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G = C22×C32⋊C6  order 216 = 23·33

Direct product of C22 and C32⋊C6

direct product, metabelian, supersoluble, monomial

Aliases: C22×C32⋊C6, C62⋊4C6, C62⋊6S3, He3⋊2C23, (C3×C6)⋊2D6, C6.22(S3×C6), C32⋊(C22×C6), (C2×He3)⋊2C22, (C22×He3)⋊4C2, C32⋊2(C22×S3), C3⋊S3⋊(C2×C6), (C3×C6)⋊(C2×C6), C3.2(S3×C2×C6), (C2×C3⋊S3)⋊3C6, (C22×C3⋊S3)⋊2C3, (C2×C6).17(C3×S3), SmallGroup(216,110)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C32 — C22×C32⋊C6
Chief series
C1 — C3 — C32 — He3 — C32⋊C6 — C2×C32⋊C6 — C22×C32⋊C6
Lower central
C32 — C22×C32⋊C6
Upper central
C1 — C22

Generators and relations for C22×C32⋊C6
 G = < a,b,c,d,e | a2=b2=c3=d3=e6=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece-1=c-1d-1, ede-1=d-1 >

Subgroups: 480 in 122 conjugacy classes, 47 normal (12 characteristic)
C1, C2, C2, C3, C3, C22, C22, S3, C6, C6, C23, C32, C32, D6, C2×C6, C2×C6, C3×S3, C3⋊S3, C3×C6, C3×C6, C22×S3, C22×C6, He3, S3×C6, C2×C3⋊S3, C62, C62, C32⋊C6, C2×He3, S3×C2×C6, C22×C3⋊S3, C2×C32⋊C6, C22×He3, C22×C32⋊C6
Quotients: C1, C2, C3, C22, S3, C6, C23, D6, C2×C6, C3×S3, C22×S3, C22×C6, S3×C6, C32⋊C6, S3×C2×C6, C2×C32⋊C6, C22×C32⋊C6

Smallest permutation representation of C22×C32⋊C6
►On 36 points
Generators in S36
(1 8)(2 7)(3 11)(4 12)(5 9)(6 10)(13 27)(14 28)(15 29)(16 30)(17 25)(18 26)(19 33)(20 34)(21 35)(22 36)(23 31)(24 32)

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

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

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

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

C22×C32⋊C6 is a maximal subgroup of   C62.4D6  C62.21D6  C62⋊D6
C22×C32⋊C6 is a maximal quotient of   C62.36D6  C62.13D6  (Q8×He3)⋊C2

40 conjugacy classes

class 1 2A2B2C2D2E2F2G3A3B3C3D3E3F6A6B6C6D···6I6J···6R6S···6Z
order122222223333336666···66···66···6
size111199992336662223···36···69···9

40 irreducible representations

dim111111222266
type+++++++
imageC1C2C2C3C6C6S3D6C3×S3S3×C6C32⋊C6C2×C32⋊C6
kernelC22×C32⋊C6C2×C32⋊C6C22×He3C22×C3⋊S3C2×C3⋊S3C62C62C3×C6C2×C6C6C22C2
# reps1612122132613

Matrix representation of C22×C32⋊C6 ►in GL8(𝔽7)

60000000
06000000
00600000
00060000
00006000
00000600
00000060
00000006
,
10000000
01000000
00600000
00060000
00006000
00000600
00000060
00000006
,
61000000
60000000
00000600
00001600
00000006
00000016
00060000
00160000
,
10000000
01000000
00060000
00160000
00000600
00001600
00000006
00000016
,
03000000
30000000
00010000
00100000
00000060
00000061
00001600
00000600

G:=sub<GL(8,GF(7))| [6,0,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,6],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,6],[6,6,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,6,6,0,0,0,1,0,0,0,0,0,0,6,6,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,6,6,0,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,6,6,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,6,6,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,6,6],[0,3,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,6,6,0,0,0,0,6,6,0,0,0,0,0,0,0,1,0,0] >;
 

C22×C32⋊C6 in GAP, Magma, Sage, TeX 

C_2^2\times C_3^2\rtimes C_6
 
% in TeX
 
G:=Group("C2^2xC3^2:C6");
 
// GroupNames label
 
G:=SmallGroup(216,110);
 
// by ID
 
G=gap.SmallGroup(216,110);
 
# by ID
 
G:=PCGroup([6,-2,-2,-2,-3,-3,-3,1444,382,5189]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^2=b^2=c^3=d^3=e^6=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^-1=c^-1*d^-1,e*d*e^-1=d^-1>;
 
// generators/relations
 

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