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G = 3+ 1+4⋊C2  order 486 = 2·35

1st semidirect product of 3+ 1+4 and C2 acting faithfully

metabelian, supersoluble, monomial

Aliases: 3+ 1+4⋊1C2, He3⋊9(C3×S3), C33⋊5(C3×S3), C33⋊3(C3×C6), C33⋊3(C3⋊S3), (C3×He3)⋊17S3, (C3×He3)⋊11C6, He3⋊4S3⋊3C3, C33⋊C2⋊3C32, C32.20(S3×C32), C3.5(C32×C3⋊S3), C32.13(C3×C3⋊S3), SmallGroup(486,236)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C33 — 3+ 1+4⋊C2
Chief series
C1 — C3 — C32 — C33 — C3×He3 — 3+ 1+4 — 3+ 1+4⋊C2
Lower central
C33 — 3+ 1+4⋊C2
Upper central
C1

Generators and relations for 3+ 1+4⋊C2
 G = < a,b,c,d,e,f,g | a3=b3=c3=d3=f3=g2=1, e1=gbg=b-1, ab=ba, cac-1=ab-1, ad=da, ae=ea, af=fa, ag=ga, bc=cb, fdf-1=bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, gcg=a-1bc-1, de=ed, gdg=d-1, ef=fe, geg=b, fg=gf >

Subgroups: 1230 in 223 conjugacy classes, 43 normal (7 characteristic)
C1, C2, C3, C3, S3, C6, C32, C32, C3×S3, C3⋊S3, C3×C6, He3, He3, C33, C33, C33, C32⋊C6, S3×C32, C3×C3⋊S3, C33⋊C2, C3×He3, C3×He3, C3×C32⋊C6, He3⋊4S3, 3+ 1+4, 3+ 1+4⋊C2
Quotients: C1, C2, C3, S3, C6, C32, C3×S3, C3⋊S3, C3×C6, S3×C32, C3×C3⋊S3, C32×C3⋊S3, 3+ 1+4⋊C2

Permutation representations of 3+ 1+4⋊C2
►On 27 points - transitive group 27T162
Generators in S27
(10 11 12)(13 14 15)(16 17 18)(19 20 21)(22 23 24)(25 26 27)

(1 6 5)(2 3 9)(4 7 8)(10 11 12)(13 14 15)(16 18 17)(19 21 20)(22 23 24)(25 27 26)

(1 20 15)(2 16 12)(3 18 10)(4 26 24)(5 21 14)(6 19 13)(7 25 22)(8 27 23)(9 17 11)

(1 4 9)(2 6 7)(3 5 8)(10 14 23)(11 15 24)(12 13 22)(16 19 25)(17 20 26)(18 21 27)

(1 5 6)(2 9 3)(4 8 7)(10 12 11)(13 15 14)(16 17 18)(19 20 21)(22 24 23)(25 26 27)

(1 5 6)(4 7 8)(13 15 14)(19 20 21)(22 23 24)(25 27 26)

(1 4)(2 3)(5 7)(6 8)(10 18)(11 16)(12 17)(13 26)(14 27)(15 25)(19 24)(20 22)(21 23)
 
G:=sub<Sym(27)| (10,11,12)(13,14,15)(16,17,18)(19,20,21)(22,23,24)(25,26,27), (1,6,5)(2,3,9)(4,7,8)(10,11,12)(13,14,15)(16,18,17)(19,21,20)(22,23,24)(25,27,26), (1,20,15)(2,16,12)(3,18,10)(4,26,24)(5,21,14)(6,19,13)(7,25,22)(8,27,23)(9,17,11), (1,4,9)(2,6,7)(3,5,8)(10,14,23)(11,15,24)(12,13,22)(16,19,25)(17,20,26)(18,21,27), (1,5,6)(2,9,3)(4,8,7)(10,12,11)(13,15,14)(16,17,18)(19,20,21)(22,24,23)(25,26,27), (1,5,6)(4,7,8)(13,15,14)(19,20,21)(22,23,24)(25,27,26), (1,4)(2,3)(5,7)(6,8)(10,18)(11,16)(12,17)(13,26)(14,27)(15,25)(19,24)(20,22)(21,23)>;
 
