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G = C22×SL2(𝔽3)  order 96 = 25·3

Direct product of C22 and SL2(𝔽3)

direct product, non-abelian, soluble

Aliases: C22×SL2(𝔽3), C23.6A4, Q8⋊(C2×C6), (C2×Q8)⋊2C6, (C22×Q8)⋊1C3, C2.3(C22×A4), C22.8(C2×A4), SmallGroup(96,198)

Series: Derived Chief Lower central Upper central

C1C2Q8 — C22×SL2(𝔽3)
C1C2Q8SL2(𝔽3)C2×SL2(𝔽3) — C22×SL2(𝔽3)
Q8 — C22×SL2(𝔽3)
C1C23

Generators and relations for C22×SL2(𝔽3)
 G = < a,b,c,d,e | a2=b2=c4=e3=1, d2=c2, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd-1=c-1, ece-1=d, ede-1=cd >

Subgroups: 147 in 61 conjugacy classes, 26 normal (7 characteristic)
C1, C2, C2 [×6], C3, C4 [×4], C22 [×7], C6 [×7], C2×C4 [×6], Q8, Q8 [×5], C23, C2×C6 [×7], C22×C4, C2×Q8 [×3], C2×Q8 [×3], SL2(𝔽3), C22×C6, C22×Q8, C2×SL2(𝔽3) [×3], C22×SL2(𝔽3)
Quotients: C1, C2 [×3], C3, C22, C6 [×3], A4, C2×C6, SL2(𝔽3) [×4], C2×A4 [×3], C2×SL2(𝔽3) [×6], C22×A4, C22×SL2(𝔽3)

Character table of C22×SL2(𝔽3)

 class 12A2B2C2D2E2F2G3A3B4A4B4C4D6A6B6C6D6E6F6G6H6I6J6K6L6M6N
 size 1111111144666644444444444444
ρ11111111111111111111111111111    trivial
ρ211-1-111-1-111-111-1-1-1-111111-1-1-1-1-11    linear of order 2
ρ31-1111-1-1-111-1-111-111-1-1-1-1111-1-1-11    linear of order 2
ρ41-1-1-11-111111-11-11-1-1-1-1-1-11-1-11111    linear of order 2
ρ51-1-1-11-111ζ3ζ321-11-1ζ32ζ65ζ65ζ65ζ65ζ6ζ6ζ32ζ6ζ6ζ3ζ3ζ32ζ3    linear of order 6
ρ611-1-111-1-1ζ32ζ3-111-1ζ65ζ6ζ6ζ32ζ32ζ3ζ3ζ3ζ65ζ65ζ6ζ6ζ65ζ32    linear of order 6
ρ71-1-1-11-111ζ32ζ31-11-1ζ3ζ6ζ6ζ6ζ6ζ65ζ65ζ3ζ65ζ65ζ32ζ32ζ3ζ32    linear of order 6
ρ81-1111-1-1-1ζ3ζ32-1-111ζ6ζ3ζ3ζ65ζ65ζ6ζ6ζ32ζ32ζ32ζ65ζ65ζ6ζ3    linear of order 6
ρ911111111ζ3ζ321111ζ32ζ3ζ3ζ3ζ3ζ32ζ32ζ32ζ32ζ32ζ3ζ3ζ32ζ3    linear of order 3
ρ1011111111ζ32ζ31111ζ3ζ32ζ32ζ32ζ32ζ3ζ3ζ3ζ3ζ3ζ32ζ32ζ3ζ32    linear of order 3
ρ111-1111-1-1-1ζ32ζ3-1-111ζ65ζ32ζ32ζ6ζ6ζ65ζ65ζ3ζ3ζ3ζ6ζ6ζ65ζ32    linear of order 6
ρ1211-1-111-1-1ζ3ζ32-111-1ζ6ζ65ζ65ζ3ζ3ζ32ζ32ζ32ζ6ζ6ζ65ζ65ζ6ζ3    linear of order 6
ρ132-2-22-222-2-1-1000011-11-11-111-1-11-11    symplectic lifted from SL2(𝔽3), Schur index 2
ρ1422-22-2-2-22-1-10000-11-1-11-1111-11-111    symplectic lifted from SL2(𝔽3), Schur index 2
ρ15222-2-2-22-2-1-100001-11-11-111-11-11-11    symplectic lifted from SL2(𝔽3), Schur index 2
ρ162-22-2-22-22-1-10000-1-111-11-11-111-111    symplectic lifted from SL2(𝔽3), Schur index 2
ρ17222-2-2-22-2ζ6ζ650000ζ3ζ6ζ32ζ6ζ32ζ65ζ3ζ3ζ65ζ3ζ6ζ32ζ65ζ32    complex lifted from SL2(𝔽3)
ρ182-22-2-22-22ζ65ζ60000ζ6ζ65ζ3ζ3ζ65ζ32ζ6ζ32ζ6ζ32ζ3ζ65ζ32ζ3    complex lifted from SL2(𝔽3)
ρ1922-22-2-2-22ζ6ζ650000ζ65ζ32ζ6ζ6ζ32ζ65ζ3ζ3ζ3ζ65ζ32ζ6ζ3ζ32    complex lifted from SL2(𝔽3)
ρ20222-2-2-22-2ζ65ζ60000ζ32ζ65ζ3ζ65ζ3ζ6ζ32ζ32ζ6ζ32ζ65ζ3ζ6ζ3    complex lifted from SL2(𝔽3)
ρ212-22-2-22-22ζ6ζ650000ζ65ζ6ζ32ζ32ζ6ζ3ζ65ζ3ζ65ζ3ζ32ζ6ζ3ζ32    complex lifted from SL2(𝔽3)
ρ222-2-22-222-2ζ65ζ60000ζ32ζ3ζ65ζ3ζ65ζ32ζ6ζ32ζ32ζ6ζ65ζ3ζ6ζ3    complex lifted from SL2(𝔽3)
ρ232-2-22-222-2ζ6ζ650000ζ3ζ32ζ6ζ32ζ6ζ3ζ65ζ3ζ3ζ65ζ6ζ32ζ65ζ32    complex lifted from SL2(𝔽3)
ρ2422-22-2-2-22ζ65ζ60000ζ6ζ3ζ65ζ65ζ3ζ6ζ32ζ32ζ32ζ6ζ3ζ65ζ32ζ3    complex lifted from SL2(𝔽3)
ρ253333333300-1-1-1-100000000000000    orthogonal lifted from A4
ρ263-3-3-33-33300-11-1100000000000000    orthogonal lifted from C2×A4
ρ2733-3-333-3-3001-1-1100000000000000    orthogonal lifted from C2×A4
ρ283-3333-3-3-30011-1-100000000000000    orthogonal lifted from C2×A4

