Copied to
clipboard

G = He34D4order 216 = 23·33

1st semidirect product of He3 and D4 acting via D4/C4=C2

metabelian, supersoluble, monomial

Aliases: He34D4, C323D12, C12⋊S3⋊C3, (C3×C12)⋊1C6, (C3×C12)⋊1S3, C4⋊(C32⋊C6), (C3×C6).8D6, C6.11(S3×C6), C12.4(C3×S3), (C4×He3)⋊1C2, C3.2(C3×D12), C322(C3×D4), (C2×He3).8C22, (C2×C3⋊S3)⋊1C6, (C3×C6).3(C2×C6), (C2×C32⋊C6)⋊3C2, C2.4(C2×C32⋊C6), SmallGroup(216,51)

Series: Derived Chief Lower central Upper central

C1C3×C6 — He34D4
C1C3C32C3×C6C2×He3C2×C32⋊C6 — He34D4
C32C3×C6 — He34D4
C1C2C4

Generators and relations for He34D4
 G = < a,b,c,d,e | a3=b3=c3=d4=e2=1, ab=ba, cac-1=ab-1, ad=da, eae=a-1, bc=cb, bd=db, ebe=b-1, cd=dc, ce=ec, ede=d-1 >

Subgroups: 302 in 62 conjugacy classes, 21 normal (17 characteristic)
C1, C2, C2 [×2], C3, C3 [×3], C4, C22 [×2], S3 [×4], C6, C6 [×5], D4, C32 [×2], C32, C12, C12 [×3], D6 [×4], C2×C6 [×2], C3×S3 [×2], C3⋊S3 [×2], C3×C6 [×2], C3×C6, D12 [×2], C3×D4, He3, C3×C12 [×2], C3×C12, S3×C6 [×2], C2×C3⋊S3 [×2], C32⋊C6 [×2], C2×He3, C3×D12, C12⋊S3, C4×He3, C2×C32⋊C6 [×2], He34D4
Quotients: C1, C2 [×3], C3, C22, S3, C6 [×3], D4, D6, C2×C6, C3×S3, D12, C3×D4, S3×C6, C32⋊C6, C3×D12, C2×C32⋊C6, He34D4

Smallest permutation representation of He34D4
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 22 30)(3 23 31)(4 24 32)(5 35 11)(6 36 12)(7 33 9)(8 34 10)(13 28 19)(14 25 20)(15 26 17)(16 27 18)
(5 11 35)(6 12 36)(7 9 33)(8 10 34)(13 28 19)(14 25 20)(15 26 17)(16 27 18)
(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 3)(5 16)(6 15)(7 14)(8 13)(9 25)(10 28)(11 27)(12 26)(17 36)(18 35)(19 34)(20 33)(21 31)(22 30)(23 29)(24 32)

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

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

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

He34D4 is a maximal subgroup of
He33SD16  He32D8  He33D8  He35SD16  He36SD16  He34D8  He36D8  He310SD16  C12⋊S3⋊S3  C12.84S32  C3⋊S3⋊D12  C12.86S32  C62.36D6  D4×C32⋊C6  (Q8×He3)⋊C2
He34D4 is a maximal quotient of
He34Q16  He36SD16  He34D8  C62.20D6  C62.21D6

31 conjugacy classes

class 1 2A2B2C3A3B3C3D3E3F 4 6A6B6C6D6E6F6G6H6I6J12A12B12C···12J
order122233333346666666666121212···12
size111818233666223366618181818226···6

31 irreducible representations

dim11111122222222666
type++++++++++
imageC1C2C2C3C6C6S3D4D6C3×S3D12C3×D4S3×C6C3×D12C32⋊C6C2×C32⋊C6He34D4
kernelHe34D4C4×He3C2×C32⋊C6C12⋊S3C3×C12C2×C3⋊S3C3×C12He3C3×C6C12C32C32C6C3C4C2C1
# reps11222411122224112

Matrix representation of He34D4 in GL6(𝔽13)

001000
000100
000010
000001
100000
010000
,
1210000
1200000
0012100
0012000
0000121
0000120
,
100000
010000
0001200
0011200
0000121
0000120
,
370000
6100000
003700
0061000
000037
0000610
,
0120000
1200000
0000012
0000120
0001200
0012000

G:=sub<GL(6,GF(13))| [0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0],[12,12,0,0,0,0,1,0,0,0,0,0,0,0,12,12,0,0,0,0,1,0,0,0,0,0,0,0,12,12,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,12,12,0,0,0,0,0,0,12,12,0,0,0,0,1,0],[3,6,0,0,0,0,7,10,0,0,0,0,0,0,3,6,0,0,0,0,7,10,0,0,0,0,0,0,3,6,0,0,0,0,7,10],[0,12,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,12,0,0,0,0,12,0,0,0,0,12,0,0,0] >;

He34D4 in GAP, Magma, Sage, TeX

{\rm He}_3\rtimes_4D_4
% in TeX

G:=Group("He3:4D4");
// GroupNames label

G:=SmallGroup(216,51);
// by ID

G=gap.SmallGroup(216,51);
# by ID

G:=PCGroup([6,-2,-2,-3,-2,-3,-3,169,79,1444,736,5189]);
// Polycyclic

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

׿
×
𝔽