G:=Group( (10,11,12)(13,14,15)(16,17,18)(19,20,21)(22,23,24)(25,26,27), (1,6,5)(2,3,9)(4,7,8)(10,11,12)(13,14,15)(16,18,17)(19,21,20)(22,23,24)(25,27,26), (1,20,15)(2,16,12)(3,18,10)(4,26,24)(5,21,14)(6,19,13)(7,25,22)(8,27,23)(9,17,11), (1,4,9)(2,6,7)(3,5,8)(10,14,23)(11,15,24)(12,13,22)(16,19,25)(17,20,26)(18,21,27), (1,5,6)(2,9,3)(4,8,7)(10,12,11)(13,15,14)(16,17,18)(19,20,21)(22,24,23)(25,26,27), (1,5,6)(4,7,8)(13,15,14)(19,20,21)(22,23,24)(25,27,26), (1,4)(2,3)(5,7)(6,8)(10,18)(11,16)(12,17)(13,26)(14,27)(15,25)(19,24)(20,22)(21,23) );
 
G=PermutationGroup([[(10,11,12),(13,14,15),(16,17,18),(19,20,21),(22,23,24),(25,26,27)], [(1,6,5),(2,3,9),(4,7,8),(10,11,12),(13,14,15),(16,18,17),(19,21,20),(22,23,24),(25,27,26)], [(1,20,15),(2,16,12),(3,18,10),(4,26,24),(5,21,14),(6,19,13),(7,25,22),(8,27,23),(9,17,11)], [(1,4,9),(2,6,7),(3,5,8),(10,14,23),(11,15,24),(12,13,22),(16,19,25),(17,20,26),(18,21,27)], [(1,5,6),(2,9,3),(4,8,7),(10,12,11),(13,15,14),(16,17,18),(19,20,21),(22,24,23),(25,26,27)], [(1,5,6),(4,7,8),(13,15,14),(19,20,21),(22,23,24),(25,27,26)], [(1,4),(2,3),(5,7),(6,8),(10,18),(11,16),(12,17),(13,26),(14,27),(15,25),(19,24),(20,22),(21,23)]])
 
G:=TransitiveGroup(27,162);
 

►On 27 points - transitive group 27T181
Generators in S27
(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)

(1 8 26)(2 9 27)(3 7 25)(4 23 19)(5 24 20)(6 22 21)(10 15 18)(11 13 16)(12 14 17)

(1 9 3)(2 25 26)(4 20 21)(5 6 23)(7 8 27)(10 16 17)(11 12 15)(13 14 18)(19 24 22)

(4 19 23)(5 20 24)(6 21 22)(10 15 18)(11 13 16)(12 14 17)

(1 26 8)(2 27 9)(3 25 7)(4 19 23)(5 20 24)(6 21 22)(10 18 15)(11 16 13)(12 17 14)

(1 14 6)(2 15 4)(3 13 5)(7 16 24)(8 17 22)(9 18 23)(10 19 27)(11 20 25)(12 21 26)

(7 25)(8 26)(9 27)(10 18)(11 16)(12 17)(19 23)(20 24)(21 22)
 
G:=sub<Sym(27)| (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), (1,8,26)(2,9,27)(3,7,25)(4,23,19)(5,24,20)(6,22,21)(10,15,18)(11,13,16)(12,14,17), (1,9,3)(2,25,26)(4,20,21)(5,6,23)(7,8,27)(10,16,17)(11,12,15)(13,14,18)(19,24,22), (4,19,23)(5,20,24)(6,21,22)(10,15,18)(11,13,16)(12,14,17), (1,26,8)(2,27,9)(3,25,7)(4,19,23)(5,20,24)(6,21,22)(10,18,15)(11,16,13)(12,17,14), (1,14,6)(2,15,4)(3,13,5)(7,16,24)(8,17,22)(9,18,23)(10,19,27)(11,20,25)(12,21,26), (7,25)(8,26)(9,27)(10,18)(11,16)(12,17)(19,23)(20,24)(21,22)>;
 
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), (1,8,26)(2,9,27)(3,7,25)(4,23,19)(5,24,20)(6,22,21)(10,15,18)(11,13,16)(12,14,17), (1,9,3)(2,25,26)(4,20,21)(5,6,23)(7,8,27)(10,16,17)(11,12,15)(13,14,18)(19,24,22), (4,19,23)(5,20,24)(6,21,22)(10,15,18)(11,13,16)(12,14,17), (1,26,8)(2,27,9)(3,25,7)(4,19,23)(5,20,24)(6,21,22)(10,18,15)(11,16,13)(12,17,14), (1,14,6)(2,15,4)(3,13,5)(7,16,24)(8,17,22)(9,18,23)(10,19,27)(11,20,25)(12,21,26), (7,25)(8,26)(9,27)(10,18)(11,16)(12,17)(19,23)(20,24)(21,22) );
 
