2    complex lifted from C3⋊F5
ρ194400-24000-1-2000000-11+-15/21--15/2-11--15/21+-15/2    complex lifted from C3⋊F5
ρ208-800-40000-240000002110-1-1    symplectic faithful, Schur index 2
ρ218-80020000-2-20000002-1+-15/2-1--15/201+-15/21--15/2    complex faithful
ρ228-80020000-2-20000002-1--15/2-1+-15/201--15/21+-15/2    complex faithful
ρ231212000-4000-30000000-300100    orthogonal lifted from A4⋊F5

Smallest permutation representation of D10.S4
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 10)(2 9)(3 8)(4 7)(5 6)(11 20)(12 19)(13 18)(14 17)(15 16)(21 22)(23 30)(24 29)(25 28)(26 27)(31 32)(33 40)(34 39)(35 38)(36 37)
(1 22 6 27)(2 23 7 28)(3 24 8 29)(4 25 9 30)(5 26 10 21)(11 32 16 37)(12 33 17 38)(13 34 18 39)(14 35 19 40)(15 36 20 31)
(1 11 6 16)(2 12 7 17)(3 13 8 18)(4 14 9 19)(5 15 10 20)(21 36 26 31)(22 37 27 32)(23 38 28 33)(24 39 29 34)(25 40 30 35)
(11 32 22)(12 33 23)(13 34 24)(14 35 25)(15 36 26)(16 37 27)(17 38 28)(18 39 29)(19 40 30)(20 31 21)
(1 6)(2 9 10 3)(4 5 8 7)(12 14 20 18)(13 17 19 15)(21 39 23 35)(22 32)(24 38 30 36)(25 31 29 33)(26 34 28 40)(27 37)

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

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

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

Matrix representation of D10.S4 in GL6(𝔽241)

24000000
02400000
002402402394
00002400
00102400
0000591
,
100000
010000
00012400
00102400
00002400
0000591
,
68670000
1361730000
001000
000100
000010
000001
,
106670000
1741350000
001000
000100
000010
000001
,
24010000
24000000
001000
000100
000010
000001
,
01770000
17700000
000100
002402402404
001000
000001

G:=sub<GL(6,GF(241))| [240,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,1,0,0,0,240,0,0,0,0,0,239,240,240,59,0,0,4,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,240,240,240,59,0,0,0,0,0,1],[68,136,0,0,0,0,67,173,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[106,174,0,0,0,0,67,135,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[240,240,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,177,0,0,0,0,177,0,0,0,0,0,0,0,0,240,1,0,0,0,1,240,0,0,0,0,0,240,0,0,0,0,0,4,0,1] >;

D10.S4 in GAP, Magma, Sage, TeX

D_{10}.S_4
% in TeX

G:=Group("D10.S4");
// GroupNames label

G:=SmallGroup(480,962);
// by ID

G=gap.SmallGroup(480,962);
# by ID

G:=PCGroup([7,-2,-2,-3,-5,-2,2,-2,14,170,1011,682,4204,3168,172,2525,1909,285,124]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^10=b^2=e^3=1,c^2=d^2=a^5,f^2=a^-1*b,b*a*b=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,f*a*f^-1=a^7,b*c=c*b,b*d=d*b,b*e=e*b,f*b*f^-1=a^6*b,d*c*d^-1=a^5*c,e*c*e^-1=a^5*c*d,f*c*f^-1=c*d,e*d*e^-1=c,f*d*f^-1=a^5*d,f*e*f^-1=e^-1>;
// generators/relations

Export

Subgroup lattice of D10.S4 in TeX
Character table of D10.S4 in TeX

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