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## G = C22⋊(He3⋊C4)  order 432 = 24·33

### The semidirect product of C22 and He3⋊C4 acting via He3⋊C4/He3⋊C2=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — C2×He3 — C22⋊(He3⋊C4)
 Chief series C1 — C3 — He3 — He3⋊C2 — C2×He3⋊C2 — C2×He3⋊C4 — C22⋊(He3⋊C4)
 Lower central He3 — C2×He3 — C22⋊(He3⋊C4)
 Upper central C1 — C6 — C2×C6

Generators and relations for C22⋊(He3⋊C4)
G = < a,b,c,d,e,f | a2=b2=c3=d3=e3=f4=1, faf-1=ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ece-1=cd-1, fcf-1=cde, de=ed, df=fd, fef-1=ce-1 >

Subgroups: 701 in 115 conjugacy classes, 17 normal (13 characteristic)
C1, C2, C2, C3, C3, C4, C22, C22, S3, C6, C6, C2×C4, C23, C32, C12, D6, C2×C6, C2×C6, C22⋊C4, C3×S3, C3×C6, C2×C12, C22×S3, C22×C6, He3, S3×C6, C62, C3×C22⋊C4, He3⋊C2, He3⋊C2, C2×He3, C2×He3, S3×C2×C6, He3⋊C4, C2×He3⋊C2, C2×He3⋊C2, C22×He3, C2×He3⋊C4, C22×He3⋊C2, C22⋊(He3⋊C4)
Quotients: C1, C2, C4, C22, C2×C4, D4, C22⋊C4, C32⋊C4, C2×C32⋊C4, He3⋊C4, C62⋊C4, C2×He3⋊C4, C22⋊(He3⋊C4)

Smallest permutation representation of C22⋊(He3⋊C4)
On 36 points
Generators in S36
(2 7)(4 11)(6 9)(14 17)(16 19)(21 33)(23 35)(26 29)(28 31)
(1 8)(2 7)(3 12)(4 11)(5 10)(6 9)(13 20)(14 17)(15 18)(16 19)(21 33)(22 34)(23 35)(24 36)(25 32)(26 29)(27 30)(28 31)
(1 25 13)(2 35 26)(3 36 27)(4 14 33)(5 15 34)(6 28 16)(7 23 29)(8 32 20)(9 31 19)(10 18 22)(11 17 21)(12 24 30)
(1 3 5)(2 4 6)(7 11 9)(8 12 10)(13 27 34)(14 28 35)(15 25 36)(16 26 33)(17 31 23)(18 32 24)(19 29 21)(20 30 22)
(1 25 27)(3 36 34)(5 15 13)(8 32 30)(10 18 20)(12 24 22)(14 28 35)(16 33 26)(17 31 23)(19 21 29)
(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)| (2,7)(4,11)(6,9)(14,17)(16,19)(21,33)(23,35)(26,29)(28,31), (1,8)(2,7)(3,12)(4,11)(5,10)(6,9)(13,20)(14,17)(15,18)(16,19)(21,33)(22,34)(23,35)(24,36)(25,32)(26,29)(27,30)(28,31), (1,25,13)(2,35,26)(3,36,27)(4,14,33)(5,15,34)(6,28,16)(7,23,29)(8,32,20)(9,31,19)(10,18,22)(11,17,21)(12,24,30), (1,3,5)(2,4,6)(7,11,9)(8,12,10)(13,27,34)(14,28,35)(15,25,36)(16,26,33)(17,31,23)(18,32,24)(19,29,21)(20,30,22), (1,25,27)(3,36,34)(5,15,13)(8,32,30)(10,18,20)(12,24,22)(14,28,35)(16,33,26)(17,31,23)(19,21,29), (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( (2,7)(4,11)(6,9)(14,17)(16,19)(21,33)(23,35)(26,29)(28,31), (1,8)(2,7)(3,12)(4,11)(5,10)(6,9)(13,20)(14,17)(15,18)(16,19)(21,33)(22,34)(23,35)(24,36)(25,32)(26,29)(27,30)(28,31), (1,25,13)(2,35,26)(3,36,27)(4,14,33)(5,15,34)(6,28,16)(7,23,29)(8,32,20)(9,31,19)(10,18,22)(11,17,21)(12,24,30), (1,3,5)(2,4,6)(7,11,9)(8,12,10)(13,27,34)(14,28,35)(15,25,36)(16,26,33)(17,31,23)(18,32,24)(19,29,21)(20,30,22), (1,25,27)(3,36,34)(5,15,13)(8,32,30)(10,18,20)(12,24,22)(14,28,35)(16,33,26)(17,31,23)(19,21,29), (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([[(2,7),(4,11),(6,9),(14,17),(16,19),(21,33),(23,35),(26,29),(28,31)], [(1,8),(2,7),(3,12),(4,11),(5,10),(6,9),(13,20),(14,17),(15,18),(16,19),(21,33),(22,34),(23,35),(24,36),(25,32),(26,29),(27,30),(28,31)], [(1,25,13),(2,35,26),(3,36,27),(4,14,33),(5,15,34),(6,28,16),(7,23,29),(8,32,20),(9,31,19),(10,18,22),(11,17,21),(12,24,30)], [(1,3,5),(2,4,6),(7,11,9),(8,12,10),(13,27,34),(14,28,35),(15,25,36),(16,26,33),(17,31,23),(18,32,24),(19,29,21),(20,30,22)], [(1,25,27),(3,36,34),(5,15,13),(8,32,30),(10,18,20),(12,24,22),(14,28,35),(16,33,26),(17,31,23),(19,21,29)], [(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)]])