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)], [(1,8,26),(2,9,27),(3,7,25),(4,23,19),(5,24,20),(6,22,21),(10,15,18),(11,13,16),(12,14,17)], [(1,9,3),(2,25,26),(4,20,21),(5,6,23),(7,8,27),(10,16,17),(11,12,15),(13,14,18),(19,24,22)], [(4,19,23),(5,20,24),(6,21,22),(10,15,18),(11,13,16),(12,14,17)], [(1,26,8),(2,27,9),(3,25,7),(4,19,23),(5,20,24),(6,21,22),(10,18,15),(11,16,13),(12,17,14)], [(1,14,6),(2,15,4),(3,13,5),(7,16,24),(8,17,22),(9,18,23),(10,19,27),(11,20,25),(12,21,26)], [(7,25),(8,26),(9,27),(10,18),(11,16),(12,17),(19,23),(20,24),(21,22)]])
 
G:=TransitiveGroup(27,181);
 

►On 27 points - transitive group 27T196
Generators in S27
(10 11 12)(13 14 15)(16 17 18)(19 20 21)(22 23 24)(25 26 27)

(1 9 7)(2 3 8)(4 6 5)(10 11 12)(13 14 15)(16 18 17)(19 21 20)(22 23 24)(25 27 26)

(1 16 12)(2 21 15)(3 20 13)(4 27 23)(5 25 22)(6 26 24)(7 17 11)(8 19 14)(9 18 10)

(1 9 7)(4 5 6)(10 11 12)(16 18 17)(22 24 23)(25 26 27)

(1 7 9)(2 8 3)(4 5 6)(10 12 11)(13 15 14)(16 17 18)(19 20 21)(22 24 23)(25 26 27)

(1 6 8)(2 9 5)(3 7 4)(10 22 15)(11 23 13)(12 24 14)(16 26 19)(17 27 20)(18 25 21)

(2 3)(4 5)(7 9)(10 16)(11 17)(12 18)(13 20)(14 21)(15 19)(22 26)(23 27)(24 25)
 
G:=sub<Sym(27)| (10,11,12)(13,14,15)(16,17,18)(19,20,21)(22,23,24)(25,26,27), (1,9,7)(2,3,8)(4,6,5)(10,11,12)(13,14,15)(16,18,17)(19,21,20)(22,23,24)(25,27,26), (1,16,12)(2,21,15)(3,20,13)(4,27,23)(5,25,22)(6,26,24)(7,17,11)(8,19,14)(9,18,10), (1,9,7)(4,5,6)(10,11,12)(16,18,17)(22,24,23)(25,26,27), (1,7,9)(2,8,3)(4,5,6)(10,12,11)(13,15,14)(16,17,18)(19,20,21)(22,24,23)(25,26,27), (1,6,8)(2,9,5)(3,7,4)(10,22,15)(11,23,13)(12,24,14)(16,26,19)(17,27,20)(18,25,21), (2,3)(4,5)(7,9)(10,16)(11,17)(12,18)(13,20)(14,21)(15,19)(22,26)(23,27)(24,25)>;
 
G:=Group( (10,11,12)(13,14,15)(16,17,18)(19,20,21)(22,23,24)(25,26,27), (1,9,7)(2,3,8)(4,6,5)(10,11,12)(13,14,15)(16,18,17)(19,21,20)(22,23,24)(25,27,26), (1,16,12)(2,21,15)(3,20,13)(4,27,23)(5,25,22)(6,26,24)(7,17,11)(8,19,14)(9,18,10), (1,9,7)(4,5,6)(10,11,12)(16,18,17)(22,24,23)(25,26,27), (1,7,9)(2,8,3)(4,5,6)(10,12,11)(13,15,14)(16,17,18)(19,20,21)(22,24,23)(25,26,27), (1,6,8)(2,9,5)(3,7,4)(10,22,15)(11,23,13)(12,24,14)(16,26,19)(17,27,20)(18,25,21), (2,3)(4,5)(7,9)(10,16)(11,17)(12,18)(13,20)(14,21)(15,19)(22,26)(23,27)(24,25) );
 
G=PermutationGroup([[(10,11,12),(13,14,15),(16,17,18),(19,20,21),(22,23,24),(25,26,27)], [(1,9,7),(2,3,8),(4,6,5),(10,11,12),(13,14,15),(16,18,17),(19,21,20),(22,23,24),(25,27,26)], [(1,16,12),(2,21,15),(3,20,13),(4,27,23),(5,25,22),(6,26,24),(7,17,11),(8,19,14),(9,18,10)], [(1,9,7),(4,5,6),(10,11,12),(16,18,17),(22,24,23),(25,26,27)], [(1,7,9),(2,8,3),(4,5,6),(10,12,11),(13,15,14),(16,17,18),(19,20,21),(22,24,23),(25,26,27)], [(1,6,8),(2,9,5),(3,7,4),(10,22,15),(11,23,13),(12,24,14),(16,26,19),(17,27,20),(18,25,21)], [(2,3),(4,5),(7,9),(10,16),(11,17),(12,18),(13,20),(14,21),(15,19),(22,26),(23,27),(24,25)]])
 
