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G = C22×He3⋊C2order 216 = 23·33

Direct product of C22 and He3⋊C2

direct product, non-abelian, supersoluble, monomial

Aliases: C22×He3⋊C2, C627S3, He33C23, (C3×C6)⋊3D6, (C2×He3)⋊3C22, (C22×He3)⋊5C2, C323(C22×S3), C6.21(C2×C3⋊S3), C3.2(C22×C3⋊S3), (C2×C6).10(C3⋊S3), SmallGroup(216,113)

Series: Derived Chief Lower central Upper central

C1C3He3 — C22×He3⋊C2
C1C3C32He3He3⋊C2C2×He3⋊C2 — C22×He3⋊C2
He3 — C22×He3⋊C2
C1C2×C6

Generators and relations for C22×He3⋊C2
 G = < a,b,c,d,e,f | a2=b2=c3=d3=e3=f2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ece-1=cd-1, fcf=c-1, de=ed, df=fd, fef=e-1 >

Subgroups: 568 in 176 conjugacy classes, 46 normal (7 characteristic)
C1, C2 [×3], C2 [×4], C3, C3 [×4], C22, C22 [×6], S3 [×16], C6 [×3], C6 [×16], C23, C32 [×4], D6 [×24], C2×C6, C2×C6 [×10], C3×S3 [×16], C3×C6 [×12], C22×S3 [×4], C22×C6, He3, S3×C6 [×24], C62 [×4], He3⋊C2 [×4], C2×He3 [×3], S3×C2×C6 [×4], C2×He3⋊C2 [×6], C22×He3, C22×He3⋊C2
Quotients: C1, C2 [×7], C22 [×7], S3 [×4], C23, D6 [×12], C3⋊S3, C22×S3 [×4], C2×C3⋊S3 [×3], He3⋊C2, C22×C3⋊S3, C2×He3⋊C2 [×3], C22×He3⋊C2

Smallest permutation representation of C22×He3⋊C2
On 36 points
Generators in S36
(1 34)(2 35)(3 36)(4 12)(5 10)(6 11)(7 15)(8 13)(9 14)(16 25)(17 26)(18 27)(19 28)(20 29)(21 30)(22 31)(23 32)(24 33)
(1 16)(2 17)(3 18)(4 30)(5 28)(6 29)(7 33)(8 31)(9 32)(10 19)(11 20)(12 21)(13 22)(14 23)(15 24)(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 14 10)(2 15 11)(3 13 12)(4 36 8)(5 34 9)(6 35 7)(16 23 19)(17 24 20)(18 22 21)(25 32 28)(26 33 29)(27 31 30)
(2 15 11)(3 12 13)(4 8 36)(6 35 7)(17 24 20)(18 21 22)(26 33 29)(27 30 31)
(1 34)(2 36)(3 35)(4 11)(5 10)(6 12)(7 13)(8 15)(9 14)(16 25)(17 27)(18 26)(19 28)(20 30)(21 29)(22 33)(23 32)(24 31)

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

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

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

C22×He3⋊C2 is a maximal subgroup of   C62.5D6  C62.31D6  C22⋊(He3⋊C4)  C622D6
C22×He3⋊C2 is a maximal quotient of   C62.47D6  C62.16D6  He35D4⋊C2

40 conjugacy classes

class 1 2A2B2C2D2E2F2G3A3B3C3D3E3F6A···6F6G···6R6S···6Z
order122222223333336···66···66···6
size111199991166661···16···69···9

40 irreducible representations

dim1112233
type+++++
imageC1C2C2S3D6He3⋊C2C2×He3⋊C2
kernelC22×He3⋊C2C2×He3⋊C2C22×He3C62C3×C6C22C2
# reps161412412

Matrix representation of C22×He3⋊C2 in GL5(𝔽7)

10000
01000
00600
00060
00006
,
60000
06000
00100
00010
00001
,
01000
66000
00010
00661
00001
,
10000
01000
00200
00020
00002
,
66000
10000
00100
00040
00622
,
10000
66000
00600
00116
00006

G:=sub<GL(5,GF(7))| [1,0,0,0,0,0,1,0,0,0,0,0,6,0,0,0,0,0,6,0,0,0,0,0,6],[6,0,0,0,0,0,6,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[0,6,0,0,0,1,6,0,0,0,0,0,0,6,0,0,0,1,6,0,0,0,0,1,1],[1,0,0,0,0,0,1,0,0,0,0,0,2,0,0,0,0,0,2,0,0,0,0,0,2],[6,1,0,0,0,6,0,0,0,0,0,0,1,0,6,0,0,0,4,2,0,0,0,0,2],[1,6,0,0,0,0,6,0,0,0,0,0,6,1,0,0,0,0,1,0,0,0,0,6,6] >;

C22×He3⋊C2 in GAP, Magma, Sage, TeX

C_2^2\times {\rm He}_3\rtimes C_2
% in TeX

G:=Group("C2^2xHe3:C2");
// GroupNames label

G:=SmallGroup(216,113);
// by ID

G=gap.SmallGroup(216,113);
# by ID

G:=PCGroup([6,-2,-2,-2,-3,-3,-3,387,1444,382]);
// Polycyclic

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

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