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## G = C2×He3⋊D4order 432 = 24·33

### Direct product of C2 and He3⋊D4

Aliases: C2×He3⋊D4, He3⋊(C2×D4), (C2×He3)⋊D4, C6.6S3≀C2, He3⋊C2⋊D4, He3⋊C4⋊C22, C32⋊D6⋊C22, He3⋊C2.2C23, C3.(C2×S3≀C2), (C2×He3⋊C4)⋊3C2, (C2×C32⋊D6)⋊5C2, (C2×He3⋊C2).6C22, SmallGroup(432,530)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — He3 — He3⋊C2 — C2×He3⋊D4
 Chief series C1 — C3 — He3 — He3⋊C2 — C32⋊D6 — He3⋊D4 — C2×He3⋊D4
 Lower central He3 — He3⋊C2 — C2×He3⋊D4
 Upper central C1 — C2

Generators and relations for C2×He3⋊D4
G = < a,b,c,d,e,f | a2=b3=c3=d3=e4=f2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, fbf=bc=cb, dbd-1=ede-1=bc-1, ebe-1=cd-1, cd=dc, ce=ec, fcf=c-1, fdf=d-1, fef=e-1 >

Subgroups: 1439 in 145 conjugacy classes, 23 normal (11 characteristic)
C1, C2, C2, C3, C3, C4, C22, S3, C6, C6, C2×C4, D4, C23, C32, C12, D6, C2×C6, C2×D4, C3×S3, C3⋊S3, C3×C6, D12, C2×C12, C22×S3, He3, S32, S3×C6, C2×C3⋊S3, C2×D12, C32⋊C6, He3⋊C2, C2×He3, C2×S32, He3⋊C4, C32⋊D6, C32⋊D6, C2×C32⋊C6, C2×He3⋊C2, He3⋊D4, C2×He3⋊C4, C2×C32⋊D6, C2×He3⋊D4
Quotients: C1, C2, C22, D4, C23, C2×D4, S3≀C2, C2×S3≀C2, He3⋊D4, C2×He3⋊D4

Character table of C2×He3⋊D4

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 3C 4A 4B 6A 6B 6C 6D 6E 6F 6G 6H 6I 12A 12B 12C 12D size 1 1 9 9 18 18 18 18 2 12 12 18 18 2 12 12 18 18 36 36 36 36 18 18 18 18 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 1 1 -1 -1 -1 -1 1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 1 1 1 1 linear of order 2 ρ3 1 1 1 1 -1 1 1 -1 1 1 1 -1 -1 1 1 1 1 1 1 -1 -1 1 -1 -1 -1 -1 linear of order 2 ρ4 1 1 1 1 1 -1 -1 1 1 1 1 -1 -1 1 1 1 1 1 -1 1 1 -1 -1 -1 -1 -1 linear of order 2 ρ5 1 -1 1 -1 -1 -1 1 1 1 1 1 1 -1 -1 -1 -1 -1 1 -1 -1 1 1 -1 -1 1 1 linear of order 2 ρ6 1 -1 1 -1 1 1 -1 -1 1 1 1 1 -1 -1 -1 -1 -1 1 1 1 -1 -1 -1 -1 1 1 linear of order 2 ρ7 1 -1 1 -1 1 -1 1 -1 1 1 1 -1 1 -1 -1 -1 -1 1 -1 1 -1 1 1 1 -1 -1 linear of order 2 ρ8 1 -1 1 -1 -1 1 -1 1 1 1 1 -1 1 -1 -1 -1 -1 1 1 -1 1 -1 1 1 -1 -1 linear of order 2 ρ9 2 -2 -2 2 0 0 0 0 2 2 2 0 0 -2 -2 -2 2 -2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ10 2 2 -2 -2 0 0 0 0 2 2 2 0 0 2 2 2 -2 -2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ11 4 -4 0 0 -2 0 0 2 4 -2 1 0 0 -4 2 -1 0 0 0 1 -1 0 0 0 0 0 orthogonal lifted from C2×S3≀C2 ρ12 4 4 0 0 2 0 0 2 4 -2 1 0 0 4 -2 1 0 0 0 -1 -1 0 0 0 0 0 orthogonal lifted from S3≀C2 ρ13 4 4 0 0 0 2 2 0 4 1 -2 0 0 4 1 -2 0 0 -1 0 0 -1 0 0 0 0 orthogonal lifted from S3≀C2 ρ14 4 -4 0 0 0 -2 2 0 4 1 -2 0 0 -4 -1 2 0 0 1 0 0 -1 0 0 0 0 orthogonal lifted from C2×S3≀C2 ρ15 4 -4 0 0 0 2 -2 0 4 1 -2 0 0 -4 -1 2 0 0 -1 0 0 1 0 0 0 0 orthogonal lifted from C2×S3≀C2 ρ16 4 -4 0 0 2 0 0 -2 4 -2 1 0 0 -4 2 -1 0 0 0 -1 1 0 0 0 0 0 orthogonal lifted from C2×S3≀C2 ρ17 4 4 0 0 0 -2 -2 0 4 1 -2 0 0 4 1 -2 0 0 1 0 0 1 0 0 0 0 orthogonal lifted from S3≀C2 ρ18 4 4 0 0 -2 0 0 -2 4 -2 1 0 0 4 -2 1 0 0 0 1 1 0 0 0 0 0 orthogonal lifted from S3≀C2 ρ19 6 6 -2 -2 0 0 0 0 -3 0 0 2 2 -3 0 0 1 1 0 0 0 0 -1 -1 -1 -1 orthogonal lifted from He3⋊D4 ρ20 6 -6 -2 2 0 0 0 0 -3 0 0 2 -2 3 0 0 -1 1 0 0 0 0 1 1 -1 -1 orthogonal faithful ρ21 6 -6 -2 2 0 0 0 0 -3 0 0 -2 2 3 0 0 -1 1 0 0 0 0 -1 -1 1 1 orthogonal faithful ρ22 6 6 -2 -2 0 0 0 0 -3 0 0 -2 -2 -3 0 0 1 1 0 0 0 0 1 1 1 1 orthogonal lifted from He3⋊D4 ρ23 6 6 2 2 0 0 0 0 -3 0 0 0 0 -3 0 0 -1 -1 0 0 0 0 -√3 √3 √3 -√3 orthogonal lifted from He3⋊D4 ρ24 6 6 2 2 0 0 0 0 -3 0 0 0 0 -3 0 0 -1 -1 0 0 0 0 √3 -√3 -√3 √3 orthogonal lifted from He3⋊D4 ρ25 6 -6 2 -2 0 0 0 0 -3 0 0 0 0 3 0 0 1 -1 0 0 0 0 √3 -√3 √3 -√3 orthogonal faithful ρ26 6 -6 2 -2 0 0 0 0 -3 0 0 0 0 3 0 0 1 -1 0 0 0 0 -√3 √3 -√3 √3 orthogonal faithful