38 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3A 3B 3C 3D 4A 4B 4C 4D 6A 6B 6C 6D 6E 6F 6G 6H 6I ··· 6N 6O 6P 12A ··· 12H order 1 2 2 2 2 2 3 3 3 3 4 4 4 4 6 6 6 6 6 6 6 6 6 ··· 6 6 6 12 ··· 12 size 1 1 2 9 9 18 1 1 12 12 18 18 18 18 1 1 2 2 9 9 9 9 12 ··· 12 18 18 18 ··· 18

38 irreducible representations

 dim 1 1 1 1 1 2 3 3 4 4 4 6 type + + + + + + + image C1 C2 C2 C4 C4 D4 He3⋊C4 C2×He3⋊C4 C32⋊C4 C2×C32⋊C4 C62⋊C4 C22⋊(He3⋊C4) kernel C22⋊(He3⋊C4) C2×He3⋊C4 C22×He3⋊C2 C2×He3⋊C2 C22×He3 He3⋊C2 C22 C2 C2×C6 C6 C3 C1 # reps 1 2 1 2 2 2 8 8 2 2 4 4

Matrix representation of C22⋊(He3⋊C4) in GL5(𝔽13)

 12 0 0 0 0 4 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 12 0 0 0 0 0 12 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 3 0 0 0 0 0 9 0
,
 1 0 0 0 0 0 1 0 0 0 0 0 9 0 0 0 0 0 9 0 0 0 0 0 9
,
 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 9 0 0 0 0 0 3
,
 8 4 0 0 0 0 5 0 0 0 0 0 3 9 3 0 0 1 1 3 0 0 3 1 1

G:=sub<GL(5,GF(13))| [12,4,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[12,0,0,0,0,0,12,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,0,3,0,0,0,0,0,9,0,0,1,0,0],[1,0,0,0,0,0,1,0,0,0,0,0,9,0,0,0,0,0,9,0,0,0,0,0,9],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,9,0,0,0,0,0,3],[8,0,0,0,0,4,5,0,0,0,0,0,3,1,3,0,0,9,1,1,0,0,3,3,1] >;

C22⋊(He3⋊C4) in GAP, Magma, Sage, TeX

C_2^2\rtimes ({\rm He}_3\rtimes C_4)
% in TeX

G:=Group("C2^2:(He3:C4)");
// GroupNames label

G:=SmallGroup(432,279);
// by ID

G=gap.SmallGroup(432,279);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,3,-3,28,141,3924,298,5381,2539,537]);
// Polycyclic

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

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