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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

 Derived series C1 — C3 — C22×He3
 Chief series C1 — C3 — C32 — He3 — C2×He3 — C22×He3
 Lower central C1 — C3 — C22×He3
 Upper central C1 — C2×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, C3, C3, C22, C6, C6, C32, C2×C6, C2×C6, C3×C6, He3, C62, C2×He3, C22×He3
Quotients: C1, C2, C3, C22, C6, C32, C2×C6, C3×C6, He3, C62, C2×He3, 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 2A 2B 2C 3A 3B 3C ··· 3J 6A ··· 6F 6G ··· 6AD order 1 2 2 2 3 3 3 ··· 3 6 ··· 6 6 ··· 6 size 1 1 1 1 1 1 3 ··· 3 1 ··· 1 3 ··· 3

44 irreducible representations

 dim 1 1 1 1 3 3 type + + image C1 C2 C3 C6 He3 C2×He3 kernel C22×He3 C2×He3 C62 C3×C6 C22 C2 # reps 1 3 8 24 2 6

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

 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 1 0 0 5 3 3 0 0 0 4
,
 1 0 0 0 0 4 0 0 0 0 4 0 0 0 0 4
,
 2 0 0 0 0 6 5 5 0 1 0 0 0 3 4 1
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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