Smallest permutation representation of C2×He3⋊D4
On 36 points
Generators in S36
(1 4)(2 3)(5 7)(6 8)(9 12)(10 11)(13 29)(14 30)(15 31)(16 32)(17 25)(18 26)(19 27)(20 28)(21 35)(22 36)(23 33)(24 34)
(1 7 9)(2 14 18)(3 30 26)(4 5 12)(6 28 33)(8 20 23)(10 21 16)(11 35 32)(13 24 19)(27 29 34)
(1 7 9)(2 8 10)(3 6 11)(4 5 12)(13 19 24)(14 20 21)(15 17 22)(16 18 23)(25 36 31)(26 33 32)(27 34 29)(28 35 30)
(1 22 24)(4 36 34)(5 31 29)(7 15 13)(9 17 19)(12 25 27)(14 20 21)(16 23 18)(26 32 33)(28 35 30)
(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 9)(2 10)(3 11)(4 12)(13 15)(17 24)(18 23)(19 22)(20 21)(25 34)(26 33)(27 36)(28 35)(29 31)

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

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

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

Matrix representation of C2×He3⋊D4 in GL6(ℤ)

 -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 0 -1
,
 0 0 -1 1 0 0 0 0 -1 0 0 0 -1 -1 -1 -1 -1 -2 0 0 0 0 1 -1 1 0 1 0 0 1 0 1 1 0 0 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 1 0 1 0 0 1 0 -1 0 -1 -1 -1
,
 1 1 1 1 2 1 1 1 1 1 1 2 -1 1 0 0 0 0 -1 0 0 0 0 0 0 -1 -1 -1 -1 -1 0 -1 0 -1 -1 -1
,
 0 -1 0 0 -1 0 0 -1 -1 0 -1 -1 -1 0 -1 0 -1 0 -1 -1 -1 -1 -1 -1 1 1 1 0 1 1 0 1 1 0 1 0
,
 0 1 1 0 1 1 0 1 0 0 1 0 1 0 0 1 1 1 0 0 0 1 1 0 0 0 0 0 -1 0 0 -1 0 -1 -1 -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,-1,0,0,0,0,0,0,-1],[0,0,-1,0,1,0,0,0,-1,0,0,1,-1,-1,-1,0,1,1,1,0,-1,0,0,0,0,0,-1,1,0,0,0,0,-2,-1,1,1],[-1,-1,0,0,1,0,1,0,0,0,0,-1,0,0,-1,-1,1,0,0,0,1,0,0,-1,0,0,0,0,0,-1,0,0,0,0,1,-1],[1,1,-1,-1,0,0,1,1,1,0,-1,-1,1,1,0,0,-1,0,1,1,0,0,-1,-1,2,1,0,0,-1,-1,1,2,0,0,-1,-1],[0,0,-1,-1,1,0,-1,-1,0,-1,1,1,0,-1,-1,-1,1,1,0,0,0,-1,0,0,-1,-1,-1,-1,1,1,0,-1,0,-1,1,0],[0,0,1,0,0,0,1,1,0,0,0,-1,1,0,0,0,0,0,0,0,1,1,0,-1,1,1,1,1,-1,-1,1,0,1,0,0,-1] >;

C2×He3⋊D4 in GAP, Magma, Sage, TeX

C_2\times {\rm He}_3\rtimes D_4
% in TeX

G:=Group("C2xHe3:D4");
// GroupNames label

G:=SmallGroup(432,530);
// by ID

G=gap.SmallGroup(432,530);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,3,-3,141,1124,851,165,348,530,537,14118,7069]);
// Polycyclic

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

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