Copied to
clipboard

G = C2×He3⋊D4order 432 = 24·33

Direct product of C2 and He3⋊D4

direct product, non-abelian, soluble

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

C1C3He3He3⋊C2 — C2×He3⋊D4
C1C3He3He3⋊C2C32⋊D6He3⋊D4 — C2×He3⋊D4
He3He3⋊C2 — C2×He3⋊D4
C1C2

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 [×6], C3, C3 [×2], C4 [×2], C22 [×9], S3 [×12], C6, C6 [×8], C2×C4, D4 [×4], C23 [×2], C32 [×2], C12 [×2], D6 [×20], C2×C6 [×3], C2×D4, C3×S3 [×8], C3⋊S3 [×4], C3×C6 [×2], D12 [×4], C2×C12, C22×S3 [×4], He3, S32 [×8], S3×C6 [×4], C2×C3⋊S3 [×2], C2×D12, C32⋊C6 [×4], He3⋊C2 [×2], C2×He3, C2×S32 [×2], He3⋊C4 [×2], C32⋊D6 [×4], C32⋊D6 [×2], C2×C32⋊C6 [×2], C2×He3⋊C2, He3⋊D4 [×4], C2×He3⋊C4, C2×C32⋊D6 [×2], C2×He3⋊D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, C2×D4, S3≀C2, C2×S3≀C2, He3⋊D4, C2×He3⋊D4

Character table of C2×He3⋊D4

 class 12A2B2C2D2E2F2G3A3B3C4A4B6A6B6C6D6E6F6G6H6I12A12B12C12D
 size 1199181818182121218182121218183636363618181818
ρ111111111111111111111111111    trivial
ρ21111-1-1-1-11111111111-1-1-1-11111    linear of order 2
ρ31111-111-1111-1-1111111-1-11-1-1-1-1    linear of order 2
ρ411111-1-11111-1-111111-111-1-1-1-1-1    linear of order 2
ρ51-11-1-1-1111111-1-1-1-1-11-1-111-1-111    linear of order 2
ρ61-11-111-1-11111-1-1-1-1-1111-1-1-1-111    linear of order 2
ρ71-11-11-11-1111-11-1-1-1-11-11-1111-1-1    linear of order 2
ρ81-11-1-11-11111-11-1-1-1-111-11-111-1-1    linear of order 2
ρ92-2-22000022200-2-2-22-200000000    orthogonal lifted from D4
ρ1022-2-2000022200222-2-200000000    orthogonal lifted from D4
ρ114-400-20024-2100-42-10001-100000    orthogonal lifted from C2×S3≀C2
ρ12440020024-21004-21000-1-100000    orthogonal lifted from S3≀C2
ρ134400022041-20041-200-100-10000    orthogonal lifted from S3≀C2
ρ144-4000-22041-200-4-1200100-10000    orthogonal lifted from C2×S3≀C2
ρ154-40002-2041-200-4-1200-10010000    orthogonal lifted from C2×S3≀C2
ρ164-400200-24-2100-42-1000-1100000    orthogonal lifted from C2×S3≀C2
ρ1744000-2-2041-20041-20010010000    orthogonal lifted from S3≀C2
ρ184400-200-24-21004-210001100000    orthogonal lifted from S3≀C2
ρ1966-2-20000-30022-300110000-1-1-1-1    orthogonal lifted from He3⋊D4
ρ206-6-220000-3002-2300-11000011-1-1    orthogonal faithful
ρ216-6-220000-300-22300-110000-1-111    orthogonal faithful
ρ2266-2-20000-300-2-2-3001100001111    orthogonal lifted from He3⋊D4
ρ2366220000-30000-300-1-10000-333-3    orthogonal lifted from He3⋊D4
ρ2466220000-30000-300-1-100003-3-33    orthogonal lifted from He3⋊D4
ρ256-62-20000-300003001-100003-33-3    orthogonal faithful
ρ266-62-20000-300003001-10000-33-33    orthogonal faithful

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

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

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

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

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

-100000
0-10000
00-1000
000-100
0000-10
00000-1
,
00-1100
00-1000
-1-1-1-1-1-2
00001-1
101001
011001
,
-110000
-100000
00-1100
00-1000
101001
0-10-1-1-1
,
111121
111112
-110000
-100000
0-1-1-1-1-1
0-10-1-1-1
,
0-100-10
0-1-10-1-1
-10-10-10
-1-1-1-1-1-1
111011
011010
,
011011
010010
100111
000110
0000-10
0-10-1-1-1

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

Export

Character table of C2×He3⋊D4 in TeX

׿
×
𝔽