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G = S3×C4⋊C4order 96 = 25·3

Direct product of S3 and C4⋊C4

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

Aliases: S3×C4⋊C4, D6.Q8, D6.11D4, C43(C4×S3), (C4×S3)⋊1C4, C121(C2×C4), C2.3(S3×D4), C2.2(S3×Q8), D6.8(C2×C4), C6.23(C2×D4), (C2×C4).30D6, C6.12(C2×Q8), C4⋊Dic311C2, Dic33(C2×C4), C6.9(C22×C4), Dic3⋊C411C2, (C2×C6).32C23, (C2×C12).23C22, C22.16(C22×S3), (C22×S3).34C22, (C2×Dic3).29C22, C31(C2×C4⋊C4), (C3×C4⋊C4)⋊2C2, (S3×C2×C4).1C2, C2.11(S3×C2×C4), SmallGroup(96,98)

Series: Derived Chief Lower central Upper central

C1C6 — S3×C4⋊C4
C1C3C6C2×C6C22×S3S3×C2×C4 — S3×C4⋊C4
C3C6 — S3×C4⋊C4
C1C22C4⋊C4

Generators and relations for S3×C4⋊C4
 G = < a,b,c,d | a3=b2=c4=d4=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 186 in 92 conjugacy classes, 49 normal (19 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×2], C4 [×6], C22, C22 [×6], S3 [×4], C6 [×3], C2×C4, C2×C4 [×2], C2×C4 [×11], C23, Dic3 [×2], Dic3 [×2], C12 [×2], C12 [×2], D6 [×6], C2×C6, C4⋊C4, C4⋊C4 [×3], C22×C4 [×3], C4×S3 [×4], C4×S3 [×4], C2×Dic3, C2×Dic3 [×2], C2×C12, C2×C12 [×2], C22×S3, C2×C4⋊C4, Dic3⋊C4 [×2], C4⋊Dic3, C3×C4⋊C4, S3×C2×C4, S3×C2×C4 [×2], S3×C4⋊C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], D4 [×2], Q8 [×2], C23, D6 [×3], C4⋊C4 [×4], C22×C4, C2×D4, C2×Q8, C4×S3 [×2], C22×S3, C2×C4⋊C4, S3×C2×C4, S3×D4, S3×Q8, S3×C4⋊C4

Character table of S3×C4⋊C4

 class 12A2B2C2D2E2F2G34A4B4C4D4E4F4G4H4I4J4K4L6A6B6C12A12B12C12D12E12F
 size 111133332222222666666222444444
ρ1111111111111111111111111111111    trivial
ρ21111-1-1-1-11-111-1-1-1-1111-11111-1-11-11-1    linear of order 2
ρ31111-1-1-1-11111111-1-1-1-1-1-1111111111    linear of order 2
ρ4111111111-111-1-1-11-1-1-11-1111-1-11-11-1    linear of order 2
ρ51111-1-1-1-11-1-1-111-11-1111-111111-1-1-1-1    linear of order 2
ρ61111111111-1-1-1-11-1-111-1-1111-1-1-11-11    linear of order 2
ρ7111111111-1-1-111-1-11-1-1-1111111-1-1-1-1    linear of order 2
ρ81111-1-1-1-111-1-1-1-1111-1-111111-1-1-11-11    linear of order 2
ρ911-1-1-11-111-1-iii-i1-i-i1-1ii-1-11-iii1-i-1    linear of order 4
ρ1011-1-11-11-111-ii-ii-1i-i1-1-ii-1-11i-ii-1-i1    linear of order 4
ρ1111-1-11-11-11-1-iii-i1ii-11-i-i-1-11-iii1-i-1    linear of order 4
ρ1211-1-1-11-1111-ii-ii-1-ii-11i-i-1-11i-ii-1-i1    linear of order 4
ρ1311-1-1-11-111-1i-i-ii1ii1-1-i-i-1-11i-i-i1i-1    linear of order 4
ρ1411-1-11-11-111i-ii-i-1-ii1-1i-i-1-11-ii-i-1i1    linear of order 4
ρ1511-1-11-11-11-1i-i-ii1-i-i-11ii-1-11i-i-i1i-1    linear of order 4
ρ1611-1-1-11-1111i-ii-i-1i-i-11-ii-1-11-ii-i-1i1    linear of order 4
ρ172-22-222-2-22000000000000-22-2000000    orthogonal lifted from D4
ρ1822220000-1-222-2-2-2000000-1-1-111-11-11    orthogonal lifted from D6
ρ1922220000-1-2-2-222-2000000-1-1-1-1-11111    orthogonal lifted from D6
ρ202-22-2-2-2222000000000000-22-2000000    orthogonal lifted from D4
ρ2122220000-12-2-2-2-22000000-1-1-1111-11-1    orthogonal lifted from D6
ρ2222220000-1222222000000-1-1-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ232-2-22-222-220000000000002-2-2000000    symplectic lifted from Q8, Schur index 2
ρ242-2-222-2-2220000000000002-2-2000000    symplectic lifted from Q8, Schur index 2
ρ2522-2-20000-122i-2i2i-2i-200000011-1i-ii1-i-1    complex lifted from C4×S3
ρ2622-2-20000-12-2i2i-2i2i-200000011-1-ii-i1i-1    complex lifted from C4×S3
ρ2722-2-20000-1-2-2i2i2i-2i200000011-1i-i-i-1i1    complex lifted from C4×S3
ρ2822-2-20000-1-22i-2i-2i2i200000011-1-iii-1-i1    complex lifted from C4×S3
ρ294-44-40000-20000000000002-22000000    orthogonal lifted from S3×D4
ρ304-4-440000-2000000000000-222000000    symplectic lifted from S3×Q8, Schur index 2

