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## G = He3⋊(C2×C4)  order 216 = 23·33

### 2nd semidirect product of He3 and C2×C4 acting via C2×C4/C2=C22

Aliases: C6.19S32, He33(C2×C4), (C3×C6).4D6, C3⋊Dic32S3, C322(C4×S3), C32⋊C123C2, He3⋊C22C4, C2.2(C32⋊D6), C3.2(C6.D6), (C2×He3).4C22, (C2×He3⋊C2).1C2, SmallGroup(216,36)

Series: Derived Chief Lower central Upper central

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

Generators and relations for He3⋊(C2×C4)
G = < a,b,c,d,e | a3=b3=c3=d2=e4=1, ab=ba, cac-1=ab-1, dad=a-1, ae=ea, bc=cb, bd=db, ebe-1=b-1, dcd=ece-1=c-1, de=ed >

Subgroups: 298 in 66 conjugacy classes, 18 normal (8 characteristic)
C1, C2, C2, C3, C3, C4, C22, S3, C6, C6, C2×C4, C32, C32, Dic3, C12, D6, C2×C6, C3×S3, C3×C6, C3×C6, C4×S3, C2×Dic3, He3, C3×Dic3, C3⋊Dic3, S3×C6, He3⋊C2, C2×He3, S3×Dic3, C32⋊C12, C2×He3⋊C2, He3⋊(C2×C4)
Quotients: C1, C2, C4, C22, S3, C2×C4, D6, C4×S3, S32, C6.D6, C32⋊D6, He3⋊(C2×C4)

Character table of He3⋊(C2×C4)

 class 1 2A 2B 2C 3A 3B 3C 3D 4A 4B 4C 4D 6A 6B 6C 6D 6E 6F 12A 12B 12C 12D size 1 1 9 9 2 6 6 12 9 9 9 9 2 6 6 12 18 18 18 18 18 18 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 -1 -1 1 1 1 1 -1 -1 1 1 1 1 1 1 -1 -1 -1 1 -1 1 linear of order 2 ρ3 1 1 1 1 1 1 1 1 -1 -1 -1 -1 1 1 1 1 1 1 -1 -1 -1 -1 linear of order 2 ρ4 1 1 -1 -1 1 1 1 1 1 1 -1 -1 1 1 1 1 -1 -1 1 -1 1 -1 linear of order 2 ρ5 1 -1 -1 1 1 1 1 1 -i i i -i -1 -1 -1 -1 -1 1 -i i i -i linear of order 4 ρ6 1 -1 1 -1 1 1 1 1 i -i i -i -1 -1 -1 -1 1 -1 i i -i -i linear of order 4 ρ7 1 -1 -1 1 1 1 1 1 i -i -i i -1 -1 -1 -1 -1 1 i -i -i i linear of order 4 ρ8 1 -1 1 -1 1 1 1 1 -i i -i i -1 -1 -1 -1 1 -1 -i -i i i linear of order 4 ρ9 2 2 0 0 2 -1 2 -1 0 0 -2 -2 2 -1 2 -1 0 0 0 1 0 1 orthogonal lifted from D6 ρ10 2 2 0 0 2 -1 2 -1 0 0 2 2 2 -1 2 -1 0 0 0 -1 0 -1 orthogonal lifted from S3 ρ11 2 2 0 0 2 2 -1 -1 -2 -2 0 0 2 2 -1 -1 0 0 1 0 1 0 orthogonal lifted from D6 ρ12 2 2 0 0 2 2 -1 -1 2 2 0 0 2 2 -1 -1 0 0 -1 0 -1 0 orthogonal lifted from S3 ρ13 2 -2 0 0 2 -1 2 -1 0 0 2i -2i -2 1 -2 1 0 0 0 -i 0 i complex lifted from C4×S3 ρ14 2 -2 0 0 2 2 -1 -1 2i -2i 0 0 -2 -2 1 1 0 0 -i 0 i 0 complex lifted from C4×S3 ρ15 2 -2 0 0 2 2 -1 -1 -2i 2i 0 0 -2 -2 1 1 0 0 i 0 -i 0 complex lifted from C4×S3 ρ16 2 -2 0 0 2 -1 2 -1 0 0 -2i 2i -2 1 -2 1 0 0 0 i 0 -i complex lifted from C4×S3 ρ17 4 -4 0 0 4 -2 -2 1 0 0 0 0 -4 2 2 -1 0 0 0 0 0 0 orthogonal lifted from C6.D6 ρ18 4 4 0 0 4 -2 -2 1 0 0 0 0 4 -2 -2 1 0 0 0 0 0 0 orthogonal lifted from S32 ρ19 6 6 -2 -2 -3 0 0 0 0 0 0 0 -3 0 0 0 1 1 0 0 0 0 orthogonal lifted from C32⋊D6 ρ20 6 6 2 2 -3 0 0 0 0 0 0 0 -3 0 0 0 -1 -1 0 0 0 0 orthogonal lifted from C32⋊D6 ρ21 6 -6 -2 2 -3 0 0 0 0 0 0 0 3 0 0 0 1 -1 0 0 0 0 symplectic faithful, Schur index 2 ρ22 6 -6 2 -2 -3 0 0 0 0 0 0 0 3 0 0 0 -1 1 0 0 0 0 symplectic faithful, Schur index 2

