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G = D4×SL2(𝔽3)  order 192 = 26·3

Direct product of D4 and SL2(𝔽3)

direct product, non-abelian, soluble

Aliases: D4×SL2(𝔽3), (D4×Q8)⋊C3, Q8⋊(C3×D4), (C4×Q8)⋊3C6, C2.4(D4×A4), (C2×D4).1A4, C4⋊(C2×SL2(𝔽3)), (C22×Q8)⋊2C6, C23.27(C2×A4), C2.4(D4.A4), (C4×SL2(𝔽3))⋊8C2, C22.23(C22×A4), C222(C2×SL2(𝔽3)), (C22×SL2(𝔽3))⋊2C2, C2.2(C22×SL2(𝔽3)), (C2×SL2(𝔽3)).28C22, (C2×C4).6(C2×A4), (C2×Q8).39(C2×C6), SmallGroup(192,1004)

Series: Derived Chief Lower central Upper central

C1C2C2×Q8 — D4×SL2(𝔽3)
C1C2Q8C2×Q8C2×SL2(𝔽3)C22×SL2(𝔽3) — D4×SL2(𝔽3)
Q8C2×Q8 — D4×SL2(𝔽3)
C1C22C2×D4

Generators and relations for D4×SL2(𝔽3)
 G = < a,b,c,d,e | a4=b2=c4=e3=1, d2=c2, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd-1=c-1, ece-1=d, ede-1=cd >

Subgroups: 339 in 105 conjugacy classes, 31 normal (15 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C22, C6, C2×C4, C2×C4, D4, Q8, Q8, C23, C12, C2×C6, C42, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×Q8, C2×Q8, SL2(𝔽3), C2×C12, C3×D4, C22×C6, C4×D4, C4×Q8, C22⋊Q8, C4⋊Q8, C22×Q8, C2×SL2(𝔽3), C2×SL2(𝔽3), C6×D4, D4×Q8, C4×SL2(𝔽3), C22×SL2(𝔽3), D4×SL2(𝔽3)
Quotients: C1, C2, C3, C22, C6, D4, A4, C2×C6, SL2(𝔽3), C3×D4, C2×A4, C2×SL2(𝔽3), C22×A4, D4×A4, C22×SL2(𝔽3), D4.A4, D4×SL2(𝔽3)

Smallest permutation representation of D4×SL2(𝔽3)
On 32 points
Generators in S32
(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)
(2 4)(6 8)(9 11)(14 16)(17 19)(22 24)(25 27)(30 32)
(1 13 5 18)(2 14 6 19)(3 15 7 20)(4 16 8 17)(9 25 30 24)(10 26 31 21)(11 27 32 22)(12 28 29 23)
(1 21 5 26)(2 22 6 27)(3 23 7 28)(4 24 8 25)(9 16 30 17)(10 13 31 18)(11 14 32 19)(12 15 29 20)
(9 25 17)(10 26 18)(11 27 19)(12 28 20)(13 31 21)(14 32 22)(15 29 23)(16 30 24)

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

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

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

35 conjugacy classes

class 1 2A2B2C2D2E2F2G3A3B4A4B4C4D4E4F4G6A···6F6G···6N12A12B12C12D
order122222223344444446···66···612121212
size111122224422661212124···48···88888

35 irreducible representations

dim1111112222333446
type++++-+++-+
imageC1C2C2C3C6C6D4SL2(𝔽3)SL2(𝔽3)C3×D4A4C2×A4C2×A4D4.A4D4.A4D4×A4
kernelD4×SL2(𝔽3)C4×SL2(𝔽3)C22×SL2(𝔽3)D4×Q8C4×Q8C22×Q8SL2(𝔽3)D4D4Q8C2×D4C2×C4C23C2C2C2
# reps1122241482112121

Matrix representation of D4×SL2(𝔽3) in GL4(𝔽13) generated by

1000
0100
00127
0091
,
1000
0100
0010
00412
,
3900
91000
0010
0001
,
0100
12000
0010
0001
,
0100
10400
0010
0001
G:=sub<GL(4,GF(13))| [1,0,0,0,0,1,0,0,0,0,12,9,0,0,7,1],[1,0,0,0,0,1,0,0,0,0,1,4,0,0,0,12],[3,9,0,0,9,10,0,0,0,0,1,0,0,0,0,1],[0,12,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[0,10,0,0,1,4,0,0,0,0,1,0,0,0,0,1] >;

D4×SL2(𝔽3) in GAP, Magma, Sage, TeX

D_4\times {\rm SL}_2({\mathbb F}_3)
% in TeX

G:=Group("D4xSL(2,3)");
// GroupNames label

G:=SmallGroup(192,1004);
// by ID

G=gap.SmallGroup(192,1004);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,2,-2,197,438,172,775,285,124]);
// Polycyclic

G:=Group<a,b,c,d,e|a^4=b^2=c^4=e^3=1,d^2=c^2,b*a*b=a^-1,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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