Smallest permutation representation of C22×SL2(𝔽3)
On 32 points
Generators in S32
(1 19)(2 20)(3 17)(4 18)(5 23)(6 24)(7 21)(8 22)(9 27)(10 28)(11 25)(12 26)(13 31)(14 32)(15 29)(16 30)
(1 11)(2 12)(3 9)(4 10)(5 15)(6 16)(7 13)(8 14)(17 27)(18 28)(19 25)(20 26)(21 31)(22 32)(23 29)(24 30)
(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)
(1 7 3 5)(2 6 4 8)(9 15 11 13)(10 14 12 16)(17 23 19 21)(18 22 20 24)(25 31 27 29)(26 30 28 32)
(2 6 7)(4 8 5)(10 14 15)(12 16 13)(18 22 23)(20 24 21)(26 30 31)(28 32 29)

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

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

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

C22×SL2(𝔽3) is a maximal subgroup of   C23.14S4  C23.15S4  C23.16S4  (C2×Q8)⋊C12  SL2(𝔽3)⋊5D4

Matrix representation of C22×SL2(𝔽3) in GL4(𝔽13) generated by

1000
01200
0010
0001
,
12000
01200
00120
00012
,
1000
0100
00012
0010
,
1000
0100
0034
00410
,
9000
0900
0010
00109
G:=sub<GL(4,GF(13))| [1,0,0,0,0,12,0,0,0,0,1,0,0,0,0,1],[12,0,0,0,0,12,0,0,0,0,12,0,0,0,0,12],[1,0,0,0,0,1,0,0,0,0,0,1,0,0,12,0],[1,0,0,0,0,1,0,0,0,0,3,4,0,0,4,10],[9,0,0,0,0,9,0,0,0,0,1,10,0,0,0,9] >;

C22×SL2(𝔽3) in GAP, Magma, Sage, TeX

C_2^2\times {\rm SL}_2({\mathbb F}_3)
% in TeX

G:=Group("C2^2xSL(2,3)");
// GroupNames label

G:=SmallGroup(96,198);
// by ID

G=gap.SmallGroup(96,198);
# by ID

G:=PCGroup([6,-2,-2,-3,-2,2,-2,159,117,286,202,88]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^4=e^3=1,d^2=c^2,a*b=b*a,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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Character table of C22×SL2(𝔽3) in TeX

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