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G = S3×C3⋊D4order 144 = 24·32

Direct product of S3 and C3⋊D4

direct product, metabelian, supersoluble, monomial

Aliases: S3×C3⋊D4, D66D6, Dic31D6, C621C22, C222S32, C35(S3×D4), (C2×C6)⋊4D6, (C3×S3)⋊2D4, C326(C2×D4), C3⋊Dic3⋊C22, C3⋊D125C2, D6⋊S35C2, (C22×S3)⋊4S3, (S3×C6)⋊3C22, (S3×Dic3)⋊3C2, C327D42C2, C6.17(C22×S3), (C3×C6).17C23, (C3×Dic3)⋊1C22, (C2×S32)⋊3C2, (S3×C2×C6)⋊3C2, C2.17(C2×S32), C32(C2×C3⋊D4), (C3×C3⋊D4)⋊3C2, (C2×C3⋊S3)⋊2C22, SmallGroup(144,153)

Series: Derived Chief Lower central Upper central

C1C3×C6 — S3×C3⋊D4
C1C3C32C3×C6S3×C6C2×S32 — S3×C3⋊D4
C32C3×C6 — S3×C3⋊D4
C1C2C22

Generators and relations for S3×C3⋊D4
 G = < a,b,c,d,e | a3=b2=c3=d4=e2=1, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd-1=ece=c-1, ede=d-1 >

Subgroups: 412 in 116 conjugacy classes, 36 normal (32 characteristic)
C1, C2, C2 [×6], C3 [×2], C3, C4 [×2], C22, C22 [×8], S3 [×2], S3 [×5], C6 [×2], C6 [×8], C2×C4, D4 [×4], C23 [×2], C32, Dic3, Dic3 [×3], C12, D6 [×3], D6 [×9], C2×C6 [×2], C2×C6 [×6], C2×D4, C3×S3 [×2], C3×S3 [×2], C3⋊S3, C3×C6, C3×C6, C4×S3, D12, C2×Dic3, C3⋊D4, C3⋊D4 [×6], C3×D4, C22×S3, C22×S3 [×2], C22×C6, C3×Dic3, C3⋊Dic3, S32 [×2], S3×C6 [×3], S3×C6 [×2], C2×C3⋊S3, C62, S3×D4, C2×C3⋊D4, S3×Dic3, D6⋊S3, C3⋊D12, C3×C3⋊D4, C327D4, C2×S32, S3×C2×C6, S3×C3⋊D4
Quotients: C1, C2 [×7], C22 [×7], S3 [×2], D4 [×2], C23, D6 [×6], C2×D4, C3⋊D4 [×2], C22×S3 [×2], S32, S3×D4, C2×C3⋊D4, C2×S32, S3×C3⋊D4

Character table of S3×C3⋊D4

 class 12A2B2C2D2E2F2G3A3B3C4A4B6A6B6C6D6E6F6G6H6I6J6K6L6M12
 size 1123366182246182222444466661212
ρ1111111111111111111111111111    trivial
ρ211-1-1-111-1111-11-111-1-11-1-1-111-11-1    linear of order 2
ρ311111-11-1111-1-1111111111111-1-1    linear of order 2
ρ411-1-1-1-1111111-1-111-1-11-1-1-111-1-11    linear of order 2
ρ511-111-1-1-111111-111-1-11-1-11-1-11-11    linear of order 2
ρ6111-1-1-1-11111-1111111111-1-1-1-1-1-1    linear of order 2
ρ711-1111-11111-1-1-111-1-11-1-11-1-111-1    linear of order 2
ρ8111-1-11-1-11111-111111111-1-1-1-111    linear of order 2
ρ9222-2-20-20-12-100-1-12-12-1-1-1111100    orthogonal lifted from D6
ρ102-20-22000222000-2-200-200-200200    orthogonal lifted from D4
ρ1122-2-2-2020-12-1001-121-2-1111-1-1100    orthogonal lifted from D6
ρ12222002002-1-12022-12-1-1-1-10000-1-1    orthogonal lifted from S3
ρ1322200-2002-1-1-2022-12-1-1-1-1000011    orthogonal lifted from D6
ρ142-202-2000222000-2-200-200200-200    orthogonal lifted from D4
ρ1522-2220-20-12-1001-121-2-111-111-100    orthogonal lifted from D6
ρ1622222020-12-100-1-12-12-1-1-1-1-1-1-100    orthogonal lifted from S3
ρ1722-200-2002-1-120-22-1-21-11100001-1    orthogonal lifted from D6
ρ1822-2002002-1-1-20-22-1-21-1110000-11    orthogonal lifted from D6
ρ192-202-2000-12-100--31-2-301-3--3-1--3-3100    complex lifted from C3⋊D4
ρ202-202-2000-12-100-31-2--301--3-3-1-3--3100    complex lifted from C3⋊D4
ρ212-20-22000-12-100--31-2-301-3--31-3--3-100    complex lifted from C3⋊D4
ρ222-20-22000-12-100-31-2--301--3-31--3-3-100    complex lifted from C3⋊D4
ρ2344400000-2-2100-2-2-2-2-2111000000    orthogonal lifted from S32
ρ244-40000004-2-2000-4200200000000    orthogonal lifted from S3×D4
ρ2544-400000-2-21002-2-2221-1-1000000    orthogonal lifted from C2×S32
ρ264-4000000-2-21002-322-2-30-1-3--3000000    complex faithful
ρ274-4000000-2-2100-2-3222-30-1--3-3000000    complex faithful

