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## G = C2×He3⋊C4order 216 = 23·33

### Direct product of C2 and He3⋊C4

Aliases: C2×He3⋊C4, (C2×He3)⋊C4, He31(C2×C4), He3⋊C21C4, C6.3(C32⋊C4), He3⋊C2.3C22, C3.(C2×C32⋊C4), (C2×He3⋊C2).2C2, SmallGroup(216,100)

Series: Derived Chief Lower central Upper central

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

Generators and relations for C2×He3⋊C4
G = < a,b,c,d,e | a2=b3=c3=d3=e4=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd-1=bc-1, ebe-1=bcd, cd=dc, ce=ec, ede-1=bd-1 >

Character table of C2×He3⋊C4

 class 1 2A 2B 2C 3A 3B 3C 3D 4A 4B 4C 4D 6A 6B 6C 6D 6E 6F 6G 6H 12A 12B 12C 12D 12E 12F 12G 12H size 1 1 9 9 1 1 12 12 9 9 9 9 1 1 9 9 9 9 12 12 9 9 9 9 9 9 9 9 ρ1 1 1 1 1 1 1 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 -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 -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 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 1 1 i -i -i i 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 -1 -1 -i i i i 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 -1 -1 i -i -i -i -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 1 1 -i i i -i -i -i i i linear of order 4 ρ9 3 3 -1 -1 -3+3√-3/2 -3-3√-3/2 0 0 -1 -1 -1 -1 -3-3√-3/2 -3+3√-3/2 ζ65 ζ65 ζ6 ζ6 0 0 ζ6 ζ6 ζ65 ζ65 ζ6 ζ65 ζ65 ζ6 complex lifted from He3⋊C4 ρ10 3 -3 1 -1 -3-3√-3/2 -3+3√-3/2 0 0 -1 1 1 -1 3-3√-3/2 3+3√-3/2 ζ32 ζ6 ζ3 ζ65 0 0 ζ3 ζ3 ζ32 ζ6 ζ65 ζ32 ζ6 ζ65 complex faithful ρ11 3 -3 1 -1 -3+3√-3/2 -3-3√-3/2 0 0 1 -1 -1 1 3+3√-3/2 3-3√-3/2 ζ3 ζ65 ζ32 ζ6 0 0 ζ6 ζ6 ζ65 ζ3 ζ32 ζ65 ζ3 ζ32 complex faithful ρ12 3 3 -1 -1 -3-3√-3/2 -3+3√-3/2 0 0 1 1 1 1 -3+3√-3/2 -3-3√-3/2 ζ6 ζ6 ζ65 ζ65 0 0 ζ3 ζ3 ζ32 ζ32 ζ3 ζ32 ζ32 ζ3 complex lifted from He3⋊C4 ρ13 3 3 -1 -1 -3+3√-3/2 -3-3√-3/2 0 0 1 1 1 1 -3-3√-3/2 -3+3√-3/2 ζ65 ζ65 ζ6 ζ6 0 0 ζ32 ζ32 ζ3 ζ3 ζ32 ζ3 ζ3 ζ32 complex lifted from He3⋊C4 ρ14 3 3 -1 -1 -3-3√-3/2 -3+3√-3/2 0 0 -1 -1 -1 -1 -3+3√-3/2 -3-3√-3/2 ζ6 ζ6 ζ65 ζ65 0 0 ζ65 ζ65 ζ6 ζ6 ζ65 ζ6 ζ6 ζ65 complex lifted from He3⋊C4 ρ15 3 -3 1 -1 -3-3√-3/2 -3+3√-3/2 0 0 1 -1 -1 1 3-3√-3/2 3+3√-3/2 ζ32 ζ6 ζ3 ζ65 0 0 ζ65 ζ65 ζ6 ζ32 ζ3 ζ6 ζ32 ζ3 complex faithful ρ16 3 -3 1 -1 -3+3√-3/2 -3-3√-3/2 0 0 -1 1 1 -1 3+3√-3/2 3-3√-3/2 ζ3 ζ65 ζ32 ζ6 0 0 ζ32 ζ32 ζ3 ζ65 ζ6 ζ3 ζ65 ζ6 complex faithful ρ17 3 3 1 1 -3+3√-3/2 -3-3√-3/2 