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G = C22×C3⋊S3order 72 = 23·32

Direct product of C22 and C3⋊S3

direct product, metabelian, supersoluble, monomial, A-group, rational

Aliases: C22×C3⋊S3, C62D6, C625C2, C323C23, (C2×C6)⋊5S3, C32(C22×S3), (C3×C6)⋊3C22, SmallGroup(72,49)

Series: Derived Chief Lower central Upper central

C1C32 — C22×C3⋊S3
C1C3C32C3⋊S3C2×C3⋊S3 — C22×C3⋊S3
C32 — C22×C3⋊S3
C1C22

Generators and relations for C22×C3⋊S3
 G = < a,b,c,d,e | a2=b2=c3=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece=c-1, ede=d-1 >

Subgroups: 272 in 96 conjugacy classes, 41 normal (5 characteristic)
C1, C2 [×3], C2 [×4], C3 [×4], C22, C22 [×6], S3 [×16], C6 [×12], C23, C32, D6 [×24], C2×C6 [×4], C3⋊S3 [×4], C3×C6 [×3], C22×S3 [×4], C2×C3⋊S3 [×6], C62, C22×C3⋊S3
Quotients: C1, C2 [×7], C22 [×7], S3 [×4], C23, D6 [×12], C3⋊S3, C22×S3 [×4], C2×C3⋊S3 [×3], C22×C3⋊S3

Character table of C22×C3⋊S3

 class 12A2B2C2D2E2F2G3A3B3C3D6A6B6C6D6E6F6G6H6I6J6K6L
 size 111199992222222222222222
ρ1111111111111111111111111    trivial
ρ211-1-1-11-1111111-1-11-1-1-1-1-1-111    linear of order 2
ρ31-11-1-1-1111111-111-1-1-1-1-111-1-1    linear of order 2
ρ41-1-111-1-111111-1-1-1-11111-1-1-1-1    linear of order 2
ρ51-1-11-111-11111-1-1-1-11111-1-1-1-1    linear of order 2
ρ61-11-111-1-11111-111-1-1-1-1-111-1-1    linear of order 2
ρ711-1-11-11-111111-1-11-1-1-1-1-1-111    linear of order 2
ρ81111-1-1-1-11111111111111111    linear of order 2
ρ92-22-20000-12-1-11-1-111-211-121-2    orthogonal lifted from D6
ρ1022220000-12-1-1-1-1-1-1-12-1-1-12-12    orthogonal lifted from S3
ρ112-2-220000-1-12-1-2-211-1-12-11111    orthogonal lifted from D6
ρ1222-2-20000-1-12-12-21-111-2111-1-1    orthogonal lifted from D6
ρ1322-2-20000-1-1-12-11-22111-211-1-1    orthogonal lifted from D6
ρ142-2-220000-1-1-1211-2-2-1-1-121111    orthogonal lifted from D6
ρ152-22-200002-1-1-11-1-11-21112-1-21    orthogonal lifted from D6
ρ162-2-2200002-1-1-111112-1-1-1-21-21    orthogonal lifted from D6
ρ1722220000-1-12-122-1-1-1-12-1-1-1-1-1    orthogonal lifted from S3
ρ182-22-20000-1-12-1-22-1111-21-1-111    orthogonal lifted from D6
ρ192-22-20000-1-1-121-12-2111-2-1-111    orthogonal lifted from D6
ρ2022220000-1-1-12-1-122-1-1-12-1-1-1-1    orthogonal lifted from S3
ρ2122-2-20000-12-1-1-111-11-2111-2-12    orthogonal lifted from D6
ρ222-2-220000-12-1-11111-12-1-11-21-2    orthogonal lifted from D6
ρ23222200002-1-1-1-1-1-1-12-1-1-12-12-1    orthogonal lifted from S3
ρ2422-2-200002-1-1-1-111-1-2111-212-1    orthogonal lifted from D6

Smallest permutation representation of C22×C3⋊S3
On 36 points
Generators in S36
(1 29)(2 30)(3 28)(4 31)(5 32)(6 33)(7 34)(8 35)(9 36)(10 19)(11 20)(12 21)(13 22)(14 23)(15 24)(16 25)(17 26)(18 27)
(1 11)(2 12)(3 10)(4 13)(5 14)(6 15)(7 16)(8 17)(9 18)(19 28)(20 29)(21 30)(22 31)(23 32)(24 33)(25 34)(26 35)(27 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 5 8)(2 6 9)(3 4 7)(10 13 16)(11 14 17)(12 15 18)(19 22 25)(20 23 26)(21 24 27)(28 31 34)(29 32 35)(30 33 36)
(2 3)(4 9)(5 8)(6 7)(10 12)(13 18)(14 17)(15 16)(19 21)(22 27)(23 26)(24 25)(28 30)(31 36)(32 35)(33 34)

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

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

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

C22×C3⋊S3 is a maximal subgroup of   C6.D12  C6.11D12  C62⋊C4  Dic3⋊D6  C22×S32  C62⋊C6
C22×C3⋊S3 is a maximal quotient of   C12.59D6  C12.D6  C12.26D6

Matrix representation of C22×C3⋊S3 in GL4(ℤ) generated by

1000
0100
00-10
000-1
,
-1000
0-100
0010
0001
,
1000
0100
00-11
00-10
,
0-100
1-100
00-11
00-10
,
0100
1000
0001
0010
G:=sub<GL(4,Integers())| [1,0,0,0,0,1,0,0,0,0,-1,0,0,0,0,-1],[-1,0,0,0,0,-1,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,-1,-1,0,0,1,0],[0,1,0,0,-1,-1,0,0,0,0,-1,-1,0,0,1,0],[0,1,0,0,1,0,0,0,0,0,0,1,0,0,1,0] >;

C22×C3⋊S3 in GAP, Magma, Sage, TeX

C_2^2\times C_3\rtimes S_3
% in TeX

G:=Group("C2^2xC3:S3");
// GroupNames label

G:=SmallGroup(72,49);
// by ID

G=gap.SmallGroup(72,49);
# by ID

G:=PCGroup([5,-2,-2,-2,-3,-3,323,1204]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^3=d^3=e^2=1,a*b=b*a,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 C22×C3⋊S3 in TeX

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