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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, C3, C4, C4, C22, C22, S3, C6, C6, C2×C4, D4, D4, C23, C32, Dic3, C12, D6, C2×C6, C2×D4, C3⋊S3, C3⋊S3, C3×C6, C3×C6, C4×S3, D12, C3⋊D4, C3×D4, C22×S3, C3⋊Dic3, C3×C12, C2×C3⋊S3, C2×C3⋊S3, C2×C3⋊S3, C62, S3×D4, C4×C3⋊S3, C12⋊S3, C327D4, D4×C32, C22×C3⋊S3, D4×C3⋊S3
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C3⋊S3, C22×S3, C2×C3⋊S3, S3×D4, 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 24)(22 23)(25 28)(26 27)(29 32)(30 31)(33 36)(34 35)
(1 21 29)(2 22 30)(3 23 31)(4 24 32)(5 35 11)(6 36 12)(7 33 9)(8 34 10)(13 28 19)(14 25 20)(15 26 17)(16 27 18)
(1 7 16)(2 8 13)(3 5 14)(4 6 15)(9 18 29)(10 19 30)(11 20 31)(12 17 32)(21 33 27)(22 34 28)(23 35 25)(24 36 26)
(1 3)(2 4)(5 16)(6 13)(7 14)(8 15)(9 25)(10 26)(11 27)(12 28)(17 34)(18 35)(19 36)(20 33)(21 31)(22 32)(23 29)(24 30)

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

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

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

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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