Permutation representations of S3×C3⋊D4
On 24 points - transitive group 24T204
Generators in S24
(1 16 17)(2 13 18)(3 14 19)(4 15 20)(5 23 10)(6 24 11)(7 21 12)(8 22 9)
(1 24)(2 21)(3 22)(4 23)(5 15)(6 16)(7 13)(8 14)(9 19)(10 20)(11 17)(12 18)
(1 17 16)(2 13 18)(3 19 14)(4 15 20)(5 10 23)(6 24 11)(7 12 21)(8 22 9)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)
(1 4)(2 3)(5 6)(7 8)(9 12)(10 11)(13 14)(15 16)(17 20)(18 19)(21 22)(23 24)

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

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

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

G:=TransitiveGroup(24,204);

On 24 points - transitive group 24T226
Generators in S24
(1 20 21)(2 17 22)(3 18 23)(4 19 24)(5 10 14)(6 11 15)(7 12 16)(8 9 13)
(1 6)(2 7)(3 8)(4 5)(9 23)(10 24)(11 21)(12 22)(13 18)(14 19)(15 20)(16 17)
(1 21 20)(2 17 22)(3 23 18)(4 19 24)(5 14 10)(6 11 15)(7 16 12)(8 9 13)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)
(1 8)(2 7)(3 6)(4 5)(9 20)(10 19)(11 18)(12 17)(13 21)(14 24)(15 23)(16 22)

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

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

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

G:=TransitiveGroup(24,226);

S3×C3⋊D4 is a maximal subgroup of
D1224D6  S32×D4  D1212D6  D1213D6  C32⋊2+ 1+4  C62⋊D6  C622D6  D64S32  D6⋊S32  C6224D6
S3×C3⋊D4 is a maximal quotient of
C62.9C23  C62.20C23  D6⋊Dic6  C62.49C23  C62.54C23  C62.55C23  Dic3⋊D12  D61Dic6  C62.58C23  C62.74C23  C62.75C23  D6⋊D12  C62.77C23  D64D12  Dic63D6  Dic6.19D6  D129D6  D12.22D6  D12.7D6  Dic6.20D6  D126D6  D12.11D6  D12.24D6  D12.12D6  Dic6.22D6  D12.13D6  C62.94C23  C62.100C23  C62.101C23  C623Q8  C62.111C23  C62.112C23  C62.113C23  C624D4  C625D4  C626D4  C62.121C23  C62.125C23  C62⋊D6  D64S32  D6⋊S32  C6224D6

Matrix representation of S3×C3⋊D4 in GL4(𝔽7) generated by

2656
4313
1125
1635
,
3263
6045
6652
0006
,
0046
2310
6122
4450
,
1240
5062
4336
3323
,
3350
6046
6132
0001
G:=sub<GL(4,GF(7))| [2,4,1,1,6,3,1,6,5,1,2,3,6,3,5,5],[3,6,6,0,2,0,6,0,6,4,5,0,3,5,2,6],[0,2,6,4,0,3,1,4,4,1,2,5,6,0,2,0],[1,5,4,3,2,0,3,3,4,6,3,2,0,2,6,3],[3,6,6,0,3,0,1,0,5,4,3,0,0,6,2,1] >;

S3×C3⋊D4 in GAP, Magma, Sage, TeX

S_3\times C_3\rtimes D_4
% in TeX

G:=Group("S3xC3:D4");
// GroupNames label

G:=SmallGroup(144,153);
// by ID

G=gap.SmallGroup(144,153);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-3,116,490,3461]);
// Polycyclic

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

Export

Character table of S3×C3⋊D4 in TeX

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