Smallest permutation representation of S3×C4⋊C4
On 48 points
Generators in S48
(1 16 17)(2 13 18)(3 14 19)(4 15 20)(5 29 34)(6 30 35)(7 31 36)(8 32 33)(9 25 42)(10 26 43)(11 27 44)(12 28 41)(21 37 46)(22 38 47)(23 39 48)(24 40 45)
(1 3)(2 4)(5 7)(6 8)(9 27)(10 28)(11 25)(12 26)(13 20)(14 17)(15 18)(16 19)(21 39)(22 40)(23 37)(24 38)(29 36)(30 33)(31 34)(32 35)(41 43)(42 44)(45 47)(46 48)
(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)
(1 47 43 8)(2 46 44 7)(3 45 41 6)(4 48 42 5)(9 29 15 23)(10 32 16 22)(11 31 13 21)(12 30 14 24)(17 38 26 33)(18 37 27 36)(19 40 28 35)(20 39 25 34)

G:=sub<Sym(48)| (1,16,17)(2,13,18)(3,14,19)(4,15,20)(5,29,34)(6,30,35)(7,31,36)(8,32,33)(9,25,42)(10,26,43)(11,27,44)(12,28,41)(21,37,46)(22,38,47)(23,39,48)(24,40,45), (1,3)(2,4)(5,7)(6,8)(9,27)(10,28)(11,25)(12,26)(13,20)(14,17)(15,18)(16,19)(21,39)(22,40)(23,37)(24,38)(29,36)(30,33)(31,34)(32,35)(41,43)(42,44)(45,47)(46,48), (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), (1,47,43,8)(2,46,44,7)(3,45,41,6)(4,48,42,5)(9,29,15,23)(10,32,16,22)(11,31,13,21)(12,30,14,24)(17,38,26,33)(18,37,27,36)(19,40,28,35)(20,39,25,34)>;

G:=Group( (1,16,17)(2,13,18)(3,14,19)(4,15,20)(5,29,34)(6,30,35)(7,31,36)(8,32,33)(9,25,42)(10,26,43)(11,27,44)(12,28,41)(21,37,46)(22,38,47)(23,39,48)(24,40,45), (1,3)(2,4)(5,7)(6,8)(9,27)(10,28)(11,25)(12,26)(13,20)(14,17)(15,18)(16,19)(21,39)(22,40)(23,37)(24,38)(29,36)(30,33)(31,34)(32,35)(41,43)(42,44)(45,47)(46,48), (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), (1,47,43,8)(2,46,44,7)(3,45,41,6)(4,48,42,5)(9,29,15,23)(10,32,16,22)(11,31,13,21)(12,30,14,24)(17,38,26,33)(18,37,27,36)(19,40,28,35)(20,39,25,34) );

