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## G = C4×He3order 108 = 22·33

### Direct product of C4 and He3

direct product, metabelian, nilpotent (class 2), monomial, 3-elementary

Aliases: C4×He3, C323C12, C12.1C32, (C3×C12)⋊C3, C2.(C2×He3), (C3×C6).2C6, C6.2(C3×C6), C12(C2×He3), C3.1(C3×C12), (C2×He3).3C2, SmallGroup(108,13)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — C4×He3
 Chief series C1 — C3 — C6 — C3×C6 — C2×He3 — C4×He3
 Lower central C1 — C3 — C4×He3
 Upper central C1 — C12 — C4×He3

Generators and relations for C4×He3
G = < a,b,c,d | a4=b3=c3=d3=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=bc-1, cd=dc >

Smallest permutation representation of C4×He3
On 36 points
Generators in S36
(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)
(5 33 10)(6 34 11)(7 35 12)(8 36 9)(17 22 27)(18 23 28)(19 24 25)(20 21 26)
(1 14 29)(2 15 30)(3 16 31)(4 13 32)(5 33 10)(6 34 11)(7 35 12)(8 36 9)(17 27 22)(18 28 23)(19 25 24)(20 26 21)
(1 26 35)(2 27 36)(3 28 33)(4 25 34)(5 31 18)(6 32 19)(7 29 20)(8 30 17)(9 15 22)(10 16 23)(11 13 24)(12 14 21)

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

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

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

C4×He3 is a maximal subgroup of   He33C8  He34C8  He33Q8  He34D4  He34Q8  He35D4  C4○D4⋊He3

44 conjugacy classes

 class 1 2 3A 3B 3C ··· 3J 4A 4B 6A 6B 6C ··· 6J 12A 12B 12C 12D 12E ··· 12T order 1 2 3 3 3 ··· 3 4 4 6 6 6 ··· 6 12 12 12 12 12 ··· 12 size 1 1 1 1 3 ··· 3 1 1 1 1 3 ··· 3 1 1 1 1 3 ··· 3

44 irreducible representations

 dim 1 1 1 1 1 1 3 3 3 type + + image C1 C2 C3 C4 C6 C12 He3 C2×He3 C4×He3 kernel C4×He3 C2×He3 C3×C12 He3 C3×C6 C32 C4 C2 C1 # reps 1 1 8 2 8 16 2 2 4

Matrix representation of C4×He3 in GL3(𝔽13) generated by

 5 0 0 0 5 0 0 0 5
,
 1 0 0 0 9 0 0 0 3
,
 9 0 0 0 9 0 0 0 9
,
 0 1 0 0 0 1 1 0 0
G:=sub<GL(3,GF(13))| [5,0,0,0,5,0,0,0,5],[1,0,0,0,9,0,0,0,3],[9,0,0,0,9,0,0,0,9],[0,0,1,1,0,0,0,1,0] >;

C4×He3 in GAP, Magma, Sage, TeX

C_4\times {\rm He}_3
% in TeX

G:=Group("C4xHe3");
// GroupNames label

G:=SmallGroup(108,13);
// by ID

G=gap.SmallGroup(108,13);
# by ID

G:=PCGroup([5,-2,-3,-3,-2,-3,90,322]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^3=c^3=d^3=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b*c^-1,c*d=d*c>;
// generators/relations

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