0 0 i -i i -i -3-3√-3/2 -3+3√-3/2 ζ3 ζ3 ζ32 ζ32 0 0 ζ43ζ32 ζ4ζ32 ζ4ζ3 ζ43ζ3 ζ43ζ32 ζ43ζ3 ζ4ζ3 ζ4ζ32 complex lifted from He3⋊C4 ρ18 3 -3 -1 1 -3+3√-3/2 -3-3√-3/2 0 0 i i -i -i 3+3√-3/2 3-3√-3/2 ζ65 ζ3 ζ6 ζ32 0 0 ζ4ζ32 ζ43ζ32 ζ43ζ3 ζ43ζ3 ζ43ζ32 ζ4ζ3 ζ4ζ3 ζ4ζ32 complex faithful ρ19 3 3 1 1 -3-3√-3/2 -3+3√-3/2 0 0 -i i -i i -3+3√-3/2 -3-3√-3/2 ζ32 ζ32 ζ3 ζ3 0 0 ζ4ζ3 ζ43ζ3 ζ43ζ32 ζ4ζ32 ζ4ζ3 ζ4ζ32 ζ43ζ32 ζ43ζ3 complex lifted from He3⋊C4 ρ20 3 -3 -1 1 -3+3√-3/2 -3-3√-3/2 0 0 -i -i i i 3+3√-3/2 3-3√-3/2 ζ65 ζ3 ζ6 ζ32 0 0 ζ43ζ32 ζ4ζ32 ζ4ζ3 ζ4ζ3 ζ4ζ32 ζ43ζ3 ζ43ζ3 ζ43ζ32 complex faithful ρ21 3 3 1 1 -3-3√-3/2 -3+3√-3/2 0 0 i -i i -i -3+3√-3/2 -3-3√-3/2 ζ32 ζ32 ζ3 ζ3 0 0 ζ43ζ3 ζ4ζ3 ζ4ζ32 ζ43ζ32 ζ43ζ3 ζ43ζ32 ζ4ζ32 ζ4ζ3 complex lifted from He3⋊C4 ρ22 3 3 1 1 -3+3√-3/2 -3-3√-3/2 0 0 -i i -i i -3-3√-3/2 -3+3√-3/2 ζ3 ζ3 ζ32 ζ32 0 0 ζ4ζ32 ζ43ζ32 ζ43ζ3 ζ4ζ3 ζ4ζ32 ζ4ζ3 ζ43ζ3 ζ43ζ32 complex lifted from He3⋊C4 ρ23 3 -3 -1 1 -3-3√-3/2 -3+3√-3/2 0 0 i i -i -i 3-3√-3/2 3+3√-3/2 ζ6 ζ32 ζ65 ζ3 0 0 ζ4ζ3 ζ43ζ3 ζ43ζ32 ζ43ζ32 ζ43ζ3 ζ4ζ32 ζ4ζ32 ζ4ζ3 complex faithful ρ24 3 -3 -1 1 -3-3√-3/2 -3+3√-3/2 0 0 -i -i i i 3-3√-3/2 3+3√-3/2 ζ6 ζ32 ζ65 ζ3 0 0 ζ43ζ3 ζ4ζ3 ζ4ζ32 ζ4ζ32 ζ4ζ3 ζ43ζ32 ζ43ζ32 ζ43ζ3 complex faithful ρ25 4 -4 0 0 4 4 -2 1 0 0 0 0 -4 -4 0 0 0 0 2 -1 0 0 0 0 0 0 0 0 orthogonal lifted from C2×C32⋊C4 ρ26 4 4 0 0 4 4 -2 1 0 0 0 0 4 4 0 0 0 0 -2 1 0 0 0 0 0 0 0 0 orthogonal lifted from C32⋊C4 ρ27 4 -4 0 0 4 4 1 -2 0 0 0 0 -4 -4 0 0 0 0 -1 2 0 0 0 0 0 0 0 0 orthogonal lifted from C2×C32⋊C4 ρ28 4 4 0 0 4 4 1 -2 0 0 0 0 4 4 0 0 0 0 1 -2 0 0 0 0 0 0 0 0 orthogonal lifted from C32⋊C4

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

C2×He3⋊C4 is a maximal subgroup of   C6.S3≀C2  C32⋊D6⋊C4  C2.SU3(𝔽2)  C4⋊(He3⋊C4)  C22⋊(He3⋊C4)
C2×He3⋊C4 is a maximal quotient of   He32(C2×C8)  He31M4(2)  C4⋊(He3⋊C4)  He34M4(2)  C22⋊(He3⋊C4)

Matrix representation of C2×He3⋊C4 in GL3(𝔽7) generated by

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

C2×He3⋊C4 in GAP, Magma, Sage, TeX

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

G:=Group("C2xHe3:C4");
// GroupNames label

G:=SmallGroup(216,100);
// by ID

G=gap.SmallGroup(216,100);
# by ID

G:=PCGroup([6,-2,-2,-2,-3,3,-3,24,1347,111,1924,916,382]);
// Polycyclic

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

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