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G = D4xC3:S3order 144 = 24·32

Direct product of D4 and C3:S3

direct product, metabelian, supersoluble, monomial, rational

Aliases: D4xC3:S3, C12:3D6, C62:4C22, C3:4(S3xD4), (C2xC6):6D6, (C3xD4):2S3, C12:S3:6C2, C32:11(C2xD4), (C3xC12):4C22, (D4xC32):5C2, C32:7D4:3C2, C6.34(C22xS3), (C3xC6).33C23, C3:Dic3:6C22, C4:1(C2xC3:S3), (C4xC3:S3):3C2, C22:2(C2xC3:S3), (C2xC3:S3):6C22, (C22xC3:S3):4C2, C2.6(C22xC3:S3), SmallGroup(144,172)

Series: Derived Chief Lower central Upper central

C1C3xC6 — D4xC3:S3
C1C3C32C3xC6C2xC3:S3C22xC3:S3 — D4xC3:S3
C32C3xC6 — D4xC3:S3
C1C2D4

Generators and relations for D4xC3: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, C2xC4, D4, D4, C23, C32, Dic3, C12, D6, C2xC6, C2xD4, C3:S3, C3:S3, C3xC6, C3xC6, C4xS3, D12, C3:D4, C3xD4, C22xS3, C3:Dic3, C3xC12, C2xC3:S3, C2xC3:S3, C2xC3:S3, C62, S3xD4, C4xC3:S3, C12:S3, C32:7D4, D4xC32, C22xC3:S3, D4xC3:S3
Quotients: C1, C2, C22, S3, D4, C23, D6, C2xD4, C3:S3, C22xS3, C2xC3:S3, S3xD4, C22xC3:S3, D4xC3:S3

Character table of D4xC3: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 S3xD4
ρ284-40000004-2-2-200222-4000000000000    orthogonal lifted from S3xD4
ρ294-4000000-2-24-20022-42000000000000    orthogonal lifted from S3xD4
ρ304-4000000-2-2-2400-4222000000000000    orthogonal lifted from S3xD4

Smallest permutation representation of D4xC3: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)]])

D4xC3:S3 is a maximal subgroup of
C3:S3.5D8  D12:D6  Dic6:D6  D12:5D6  C24:8D6  C24:7D6  S32xD4  Dic6:12D6  D12:13D6  C32:82+ 1+4  C62.154C23  C12:S32  C62:23D6
D4xC3:S3 is a maximal quotient of
C62:6Q8  C62.225C23  C62:12D4  C62.227C23  C62.228C23  C62.229C23  C12:2Dic6  C62.237C23  C62.238C23  C12:3D12  C62.240C23  C24:8D6  C24.26D6  C24:7D6  C24.32D6  C24.40D6  C24.35D6  C24.28D6  C62:13D4  C62.256C23  C62:14D4  C62.258C23  C12:S32  C62:23D6

Matrix representation of D4xC3: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] >;

D4xC3: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 D4xC3:S3 in TeX

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