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G = Q8×SL2(𝔽3)  order 192 = 26·3

Direct product of Q8 and SL2(𝔽3)

direct product, non-abelian, soluble

Aliases: Q8×SL2(𝔽3), Q821C3, Q8⋊(C3×Q8), C2.3(Q8×A4), (C2×Q8).2A4, (C4×Q8).4C6, C4.(C2×SL2(𝔽3)), C2.3(Q8.A4), C22.26(C22×A4), (C4×SL2(𝔽3)).9C2, C2.3(C22×SL2(𝔽3)), (C2×SL2(𝔽3)).31C22, (C2×C4).8(C2×A4), (C2×Q8).42(C2×C6), SmallGroup(192,1007)

Series: Derived Chief Lower central Upper central

C1C2C2×Q8 — Q8×SL2(𝔽3)
C1C2Q8C2×Q8C2×SL2(𝔽3)C4×SL2(𝔽3) — Q8×SL2(𝔽3)
Q8C2×Q8 — Q8×SL2(𝔽3)
C1C22C2×Q8

Generators and relations for Q8×SL2(𝔽3)
 G = < a,b,c,d,e | a4=c4=e3=1, b2=a2, d2=c2, bab-1=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: 215 in 77 conjugacy classes, 31 normal (12 characteristic)
C1, C2 [×3], C3, C4 [×6], C4 [×5], C22, C6 [×3], C2×C4 [×3], C2×C4 [×4], Q8, Q8 [×4], Q8 [×3], C12 [×6], C2×C6, C42 [×3], C4⋊C4 [×6], C2×Q8 [×2], C2×Q8 [×2], SL2(𝔽3), C2×C12 [×3], C3×Q8 [×4], C4×Q8 [×3], C4×Q8, C4⋊Q8 [×3], C2×SL2(𝔽3), C6×Q8, Q82, C4×SL2(𝔽3) [×3], Q8×SL2(𝔽3)
Quotients: C1, C2 [×3], C3, C22, C6 [×3], Q8, A4, C2×C6, SL2(𝔽3) [×4], C3×Q8, C2×A4 [×3], C2×SL2(𝔽3) [×6], C22×A4, C22×SL2(𝔽3), Q8×A4, Q8.A4, Q8×SL2(𝔽3)

Smallest permutation representation of Q8×SL2(𝔽3)
On 64 points
Generators in S64
(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)(41 42 43 44)(45 46 47 48)(49 50 51 52)(53 54 55 56)(57 58 59 60)(61 62 63 64)
(1 12 3 10)(2 11 4 9)(5 57 7 59)(6 60 8 58)(13 53 15 55)(14 56 16 54)(17 44 19 42)(18 43 20 41)(21 37 23 39)(22 40 24 38)(25 36 27 34)(26 35 28 33)(29 47 31 45)(30 46 32 48)(49 63 51 61)(50 62 52 64)
(1 5 23 35)(2 6 24 36)(3 7 21 33)(4 8 22 34)(9 58 40 25)(10 59 37 26)(11 60 38 27)(12 57 39 28)(13 29 41 49)(14 30 42 50)(15 31 43 51)(16 32 44 52)(17 62 56 46)(18 63 53 47)(19 64 54 48)(20 61 55 45)
(1 15 23 43)(2 16 24 44)(3 13 21 41)(4 14 22 42)(5 51 35 31)(6 52 36 32)(7 49 33 29)(8 50 34 30)(9 56 40 17)(10 53 37 18)(11 54 38 19)(12 55 39 20)(25 46 58 62)(26 47 59 63)(27 48 60 64)(28 45 57 61)
(5 51 15)(6 52 16)(7 49 13)(8 50 14)(17 25 46)(18 26 47)(19 27 48)(20 28 45)(29 41 33)(30 42 34)(31 43 35)(32 44 36)(53 59 63)(54 60 64)(55 57 61)(56 58 62)

G:=sub<Sym(64)| (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)(41,42,43,44)(45,46,47,48)(49,50,51,52)(53,54,55,56)(57,58,59,60)(61,62,63,64), (1,12,3,10)(2,11,4,9)(5,57,7,59)(6,60,8,58)(13,53,15,55)(14,56,16,54)(17,44,19,42)(18,43,20,41)(21,37,23,39)(22,40,24,38)(25,36,27,34)(26,35,28,33)(29,47,31,45)(30,46,32,48)(49,63,51,61)(50,62,52,64), (1,5,23,35)(2,6,24,36)(3,7,21,33)(4,8,22,34)(9,58,40,25)(10,59,37,26)(11,60,38,27)(12,57,39,28)(13,29,41,49)(14,30,42,50)(15,31,43,51)(16,32,44,52)(17,62,56,46)(18,63,53,47)(19,64,54,48)(20,61,55,45), (1,15,23,43)(2,16,24,44)(3,13,21,41)(4,14,22,42)(5,51,35,31)(6,52,36,32)(7,49,33,29)(8,50,34,30)(9,56,40,17)(10,53,37,18)(11,54,38,19)(12,55,39,20)(25,46,58,62)(26,47,59,63)(27,48,60,64)(28,45,57,61), (5,51,15)(6,52,16)(7,49,13)(8,50,14)(17,25,46)(18,26,47)(19,27,48)(20,28,45)(29,41,33)(30,42,34)(31,43,35)(32,44,36)(53,59,63)(54,60,64)(55,57,61)(56,58,62)>;

