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G = D4×C3⋊S3order 144 = 24·32

Direct product of D4 and C3⋊S3

direct product, metabelian, supersoluble, monomial, rational

Aliases: D4×C3⋊S3, C123D6, C624C22, C34(S3×D4), (C2×C6)⋊6D6, (C3×D4)⋊2S3, C12⋊S36C2, C3211(C2×D4), (C3×C12)⋊4C22, (D4×C32)⋊5C2, C327D43C2, C6.34(C22×S3), (C3×C6).33C23, C3⋊Dic36C22, C41(C2×C3⋊S3), (C4×C3⋊S3)⋊3C2, C222(C2×C3⋊S3), (C2×C3⋊S3)⋊6C22, (C22×C3⋊S3)⋊4C2, C2.6(C22×C3⋊S3), SmallGroup(144,172)

Series: Derived Chief Lower central Upper central

C1C3×C6 — D4×C3⋊S3
C1C3C32C3×C6C2×C3⋊S3C22×C3⋊S3 — D4×C3⋊S3
C32C3×C6 — D4×C3⋊S3
C1C2D4

Generators and relations for D4×C3⋊S3
 G = < a,b,c,d,e | a4=b2=c3=d3=e2=1, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece=c-1, ede=d-1 >

Subgroups: 610 in 162 conjugacy classes, 49 normal (13 characteristic)
C1, C2, C2 [×6], C3 [×4], C4, C4, C22 [×2], C22 [×7], S3 [×16], C6 [×4], C6 [×8], C2×C4, D4, D4 [×3], C23 [×2], C32, Dic3 [×4], C12 [×4], D6 [×28], C2×C6 [×8], C2×D4, C3⋊S3 [×2], C3⋊S3 [×2], C3×C6, C3×C6 [×2], C4×S3 [×4], D12 [×4], C3⋊D4 [×8], C3×D4 [×4], C22×S3 [×8], C3⋊Dic3, C3×C12, C2×C3⋊S3, C2×C3⋊S3 [×2], C2×C3⋊S3 [×4], C62 [×2], S3×D4 [×4], C4×C3⋊S3, C12⋊S3, C327D4 [×2], D4×C32, C22×C3⋊S3 [×2], D4×C3⋊S3
Quotients: C1, C2 [×7], C22 [×7], S3 [×4], D4 [×2], C23, D6 [×12], C2×D4, C3⋊S3, C22×S3 [×4], C2×C3⋊S3 [×3], S3×D4 [×4], C22×C3⋊S3, D4×C3⋊S3

Character table of D4×C3⋊S3

 class 12A2B2C2D2E2F2G3A3B3C3D4A4B6A6B6C6D6E6F6G6H6I6J6K6L12A12B12C12D
 size 112299181822222182222444444444444
ρ1111111111111111111111111111111    trivial
ρ211-11-1-11-11111-1111111-1-1-1-1111-1-1-1-1    linear of order 2
ρ311-1-1-1-11111111-11111-1-1-1-1-1-1-1-11111    linear of order 2
ρ4111-1111-11111-1-11111-11111-1-1-1-1-1-1-1    linear of order 2
ρ51111-1-1-1-111111-11111111111111111    linear of order 2
ρ611-1111-111111-1-111111-1-1-1-1111-1-1-1-1    linear of order 2
ρ711-1-111-1-11111111111-1-1-1-1-1-1-1-11111    linear of order 2
ρ8111-1-1-1-111111-111111-11111-1-1-1-1-1-1-1    linear of order 2
ρ9222-20000-1-12-1-20-1-12-11-1-1-1211-21-211    orthogonal lifted from D6
ρ1022-2-20000-1-1-12202-1-1-111-211-211-1-1-12    orthogonal lifted from D6
ρ1122220000-1-1-12202-1-1-1-1-12-1-12-1-1-1-1-12    orthogonal lifted from S3
ρ1222-220000-1-1-12-202-1-1-1-11-2112-1-1111-2    orthogonal lifted from D6
ρ13222-20000-1-1-12-202-1-1-11-12-1-1-211111-2    orthogonal lifted from D6
ρ1422220000-1-12-120-1-12-1-1-1-1-12-1-12-12-1-1    orthogonal lifted from S3
ρ15222200002-1-1-120-1-1-1222-1-1-1-1-1-1-1-12-1    orthogonal lifted from S3
ρ1622-2-200002-1-1-120-1-1-12-2-2111111-1-12-1    orthogonal lifted from D6
ρ172-200-2200222200-2-2-2-2000000000000    orthogonal lifted from D4
ρ18222-200002-1-1-1-20-1-1-12-22-1-1-111111-21    orthogonal lifted from D6
ρ1922-2200002-1-1-1-20-1-1-122-2111-1-1-111-21    orthogonal lifted from D6
ρ2022-220000-1-12-1-20-1-12-1-1111-2-1-121-211    orthogonal lifted from D6
ρ2122-2-20000-1-12-120-1-12-11111-211-2-12-1-1    orthogonal lifted from D6
ρ222-2002-200222200-2-2-2-2000000000000    orthogonal lifted from D4
ρ2322220000-12-1-120-12-1-1-1-1-12-1-12-12-1-1-1    orthogonal lifted from S3
ρ2422-2-20000-12-1-120-12-1-1111-211-212-1-1-1    orthogonal lifted from D6
ρ25222-20000-12-1-1-20-12-1-11-1-12-11-21-2111    orthogonal lifted from D6
ρ2622-220000-12-1-1-20-12-1-1-111-21-12-1-2111    orthogonal lifted from D6
ρ274-4000000-24-2-2002-422000000000000    orthogonal lifted from S3×D4
ρ284-40000004-2-2-200222-4000000000000    orthogonal lifted from S3×D4
ρ294-4000000-2-24-20022-42000000000000    orthogonal lifted from S3×D4
ρ304-4000000-2-2-2400-4222000000000000    orthogonal lifted from S3×D4

