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G = He35D4order 216 = 23·33

2nd semidirect product of He3 and D4 acting via D4/C4=C2

non-abelian, supersoluble, monomial

Aliases: He35D4, C324D12, (C3×C12)⋊2S3, C4⋊(He3⋊C2), (C4×He3)⋊2C2, (C3×C6).18D6, C12.8(C3⋊S3), C3.2(C12⋊S3), (C2×He3).13C22, C6.28(C2×C3⋊S3), (C2×He3⋊C2)⋊2C2, C2.4(C2×He3⋊C2), SmallGroup(216,68)

Series: Derived Chief Lower central Upper central

C1C3C2×He3 — He35D4
C1C3C32He3C2×He3C2×He3⋊C2 — He35D4
He3C2×He3 — He35D4
C1C6C12

Generators and relations for He35D4
 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, be=eb, cd=dc, ece=c-1, ede=d-1 >

Subgroups: 358 in 88 conjugacy classes, 24 normal (10 characteristic)
C1, C2, C2 [×2], C3, C3 [×4], C4, C22 [×2], S3 [×8], C6, C6 [×6], D4, C32 [×4], C12, C12 [×4], D6 [×8], C2×C6 [×2], C3×S3 [×8], C3×C6 [×4], D12 [×4], C3×D4, He3, C3×C12 [×4], S3×C6 [×8], He3⋊C2 [×2], C2×He3, C3×D12 [×4], C4×He3, C2×He3⋊C2 [×2], He35D4
Quotients: C1, C2 [×3], C22, S3 [×4], D4, D6 [×4], C3⋊S3, D12 [×4], C2×C3⋊S3, He3⋊C2, C12⋊S3, C2×He3⋊C2, He35D4

Smallest permutation representation of He35D4
On 36 points
Generators in S36
(1 16 24)(2 13 21)(3 14 22)(4 15 23)(5 25 35)(6 26 36)(7 27 33)(8 28 34)(9 18 29)(10 19 30)(11 20 31)(12 17 32)
(1 7 29)(2 8 30)(3 5 31)(4 6 32)(9 16 27)(10 13 28)(11 14 25)(12 15 26)(17 23 36)(18 24 33)(19 21 34)(20 22 35)
(1 33 27)(2 34 28)(3 35 25)(4 36 26)(5 20 11)(6 17 12)(7 18 9)(8 19 10)(13 30 21)(14 31 22)(15 32 23)(16 29 24)
(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 17)(10 20)(11 19)(12 18)(13 22)(14 21)(15 24)(16 23)(25 34)(26 33)(27 36)(28 35)(29 32)(30 31)

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

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

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

He35D4 is a maximal subgroup of
He33D8  He34SD16  He37SD16  He35D8  He37D8  He311SD16  C12.S32  C3⋊S3⋊D12  C62.47D6  D4×He3⋊C2  He35D4⋊C2
He35D4 is a maximal quotient of
He37SD16  He35D8  He35Q16  C62.30D6  C62.31D6

31 conjugacy classes

class 1 2A2B2C3A3B3C3D3E3F 4 6A6B6C6D6E6F6G6H6I6J12A12B12C···12J
order122233333346666666666121212···12
size111818116666211666618181818226···6

31 irreducible representations

dim1112222336
type+++++++
imageC1C2C2S3D4D6D12He3⋊C2C2×He3⋊C2He35D4
kernelHe35D4C4×He3C2×He3⋊C2C3×C12He3C3×C6C32C4C2C1
# reps1124148442

Matrix representation of He35D4 in GL5(𝔽13)

012000
112000
00010
0012128
00001
,
10000
01000
00900
00090
00009
,
012000
112000
00101011
00900
00433
,
37000
610000
00100
00010
00001
,
610000
37000
00100
0012128
00001

G:=sub<GL(5,GF(13))| [0,1,0,0,0,12,12,0,0,0,0,0,0,12,0,0,0,1,12,0,0,0,0,8,1],[1,0,0,0,0,0,1,0,0,0,0,0,9,0,0,0,0,0,9,0,0,0,0,0,9],[0,1,0,0,0,12,12,0,0,0,0,0,10,9,4,0,0,10,0,3,0,0,11,0,3],[3,6,0,0,0,7,10,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[6,3,0,0,0,10,7,0,0,0,0,0,1,12,0,0,0,0,12,0,0,0,0,8,1] >;

He35D4 in GAP, Magma, Sage, TeX

{\rm He}_3\rtimes_5D_4
% in TeX

G:=Group("He3:5D4");
// GroupNames label

G:=SmallGroup(216,68);
// by ID

G=gap.SmallGroup(216,68);
# by ID

G:=PCGroup([6,-2,-2,-2,-3,-3,-3,73,31,387,1444,382]);
// 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,b*e=e*b,c*d=d*c,e*c*e=c^-1,e*d*e=d^-1>;
// generators/relations

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