G=PermutationGroup([(1,16,17),(2,13,18),(3,14,19),(4,15,20),(5,29,34),(6,30,35),(7,31,36),(8,32,33),(9,25,42),(10,26,43),(11,27,44),(12,28,41),(21,37,46),(22,38,47),(23,39,48),(24,40,45)], [(1,3),(2,4),(5,7),(6,8),(9,27),(10,28),(11,25),(12,26),(13,20),(14,17),(15,18),(16,19),(21,39),(22,40),(23,37),(24,38),(29,36),(30,33),(31,34),(32,35),(41,43),(42,44),(45,47),(46,48)], [(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)], [(1,47,43,8),(2,46,44,7),(3,45,41,6),(4,48,42,5),(9,29,15,23),(10,32,16,22),(11,31,13,21),(12,30,14,24),(17,38,26,33),(18,37,27,36),(19,40,28,35),(20,39,25,34)])

S3×C4⋊C4 is a maximal subgroup of
D4⋊(C4×S3)  D6.D8  D6.SD16  Q87(C4×S3)  D6.1SD16  D6.Q16  C8⋊(C4×S3)  D6.2SD16  D6.4SD16  C8⋊S3⋊C4  D6.5D8  D6.2Q16  C6.82+ 1+4  C6.2- 1+4  C6.102+ 1+4  C42.91D6  C42.94D6  C42.95D6  C4×S3×D4  C42.108D6  D1224D4  C42.113D6  C4×S3×Q8  C42.126D6  D1210Q8  C42.132D6  C6.722- 1+4  C6.732- 1+4  C6.432+ 1+4  C6.172- 1+4  D1222D4  C6.1182+ 1+4  C6.522+ 1+4  C6.202- 1+4  C6.212- 1+4  C6.822- 1+4  C6.632+ 1+4  C6.642+ 1+4  C42.148D6  D127Q8  C42.150D6  C42.151D6  C42.152D6  C42.153D6  C42.161D6  C42.162D6  C42.163D6  D1212D4  C42.174D6  D129Q8  C62.53C23  C62.70C23  D30.Q8  D30.2Q8
S3×C4⋊C4 is a maximal quotient of
C6.(C4×Q8)  Dic3⋊C42  C3⋊(C428C4)  D6⋊(C4⋊C4)  D6⋊C4⋊C4  C12⋊M4(2)  C42.30D6  (S3×C8)⋊C4  C8⋊(C4×S3)  C8.27(C4×S3)  C8⋊S3⋊C4  M4(2).25D6  (C4×Dic3)⋊8C4  Dic3⋊(C4⋊C4)  C4⋊(D6⋊C4)  D6⋊C46C4  C62.53C23  C62.70C23  D30.Q8  D30.2Q8

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

1000
0100
00012
00112
,
12000
01200
0001
0010
,
9300
3400
00120
00012
,
0100
12000
0080
0008
G:=sub<GL(4,GF(13))| [1,0,0,0,0,1,0,0,0,0,0,1,0,0,12,12],[12,0,0,0,0,12,0,0,0,0,0,1,0,0,1,0],[9,3,0,0,3,4,0,0,0,0,12,0,0,0,0,12],[0,12,0,0,1,0,0,0,0,0,8,0,0,0,0,8] >;

S3×C4⋊C4 in GAP, Magma, Sage, TeX

S_3\times C_4\rtimes C_4
% in TeX

G:=Group("S3xC4:C4");
// GroupNames label

G:=SmallGroup(96,98);
// by ID

G=gap.SmallGroup(96,98);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-3,103,188,50,2309]);
// Polycyclic

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

Export

Character table of S3×C4⋊C4 in TeX

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