Smallest permutation representation of D4×C3⋊S3
On 36 points
Generators in S36
(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)
(1 4)(2 3)(5 8)(6 7)(9 12)(10 11)(13 14)(15 16)(17 18)(19 20)(21 22)(23 24)(25 28)(26 27)(29 32)(30 31)(33 36)(34 35)
(1 7 29)(2 8 30)(3 5 31)(4 6 32)(9 16 27)(10 13 28)(11 14 25)(12 15 26)(17 23 36)(18 24 33)(19 21 34)(20 22 35)
(1 24 16)(2 21 13)(3 22 14)(4 23 15)(5 35 25)(6 36 26)(7 33 27)(8 34 28)(9 29 18)(10 30 19)(11 31 20)(12 32 17)
(1 3)(2 4)(5 29)(6 30)(7 31)(8 32)(9 35)(10 36)(11 33)(12 34)(13 23)(14 24)(15 21)(16 22)(17 28)(18 25)(19 26)(20 27)

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

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), (1,4)(2,3)(5,8)(6,7)(9,12)(10,11)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,28)(26,27)(29,32)(30,31)(33,36)(34,35), (1,7,29)(2,8,30)(3,5,31)(4,6,32)(9,16,27)(10,13,28)(11,14,25)(12,15,26)(17,23,36)(18,24,33)(19,21,34)(20,22,35), (1,24,16)(2,21,13)(3,22,14)(4,23,15)(5,35,25)(6,36,26)(7,33,27)(8,34,28)(9,29,18)(10,30,19)(11,31,20)(12,32,17), (1,3)(2,4)(5,29)(6,30)(7,31)(8,32)(9,35)(10,36)(11,33)(12,34)(13,23)(14,24)(15,21)(16,22)(17,28)(18,25)(19,26)(20,27) );

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)], [(1,4),(2,3),(5,8),(6,7),(9,12),(10,11),(13,14),(15,16),(17,18),(19,20),(21,22),(23,24),(25,28),(26,27),(29,32),(30,31),(33,36),(34,35)], [(1,7,29),(2,8,30),(3,5,31),(4,6,32),(9,16,27),(10,13,28),(11,14,25),(12,15,26),(17,23,36),(18,24,33),(19,21,34),(20,22,35)], [(1,24,16),(2,21,13),(3,22,14),(4,23,15),(5,35,25),(6,36,26),(7,33,27),(8,34,28),(9,29,18),(10,30,19),(11,31,20),(12,32,17)], [(1,3),(2,4),(5,29),(6,30),(7,31),(8,32),(9,35),(10,36),(11,33),(12,34),(13,23),(14,24),(15,21),(16,22),(17,28),(18,25),(19,26),(20,27)])

D4×C3⋊S3 is a maximal subgroup of
C3⋊S3.5D8  D12⋊D6  Dic6⋊D6  D125D6  C248D6  C247D6  S32×D4  Dic612D6  D1213D6  C3282+ 1+4  C62.154C23  C12⋊S32  C6223D6
D4×C3⋊S3 is a maximal quotient of
C626Q8  C62.225C23  C6212D4  C62.227C23  C62.228C23  C62.229C23  C122Dic6  C62.237C23  C62.238C23  C123D12  C62.240C23  C248D6  C24.26D6  C247D6  C24.32D6  C24.40D6  C24.35D6  C24.28D6  C6213D4  C62.256C23  C6214D4  C62.258C23  C12⋊S32  C6223D6

Matrix representation of D4×C3⋊S3 in GL6(ℤ)

-100000
0-10000
00-1000
000-100
0000-1-2
000011
,
-100000
0-10000
00-1000
000-100
0000-1-2
000001
,
-110000
-100000
00-1100
00-1000
000010
000001
,
-110000
-100000
001000
000100
000010
000001
,
010000
100000
000100
001000
0000-10
00000-1

G:=sub<GL(6,Integers())| [-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,1,0,0,0,0,-2,1],[-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,-2,1],[-1,-1,0,0,0,0,1,0,0,0,0,0,0,0,-1,-1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[-1,-1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1] >;

D4×C3⋊S3 in GAP, Magma, Sage, TeX

D_4\times C_3\rtimes S_3
% in TeX

G:=Group("D4xC3:S3");
// GroupNames label

G:=SmallGroup(144,172);
// by ID

G=gap.SmallGroup(144,172);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-3,116,964,3461]);
// Polycyclic

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

Export

Character table of D4×C3⋊S3 in TeX

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