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G = S3×C22⋊C4order 96 = 25·3

Direct product of S3 and C22⋊C4

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: S3×C22⋊C4, D6.10D4, C23.18D6, (C2×C4)⋊5D6, D6⋊C49C2, D65(C2×C4), C2.1(S3×D4), C224(C4×S3), C6.17(C2×D4), (C22×S3)⋊2C4, (C2×C12)⋊6C22, C6.6(C22×C4), C6.D43C2, (S3×C23).1C2, (C2×C6).21C23, (C2×Dic3)⋊5C22, (C22×C6).10C22, C22.13(C22×S3), (C22×S3).33C22, (S3×C2×C4)⋊8C2, C2.8(S3×C2×C4), (C2×C6)⋊1(C2×C4), C31(C2×C22⋊C4), (C3×C22⋊C4)⋊8C2, SmallGroup(96,87)

Series: Derived Chief Lower central Upper central

C1C6 — S3×C22⋊C4
C1C3C6C2×C6C22×S3S3×C23 — S3×C22⋊C4
C3C6 — S3×C22⋊C4
C1C22C22⋊C4

Generators and relations for S3×C22⋊C4
 G = < a,b,c,d,e | a3=b2=c2=d2=e4=1, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, ece-1=cd=dc, de=ed >

Subgroups: 330 in 132 conjugacy classes, 49 normal (15 characteristic)
C1, C2, C2 [×2], C2 [×8], C3, C4 [×4], C22, C22 [×2], C22 [×20], S3 [×4], S3 [×2], C6, C6 [×2], C6 [×2], C2×C4 [×2], C2×C4 [×6], C23, C23 [×10], Dic3 [×2], C12 [×2], D6 [×8], D6 [×10], C2×C6, C2×C6 [×2], C2×C6 [×2], C22⋊C4, C22⋊C4 [×3], C22×C4 [×2], C24, C4×S3 [×4], C2×Dic3 [×2], C2×C12 [×2], C22×S3 [×2], C22×S3 [×4], C22×S3 [×4], C22×C6, C2×C22⋊C4, D6⋊C4 [×2], C6.D4, C3×C22⋊C4, S3×C2×C4 [×2], S3×C23, S3×C22⋊C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], D4 [×4], C23, D6 [×3], C22⋊C4 [×4], C22×C4, C2×D4 [×2], C4×S3 [×2], C22×S3, C2×C22⋊C4, S3×C2×C4, S3×D4 [×2], S3×C22⋊C4

Character table of S3×C22⋊C4

 class 12A2B2C2D2E2F2G2H2I2J2K34A4B4C4D4E4F4G4H6A6B6C6D6E12A12B12C12D
 size 111122333366222226666222444444
ρ1111111111111111111111111111111    trivial
ρ21111111111111-1-1-1-1-1-1-1-111111-1-1-1-1    linear of order 2
ρ3111111-1-1-1-1-1-111111-1-1-1-1111111111    linear of order 2
ρ4111111-1-1-1-1-1-11-1-1-1-1111111111-1-1-1-1    linear of order 2
ρ51111-1-11111-1-11-1-111-11-11111-1-111-1-1    linear of order 2
ρ61111-1-11111-1-1111-1-11-11-1111-1-1-1-111    linear of order 2
ρ71111-1-1-1-1-1-1111-1-1111-11-1111-1-111-1-1    linear of order 2
ρ81111-1-1-1-1-1-111111-1-1-11-11111-1-1-1-111    linear of order 2
ρ911-1-1-111-1-111-11-ii-ii-i-iii-1-11-11i-ii-i    linear of order 4
ρ1011-1-1-111-1-111-11i-ii-iii-i-i-1-11-11-ii-ii    linear of order 4
ρ1111-1-1-11-111-1-111-ii-iiii-i-i-1-11-11i-ii-i    linear of order 4
ρ1211-1-1-11-111-1-111i-ii-i-i-iii-1-11-11-ii-ii    linear of order 4
ρ1311-1-11-11-1-11-111i-i-iii-i-ii-1-111-1i-i-ii    linear of order 4
ρ1411-1-11-11-1-11-111-iii-i-iii-i-1-111-1-iii-i    linear of order 4
ρ1511-1-11-1-111-11-11i-i-ii-iii-i-1-111-1i-i-ii    linear of order 4
ρ1611-1-11-1-111-11-11-iii-ii-i-ii-1-111-1-iii-i    linear of order 4
ρ17222222000000-1-2-2-2-20000-1-1-1-1-11111    orthogonal lifted from D6
ρ182222-2-2000000-122-2-20000-1-1-11111-1-1    orthogonal lifted from D6
ρ192-2-2200-2-222002000000002-2-2000000    orthogonal lifted from D4
ρ202-22-200-22-2200200000000-22-2000000    orthogonal lifted from D4
ρ212-22-2002-22-200200000000-22-2000000    orthogonal lifted from D4
ρ22222222000000-122220000-1-1-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ232-2-220022-2-2002000000002-2-2000000    orthogonal lifted from D4
ρ242222-2-2000000-1-2-2220000-1-1-111-1-111    orthogonal lifted from D6
ρ2522-2-22-2000000-1-2i2i2i-2i000011-1-11i-i-ii    complex lifted from C4×S3
ρ2622-2-2-22000000-12i-2i2i-2i000011-11-1i-ii-i    complex lifted from C4×S3
ρ2722-2-22-2000000-12i-2i-2i2i000011-1-11-iii-i    complex lifted from C4×S3
ρ2822-2-2-22000000-1-2i2i-2i2i000011-11-1-ii-ii    complex lifted from C4×S3
ρ294-4-4400000000-200000000-222000000    orthogonal lifted from S3×D4
ρ304-44-400000000-2000000002-22000000    orthogonal lifted from S3×D4