Smallest permutation representation of He3⋊(C2×C4)
On 36 points
Generators in S36
(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 30 22)(3 23 31)(4 32 24)(5 35 11)(6 12 36)(7 33 9)(8 10 34)(13 19 28)(14 25 20)(15 17 26)(16 27 18)
(1 33 27)(2 28 34)(3 35 25)(4 26 36)(5 14 31)(6 32 15)(7 16 29)(8 30 13)(9 18 21)(10 22 19)(11 20 23)(12 24 17)
(5 14)(6 15)(7 16)(8 13)(9 18)(10 19)(11 20)(12 17)(25 35)(26 36)(27 33)(28 34)
(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)| (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)(2,30,22)(3,23,31)(4,32,24)(5,35,11)(6,12,36)(7,33,9)(8,10,34)(13,19,28)(14,25,20)(15,17,26)(16,27,18), (1,33,27)(2,28,34)(3,35,25)(4,26,36)(5,14,31)(6,32,15)(7,16,29)(8,30,13)(9,18,21)(10,22,19)(11,20,23)(12,24,17), (5,14)(6,15)(7,16)(8,13)(9,18)(10,19)(11,20)(12,17)(25,35)(26,36)(27,33)(28,34), (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( (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)(2,30,22)(3,23,31)(4,32,24)(5,35,11)(6,12,36)(7,33,9)(8,10,34)(13,19,28)(14,25,20)(15,17,26)(16,27,18), (1,33,27)(2,28,34)(3,35,25)(4,26,36)(5,14,31)(6,32,15)(7,16,29)(8,30,13)(9,18,21)(10,22,19)(11,20,23)(12,24,17), (5,14)(6,15)(7,16)(8,13)(9,18)(10,19)(11,20)(12,17)(25,35)(26,36)(27,33)(28,34), (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([[(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),(2,30,22),(3,23,31),(4,32,24),(5,35,11),(6,12,36),(7,33,9),(8,10,34),(13,19,28),(14,25,20),(15,17,26),(16,27,18)], [(1,33,27),(2,28,34),(3,35,25),(4,26,36),(5,14,31),(6,32,15),(7,16,29),(8,30,13),(9,18,21),(10,22,19),(11,20,23),(12,24,17)], [(5,14),(6,15),(7,16),(8,13),(9,18),(10,19),(11,20),(12,17),(25,35),(26,36),(27,33),(28,34)], [(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)]])

He3⋊(C2×C4) is a maximal subgroup of   C6.S3≀C2  C32⋊D6⋊C4  C12.85S32  C12.S32  C4×C32⋊D6  C62.8D6  C622D6
He3⋊(C2×C4) is a maximal quotient of   C12.89S32  He33M4(2)  He3⋊C42  C62.3D6  C62.5D6

Matrix representation of He3⋊(C2×C4) in GL10(𝔽13)

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

G:=sub<GL(10,GF(13))| [0,12,0,0,0,0,0,0,0,0,1,12,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,12,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,12,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,12],[12,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,12,0,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,12,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,12,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0],[1,12,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,1,12,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0],[0,0,12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,9,9,8,9,8,9,0,0,0,0,5,4,4,5,4,5,0,0,0,0,8,9,9,9,8,9,0,0,0,0,4,5,5,4,4,5,0,0,0,0,8,9,8,9,9,9,0,0,0,0,4,5,4,5,5,4] >;

He3⋊(C2×C4) in GAP, Magma, Sage, TeX

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

G:=Group("He3:(C2xC4)");
// GroupNames label

G:=SmallGroup(216,36);
// by ID

G=gap.SmallGroup(216,36);
# by ID

G:=PCGroup([6,-2,-2,-2,-3,-3,-3,24,31,201,1444,382,5189,2603]);
// Polycyclic

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

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