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G = D4xHe3order 216 = 23·33

Direct product of D4 and He3

direct product, metabelian, nilpotent (class 2), monomial

Aliases: D4xHe3, C62:1C6, C6.14C62, C4:(C2xHe3), (C3xC12):3C6, (D4xC32):C3, C12.4(C3xC6), (C4xHe3):5C2, C32:6(C3xD4), C22:2(C2xHe3), C3.2(D4xC32), (C22xHe3):1C2, (C3xD4).2C32, C2.2(C22xHe3), (C2xHe3).17C22, (C2xC6).7(C3xC6), (C3xC6).13(C2xC6), SmallGroup(216,77)

Series: Derived Chief Lower central Upper central

C1C6 — D4xHe3
C1C3C6C3xC6C2xHe3C22xHe3 — D4xHe3
C1C6 — D4xHe3
C1C6 — D4xHe3

Generators and relations for D4xHe3
 G = < a,b,c,d,e | a4=b2=c3=d3=e3=1, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece-1=cd-1, de=ed >

Subgroups: 190 in 88 conjugacy classes, 42 normal (12 characteristic)
C1, C2, C2, C3, C3, C4, C22, C6, C6, D4, C32, C12, C12, C2xC6, C2xC6, C3xC6, C3xC6, C3xD4, C3xD4, He3, C3xC12, C62, C2xHe3, C2xHe3, D4xC32, C4xHe3, C22xHe3, D4xHe3
Quotients: C1, C2, C3, C22, C6, D4, C32, C2xC6, C3xC6, C3xD4, He3, C62, C2xHe3, D4xC32, C22xHe3, D4xHe3

Smallest permutation representation of D4xHe3
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 4)(2 3)(5 8)(6 7)(9 10)(11 12)(13 14)(15 16)(17 18)(19 20)(21 24)(22 23)(25 26)(27 28)(29 32)(30 31)(33 36)(34 35)
(1 26 35)(2 27 36)(3 28 33)(4 25 34)(5 31 18)(6 32 19)(7 29 20)(8 30 17)(9 15 22)(10 16 23)(11 13 24)(12 14 21)
(1 14 29)(2 15 30)(3 16 31)(4 13 32)(5 33 10)(6 34 11)(7 35 12)(8 36 9)(17 27 22)(18 28 23)(19 25 24)(20 26 21)
(1 26 12)(2 27 9)(3 28 10)(4 25 11)(5 16 23)(6 13 24)(7 14 21)(8 15 22)(17 36 30)(18 33 31)(19 34 32)(20 35 29)

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

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

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

D4xHe3 is a maximal subgroup of   He3:8SD16  He3:6D8  He3:7D8  He3:9SD16  C62.13D6  C62.16D6

55 conjugacy classes

class 1 2A2B2C3A3B3C···3J 4 6A6B6C6D6E6F6G···6N6O···6AD12A12B12C···12J
order1222333···346666666···66···6121212···12
size1122113···321122223···36···6226···6

55 irreducible representations

dim111111223336
type++++
imageC1C2C2C3C6C6D4C3xD4He3C2xHe3C2xHe3D4xHe3
kernelD4xHe3C4xHe3C22xHe3D4xC32C3xC12C62He3C32D4C4C22C1
# reps1128816182242

Matrix representation of D4xHe3 in GL5(F13)

01000
120000
001200
000120
000012
,
01000
10000
001200
000120
000012
,
90000
09000
00010
004102
00003
,
10000
01000
00300
00030
00003
,
90000
09000
00090
001246
001219

G:=sub<GL(5,GF(13))| [0,12,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12],[0,1,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12],[9,0,0,0,0,0,9,0,0,0,0,0,0,4,0,0,0,1,10,0,0,0,0,2,3],[1,0,0,0,0,0,1,0,0,0,0,0,3,0,0,0,0,0,3,0,0,0,0,0,3],[9,0,0,0,0,0,9,0,0,0,0,0,0,12,12,0,0,9,4,1,0,0,0,6,9] >;

D4xHe3 in GAP, Magma, Sage, TeX

D_4\times {\rm He}_3
% in TeX

G:=Group("D4xHe3");
// GroupNames label

G:=SmallGroup(216,77);
// by ID

G=gap.SmallGroup(216,77);
# by ID

G:=PCGroup([6,-2,-2,-3,-3,-2,-3,457,519]);
// Polycyclic

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

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