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)(41,42,43,44)(45,46,47,48)(49,50,51,52)(53,54,55,56)(57,58,59,60)(61,62,63,64), (1,12,3,10)(2,11,4,9)(5,57,7,59)(6,60,8,58)(13,53,15,55)(14,56,16,54)(17,44,19,42)(18,43,20,41)(21,37,23,39)(22,40,24,38)(25,36,27,34)(26,35,28,33)(29,47,31,45)(30,46,32,48)(49,63,51,61)(50,62,52,64), (1,5,23,35)(2,6,24,36)(3,7,21,33)(4,8,22,34)(9,58,40,25)(10,59,37,26)(11,60,38,27)(12,57,39,28)(13,29,41,49)(14,30,42,50)(15,31,43,51)(16,32,44,52)(17,62,56,46)(18,63,53,47)(19,64,54,48)(20,61,55,45), (1,15,23,43)(2,16,24,44)(3,13,21,41)(4,14,22,42)(5,51,35,31)(6,52,36,32)(7,49,33,29)(8,50,34,30)(9,56,40,17)(10,53,37,18)(11,54,38,19)(12,55,39,20)(25,46,58,62)(26,47,59,63)(27,48,60,64)(28,45,57,61), (5,51,15)(6,52,16)(7,49,13)(8,50,14)(17,25,46)(18,26,47)(19,27,48)(20,28,45)(29,41,33)(30,42,34)(31,43,35)(32,44,36)(53,59,63)(54,60,64)(55,57,61)(56,58,62) );

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),(41,42,43,44),(45,46,47,48),(49,50,51,52),(53,54,55,56),(57,58,59,60),(61,62,63,64)], [(1,12,3,10),(2,11,4,9),(5,57,7,59),(6,60,8,58),(13,53,15,55),(14,56,16,54),(17,44,19,42),(18,43,20,41),(21,37,23,39),(22,40,24,38),(25,36,27,34),(26,35,28,33),(29,47,31,45),(30,46,32,48),(49,63,51,61),(50,62,52,64)], [(1,5,23,35),(2,6,24,36),(3,7,21,33),(4,8,22,34),(9,58,40,25),(10,59,37,26),(11,60,38,27),(12,57,39,28),(13,29,41,49),(14,30,42,50),(15,31,43,51),(16,32,44,52),(17,62,56,46),(18,63,53,47),(19,64,54,48),(20,61,55,45)], [(1,15,23,43),(2,16,24,44),(3,13,21,41),(4,14,22,42),(5,51,35,31),(6,52,36,32),(7,49,33,29),(8,50,34,30),(9,56,40,17),(10,53,37,18),(11,54,38,19),(12,55,39,20),(25,46,58,62),(26,47,59,63),(27,48,60,64),(28,45,57,61)], [(5,51,15),(6,52,16),(7,49,13),(8,50,14),(17,25,46),(18,26,47),(19,27,48),(20,28,45),(29,41,33),(30,42,34),(31,43,35),(32,44,36),(53,59,63),(54,60,64),(55,57,61),(56,58,62)])

35 conjugacy classes

class 1 2A2B2C3A3B4A···4F4G4H4I4J4K6A···6F12A···12L
order1222334···4444446···612···12
size1111442···2661212124···48···8

35 irreducible representations

dim1111222233446
type++--+++-
imageC1C2C3C6Q8SL2(𝔽3)SL2(𝔽3)C3×Q8A4C2×A4Q8.A4Q8.A4Q8×A4
kernelQ8×SL2(𝔽3)C4×SL2(𝔽3)Q82C4×Q8SL2(𝔽3)Q8Q8Q8C2×Q8C2×C4C2C2C2
# reps1326148213121

Matrix representation of Q8×SL2(𝔽3) in GL4(𝔽13) generated by

1000
0100
0001
00120
,
1000
0100
0005
0050
,
41000
10900
0010
0001
,
0100
12000
0010
0001
,
4100
10000
0010
0001
G:=sub<GL(4,GF(13))| [1,0,0,0,0,1,0,0,0,0,0,12,0,0,1,0],[1,0,0,0,0,1,0,0,0,0,0,5,0,0,5,0],[4,10,0,0,10,9,0,0,0,0,1,0,0,0,0,1],[0,12,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[4,10,0,0,1,0,0,0,0,0,1,0,0,0,0,1] >;

Q8×SL2(𝔽3) in GAP, Magma, Sage, TeX

Q_8\times {\rm SL}_2({\mathbb F}_3)
% in TeX

G:=Group("Q8xSL(2,3)");
// GroupNames label

G:=SmallGroup(192,1007);
// by ID

G=gap.SmallGroup(192,1007);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,2,-2,84,197,92,438,172,775,285,124]);
// Polycyclic

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