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G = C22×He3order 108 = 22·33

Direct product of C22 and He3

direct product, metabelian, nilpotent (class 2), monomial

Aliases: C22×He3, C622C3, C3.1C62, (C3×C6)⋊2C6, C6.4(C3×C6), C323(C2×C6), (C2×C6).6C32, SmallGroup(108,30)

Series: Derived Chief Lower central Upper central

C1C3 — C22×He3
C1C3C32He3C2×He3 — C22×He3
C1C3 — C22×He3
C1C2×C6 — C22×He3

Generators and relations for C22×He3
 G = < a,b,c,d,e | a2=b2=c3=d3=e3=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece-1=cd-1, de=ed >

Subgroups: 95 in 55 conjugacy classes, 35 normal (6 characteristic)
C1, C2 [×3], C3, C3 [×4], C22, C6 [×3], C6 [×12], C32 [×4], C2×C6, C2×C6 [×4], C3×C6 [×12], He3, C62 [×4], C2×He3 [×3], C22×He3
Quotients: C1, C2 [×3], C3 [×4], C22, C6 [×12], C32, C2×C6 [×4], C3×C6 [×3], He3, C62, C2×He3 [×3], C22×He3

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

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

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

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

C22×He3 is a maximal subgroup of   He36D4  He37D4  He3.A4  He3⋊A4  He32A4  He3.2A4

44 conjugacy classes

class 1 2A2B2C3A3B3C···3J6A···6F6G···6AD
order1222333···36···66···6
size1111113···31···13···3

44 irreducible representations

dim111133
type++
imageC1C2C3C6He3C2×He3
kernelC22×He3C2×He3C62C3×C6C22C2
# reps1382426

Matrix representation of C22×He3 in GL4(𝔽7) generated by

6000
0600
0060
0006
,
1000
0600
0060
0006
,
2000
0010
0533
0004
,
1000
0400
0040
0004
,
2000
0655
0100
0341
G:=sub<GL(4,GF(7))| [6,0,0,0,0,6,0,0,0,0,6,0,0,0,0,6],[1,0,0,0,0,6,0,0,0,0,6,0,0,0,0,6],[2,0,0,0,0,0,5,0,0,1,3,0,0,0,3,4],[1,0,0,0,0,4,0,0,0,0,4,0,0,0,0,4],[2,0,0,0,0,6,1,3,0,5,0,4,0,5,0,1] >;

C22×He3 in GAP, Magma, Sage, TeX

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

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

G:=SmallGroup(108,30);
// by ID

G=gap.SmallGroup(108,30);
# by ID

G:=PCGroup([5,-2,-2,-3,-3,-3,253]);
// Polycyclic

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

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