G:=TransitiveGroup(27,196);
 

55 conjugacy classes

class 1  2 3A3B···3I3J···3AS6A···6H
order1233···33···36···6
size12723···36···627···27

55 irreducible representations

dim111118222
type++++
imageC1C2C3C63+ 1+4⋊C2S3C3×S3C3×S3
kernel3+ 1+4⋊C23+ 1+4He3⋊4S3C3×He3C1C3×He3He3C33
# reps118814248

Matrix representation of 3+ 1+4⋊C2 ►in GL18(ℤ)

000000100000000000
000000010000000000
000000001000000000
000000000100000000
000000000010000000
000000000001000000
000000000000100000
000000000000010000
000000000000001000
000000000000000100
000000000000000010
000000000000000001
100000000000000000
010000000000000000
001000000000000000
000100000000000000
000010000000000000
000001000000000000
,
-1-10000000000000000
100000000000000000
00-1-100000000000000
001000000000000000
0000-1-1000000000000
000010000000000000
000000-1-10000000000
000000100000000000
00000000-1-100000000
000000001000000000
0000000000-1-1000000
000000000010000000
000000000000-1-10000
000000000000100000
00000000000000-1-100
000000000000001000
0000000000000000-1-1
000000000000000010
,
000000100000000000
000000010000000000
000000001000000000
000000000100000000
000000000010000000
000000000001000000
000000000000010000
000000000000-1-10000
000000000000000100
00000000000000-1-100
000000000000000001
0000000000000000-1-1
-1-10000000000000000
100000000000000000
00-1-100000000000000
001000000000000000
0000-1-1000000000000
000010000000000000
,
000010000000000000
000001000000000000
100000000000000000
010000000000000000
001000000000000000
000100000000000000
000000000010000000
000000000001000000
000000100000000000
000000010000000000
000000001000000000
000000000100000000
000000000000000010
000000000000000001
000000000000100000
000000000000010000
000000000000001000
000000000000000100
,
010000000000000000
-1-10000000000000000
000100000000000000
00-1-100000000000000
000001000000000000
0000-1-1000000000000
000000010000000000
000000-1-10000000000
000000000100000000
00000000-1-100000000
000000000001000000
0000000000-1-1000000
000000000000010000
000000000000-1-10000
000000000000000100
00000000000000-1-100
000000000000000001
0000000000000000-1-1
,
100000000000000000
010000000000000000
000100000000000000
00-1-100000000000000
0000-1-1000000000000
000010000000000000
000000100000000000
000000010000000000
000000000100000000
00000000-1-100000000
0000000000-1-1000000
000000000010000000
000000000000100000
000000000000010000
000000000000000100
00000000000000-1-100
0000000000000000-1-1
000000000000000010
,
100000000000000000
-1-10000000000000000
000010000000000000
0000-1-1000000000000
001000000000000000
00-1-100000000000000
000000100000000000
000000-1-10000000000
000000000010000000
0000000000-1-1000000
000000001000000000
00000000-1-100000000
000000000000100000
000000000000-1-10000
000000000000000010
0000000000000000-1-1
000000000000001000
00000000000000-1-100

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3+ 1+4⋊C2 in GAP, Magma, Sage, TeX 

3_+^{1+4}\rtimes C_2
 
% in TeX
 
G:=Group("ES+(3,2):C2");
 
// GroupNames label
 
G:=SmallGroup(486,236);
 
// by ID
 
G=gap.SmallGroup(486,236);
 
# by ID
 
G:=PCGroup([6,-2,-3,-3,-3,-3,-3,867,735,3244,3250,11669]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^3=b^3=c^3=d^3=f^3=g^2=1,e^1=g*b*g=b^-1,a*b=b*a,c*a*c^-1=a*b^-1,a*d=d*a,a*e=e*a,a*f=f*a,a*g=g*a,b*c=c*b,f*d*f^-1=b*d=d*b,b*e=e*b,b*f=f*b,c*d=d*c,c*e=e*c,c*f=f*c,g*c*g=a^-1*b*c^-1,d*e=e*d,g*d*g=d^-1,e*f=f*e,g*e*g=b,f*g=g*f>;
 
// generators/relations
 

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