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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

non-abelian, supersoluble, monomial

Aliases: C6.19S32, He3⋊3(C2×C4), (C3×C6).4D6, C3⋊Dic3⋊2S3, C32⋊2(C4×S3), C32⋊C12⋊3C2, He3⋊C2⋊2C4, 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 12A2B2C3A3B3C3D4A4B4C4D6A6B6C6D6E6F12A12B12C12D
 size 119926612999926612181818181818
ρ11111111111111111111111    trivial
ρ211-1-11111-1-1111111-1-1-11-11    linear of order 2
ρ311111111-1-1-1-1111111-1-1-1-1    linear of order 2
ρ411-1-1111111-1-11111-1-11-11-1    linear of order 2
ρ51-1-111111-iii-i-1-1-1-1-11-iii-i    linear of order 4
ρ61-11-11111i-ii-i-1-1-1-11-1ii-i-i    linear of order 4
ρ71-1-111111i-i-ii-1-1-1-1-11i-i-ii    linear of order 4
ρ81-11-11111-ii-ii-1-1-1-11-1-i-iii    linear of order 4
ρ922002-12-100-2-22-12-1000101    orthogonal lifted from D6
ρ1022002-12-100222-12-1000-10-1    orthogonal lifted from S3
ρ11220022-1-1-2-20022-1-1001010    orthogonal lifted from D6
ρ12220022-1-1220022-1-100-10-10    orthogonal lifted from S3
ρ132-2002-12-1002i-2i-21-21000-i0i    complex lifted from C4×S3
ρ142-20022-1-12i-2i00-2-21100-i0i0    complex lifted from C4×S3
ρ152-20022-1-1-2i2i00-2-21100i0-i0    complex lifted from C4×S3
ρ162-2002-12-100-2i2i-21-21000i0-i    complex lifted from C4×S3
ρ174-4004-2-210000-422-1000000    orthogonal lifted from C6.D6
ρ1844004-2-2100004-2-21000000    orthogonal lifted from S32
ρ1966-2-2-30000000-3000110000    orthogonal lifted from C32⋊D6
ρ206622-30000000-3000-1-10000    orthogonal lifted from C32⋊D6
ρ216-6-22-3000000030001-10000    symplectic faithful, Schur index 2
ρ226-62-2-300000003000-110000    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  C62⋊2D6
He3⋊(C2×C4) is a maximal quotient of   C12.89S32  He3⋊3M4(2)  He3⋊C42  C62.3D6  C62.5D6

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

0100000000
121200000000
0001000000
001212000000
0000001000
0000000100
0000000010
0000000001
0000100000
0000010000
,
1000000000
0100000000
0010000000
0001000000
0000010000
000012120000
0000000100
000000121200
0000000001
000000001212
,
121200000000
1000000000
0001000000
001212000000
000000001212
0000000010
0000010000
000012120000
0000001000
0000000100
,
1000000000
121200000000
0010000000
001212000000
0000100000
0000010000
0000000010
0000000001
0000001000
0000000100
,
0010000000
0001000000
12000000000
01200000000
0000958484
0000949595
0000849584
0000959495
0000848495
0000959594

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
 

Export

Character table of He3⋊(C2×C4) in TeX

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