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## G = C4×He3⋊C2order 216 = 23·33

### Direct product of C4 and He3⋊C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — He3 — C4×He3⋊C2
 Chief series C1 — C3 — C32 — He3 — C2×He3 — C2×He3⋊C2 — C4×He3⋊C2
 Lower central He3 — C4×He3⋊C2
 Upper central C1 — C12

Generators and relations for C4×He3⋊C2
G = < a,b,c,d,e | a4=b3=c3=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd-1=bc-1, ebe=b-1, cd=dc, ce=ec, ede=d-1 >

Subgroups: 272 in 88 conjugacy classes, 26 normal (12 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C2×C4, C32, Dic3, C12, C12, D6, C2×C6, C3×S3, C3×C6, C4×S3, C2×C12, He3, C3×Dic3, C3×C12, S3×C6, He3⋊C2, C2×He3, S3×C12, He33C4, C4×He3, C2×He3⋊C2, C4×He3⋊C2
Quotients: C1, C2, C4, C22, S3, C2×C4, D6, C3⋊S3, C4×S3, C2×C3⋊S3, He3⋊C2, C4×C3⋊S3, C2×He3⋊C2, C4×He3⋊C2

Smallest permutation representation of C4×He3⋊C2
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)
(1 16 7)(2 13 8)(3 14 5)(4 15 6)(9 29 18)(10 30 19)(11 31 20)(12 32 17)(21 27 33)(22 28 34)(23 25 35)(24 26 36)
(1 21 29)(2 22 30)(3 23 31)(4 24 32)(5 35 11)(6 36 12)(7 33 9)(8 34 10)(13 28 19)(14 25 20)(15 26 17)(16 27 18)
(1 33 27)(2 34 28)(3 35 25)(4 36 26)(5 14 31)(6 15 32)(7 16 29)(8 13 30)(9 18 21)(10 19 22)(11 20 23)(12 17 24)
(1 3)(2 4)(5 16)(6 13)(7 14)(8 15)(9 20)(10 17)(11 18)(12 19)(21 23)(22 24)(25 33)(26 34)(27 35)(28 36)(29 31)(30 32)

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

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

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

C4×He3⋊C2 is a maximal subgroup of
C12.89S32  He33M4(2)  He36M4(2)  He32(C2×C8)  He31M4(2)  C4⋊(He3⋊C4)  C12.84S32  C12.91S32  C12.85S32  C12.86S32  C62.47D6  C62.16D6  He35D4⋊C2
C4×He3⋊C2 is a maximal quotient of
He36M4(2)  C62.29D6  C62.31D6

40 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 3D 3E 3F 4A 4B 4C 4D 6A 6B 6C 6D 6E 6F 6G 6H 6I 6J 12A 12B 12C 12D 12E ··· 12L 12M 12N 12O 12P order 1 2 2 2 3 3 3 3 3 3 4 4 4 4 6 6 6 6 6 6 6 6 6 6 12 12 12 12 12 ··· 12 12 12 12 12 size 1 1 9 9 1 1 6 6 6 6 1 1 9 9 1 1 6 6 6 6 9 9 9 9 1 1 1 1 6 ··· 6 9 9 9 9

40 irreducible representations

 dim 1 1 1 1 1 2 2 2 3 3 3 type + + + + + + image C1 C2 C2 C2 C4 S3 D6 C4×S3 He3⋊C2 C2×He3⋊C2 C4×He3⋊C2 kernel C4×He3⋊C2 He3⋊3C4 C4×He3 C2×He3⋊C2 He3⋊C2 C3×C12 C3×C6 C32 C4 C2 C1 # reps 1 1 1 1 4 4 4 8 4 4 8

Matrix representation of C4×He3⋊C2 in GL3(𝔽13) generated by

 5 0 0 0 5 0 0 0 5
,
 6 7 0 1 7 12 7 4 0
,
 3 0 0 0 3 0 0 0 3
,
 2 1 5 8 11 0 8 12 0
,
 1 6 10 0 12 1 0 0 1
G:=sub<GL(3,GF(13))| [5,0,0,0,5,0,0,0,5],[6,1,7,7,7,4,0,12,0],[3,0,0,0,3,0,0,0,3],[2,8,8,1,11,12,5,0,0],[1,0,0,6,12,0,10,1,1] >;

C4×He3⋊C2 in GAP, Magma, Sage, TeX

C_4\times {\rm He}_3\rtimes C_2
% in TeX

G:=Group("C4xHe3:C2");
// GroupNames label

G:=SmallGroup(216,67);
// by ID

G=gap.SmallGroup(216,67);
# by ID

G:=PCGroup([6,-2,-2,-2,-3,-3,-3,31,387,1444,382]);
// Polycyclic

G:=Group<a,b,c,d,e|a^4=b^3=c^3=d^3=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,d*b*d^-1=b*c^-1,e*b*e=b^-1,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations

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