Permutation representations of S3×C22⋊C4
On 24 points - transitive group 24T146
Generators in S24
(1 6 9)(2 7 10)(3 8 11)(4 5 12)(13 22 17)(14 23 18)(15 24 19)(16 21 20)
(5 12)(6 9)(7 10)(8 11)(17 22)(18 23)(19 24)(20 21)
(1 3)(2 13)(4 15)(5 24)(6 8)(7 22)(9 11)(10 17)(12 19)(14 16)(18 20)(21 23)
(1 14)(2 15)(3 16)(4 13)(5 22)(6 23)(7 24)(8 21)(9 18)(10 19)(11 20)(12 17)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)

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

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

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

G:=TransitiveGroup(24,146);

S3×C22⋊C4 is a maximal subgroup of
C24.35D6  C24.38D6  C429D6  C4212D6  C4×S3×D4  C4213D6  D1223D4  C4218D6  C247D6  C24.44D6  C6.402+ 1+4  D1220D4  C6.422+ 1+4  D1221D4  C6.512+ 1+4  C6.532+ 1+4  C6.1202+ 1+4  C6.1212+ 1+4  C6.612+ 1+4  C6.1222+ 1+4  C6.622+ 1+4  D1210D4  C4222D6  C4223D6  C4225D6  C4226D6  C62.91C23  C62.116C23  D30.27D4  D30.45D4
S3×C22⋊C4 is a maximal quotient of
(C2×C12)⋊Q8  C22.58(S3×D4)  (C2×C4)⋊9D12  D6⋊C42  D6⋊(C4⋊C4)  D6⋊M4(2)  D6⋊C8⋊C2  C23⋊C45S3  M4(2).19D6  M4(2).21D6  C4⋊C419D6  D4⋊(C4×S3)  D42S3⋊C4  (S3×Q8)⋊C4  Q87(C4×S3)  C4⋊C4.150D6  C423D6  C24.55D6  C24.59D6  C24.23D6  C24.24D6  C62.91C23  C62.116C23  D30.27D4  D30.45D4

Matrix representation of S3×C22⋊C4 in GL4(𝔽13) generated by

121200
1000
0010
0001
,
1000
121200
00120
00012
,
12000
01200
00120
00121
,
1000
0100
00120
00012
,
5000
0500
00111
00112
G:=sub<GL(4,GF(13))| [12,1,0,0,12,0,0,0,0,0,1,0,0,0,0,1],[1,12,0,0,0,12,0,0,0,0,12,0,0,0,0,12],[12,0,0,0,0,12,0,0,0,0,12,12,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,12,0,0,0,0,12],[5,0,0,0,0,5,0,0,0,0,1,1,0,0,11,12] >;

S3×C22⋊C4 in GAP, Magma, Sage, TeX

S_3\times C_2^2\rtimes C_4
% in TeX

G:=Group("S3xC2^2:C4");
// GroupNames label

G:=SmallGroup(96,87);
// by ID

G=gap.SmallGroup(96,87);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-3,188,50,2309]);
// Polycyclic

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

Export

Character table of S3×C22⋊C4 in TeX

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