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G = 3- 1+4  order 243 = 35

Extraspecial group

p-group, metabelian, nilpotent (class 2), monomial

Aliases: 3- 1+4, C3.5C34, C9.2C33, He3.7C32, He3○3- 1+2, C33.17C32, C32.14C33, 3- 1+2⋊6C32, 3- 1+2○3- 1+2, C9○He3⋊3C3, (C3×C9)⋊4C32, (C3×He3).8C3, (C3×3- 1+2)⋊11C3, SmallGroup(243,66)

Series: Derived ►Chief ►Lower central ►Upper central ►Jennings

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

Generators and relations for 3- 1+4
 G = < a,b,c,d,e | a3=c3=d3=e3=1, b3=a, cbc-1=ebe-1=ab=ba, dcd-1=ac=ca, ad=da, ae=ea, bd=db, ce=ec, de=ed >

Subgroups: 288 in 230 conjugacy classes, 213 normal (5 characteristic)
C1, C3, C3, C9, C32, C32, C32, C3×C9, He3, 3- 1+2, C33, C3×He3, C3×3- 1+2, C9○He3, 3- 1+4
Quotients: C1, C3, C32, C33, C34, 3- 1+4

Permutation representations of 3- 1+4
►On 27 points - transitive group 27T113
Generators in S27
(1 4 7)(2 5 8)(3 6 9)(10 13 16)(11 14 17)(12 15 18)(19 22 25)(20 23 26)(21 24 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 17 26)(2 15 21)(3 13 25)(4 11 20)(5 18 24)(6 16 19)(7 14 23)(8 12 27)(9 10 22)

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

(2 8 5)(3 6 9)(10 13 16)(12 18 15)(19 22 25)(21 27 24)
 
G:=sub<Sym(27)| (1,4,7)(2,5,8)(3,6,9)(10,13,16)(11,14,17)(12,15,18)(19,22,25)(20,23,26)(21,24,27), (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), (1,17,26)(2,15,21)(3,13,25)(4,11,20)(5,18,24)(6,16,19)(7,14,23)(8,12,27)(9,10,22), (1,20,17)(2,21,18)(3,22,10)(4,23,11)(5,24,12)(6,25,13)(7,26,14)(8,27,15)(9,19,16), (2,8,5)(3,6,9)(10,13,16)(12,18,15)(19,22,25)(21,27,24)>;
 
G:=Group( (1,4,7)(2,5,8)(3,6,9)(10,13,16)(11,14,17)(12,15,18)(19,22,25)(20,23,26)(21,24,27), (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), (1,17,26)(2,15,21)(3,13,25)(4,11,20)(5,18,24)(6,16,19)(7,14,23)(8,12,27)(9,10,22), (1,20,17)(2,21,18)(3,22,10)(4,23,11)(5,24,12)(6,25,13)(7,26,14)(8,27,15)(9,19,16), (2,8,5)(3,6,9)(10,13,16)(12,18,15)(19,22,25)(21,27,24) );
 
G=PermutationGroup([[(1,4,7),(2,5,8),(3,6,9),(10,13,16),(11,14,17),(12,15,18),(19,22,25),(20,23,26),(21,24,27)], [(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)], [(1,17,26),(2,15,21),(3,13,25),(4,11,20),(5,18,24),(6,16,19),(7,14,23),(8,12,27),(9,10,22)], [(1,20,17),(2,21,18),(3,22,10),(4,23,11),(5,24,12),(6,25,13),(7,26,14),(8,27,15),(9,19,16)], [(2,8,5),(3,6,9),(10,13,16),(12,18,15),(19,22,25),(21,27,24)]])
 
G:=TransitiveGroup(27,113);
 

3- 1+4 is a maximal subgroup of   3- 1+4⋊C2  3- 1+4⋊2C2

83 conjugacy classes

class 1 3A3B3C···3AB9A···9BB
order1333···39···9
size1113···33···3

83 irreducible representations

dim11119
type+
imageC1C3C3C33- 1+4
kernel3- 1+4C3×He3C3×3- 1+2C9○He3C1
# reps1224542

Matrix representation of 3- 1+4 ►in GL9(𝔽19)

1100000000
0110000000
0011000000
0001100000
0000110000
0000011000
0000001100
0000000110
0000000011
,
110011711171
000010000
000001000
000000100
000000010
000000001
912188121881218
0110000000
0011000000
,
711218011780
0011000000
68121881218812
000010000
000007000
0001100000
0000000110
000000001
000000700
,
700000000
001000000
68121881218812
000010000
000001000
000100000
000000010
000000001
000000100
,
100812181117
010000000
001000000
0001100000
0000110000
0000011000
000000700
000000070
000000007

G:=sub<GL(9,GF(19))| [11,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,0,11],[11,0,0,0,0,0,9,0,0,0,0,0,0,0,0,12,11,0,0,0,0,0,0,0,18,0,11,11,0,0,0,0,0,8,0,0,7,1,0,0,0,0,12,0,0,1,0,1,0,0,0,18,0,0,11,0,0,1,0,0,8,0,0,7,0,0,0,1,0,12,0,0,1,0,0,0,0,1,18,0,0],[7,0,6,0,0,0,0,0,0,1,0,8,0,0,0,0,0,0,12,11,12,0,0,0,0,0,0,18,0,18,0,0,11,0,0,0,0,0,8,1,0,0,0,0,0,11,0,12,0,7,0,0,0,0,7,0,18,0,0,0,0,0,7,8,0,8,0,0,0,11,0,0,0,0,12,0,0,0,0,1,0],[7,0,6,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,1,12,0,0,0,0,0,0,0,0,18,0,0,1,0,0,0,0,0,8,1,0,0,0,0,0,0,0,12,0,1,0,0,0,0,0,0,18,0,0,0,0,0,1,0,0,8,0,0,0,1,0,0,0,0,12,0,0,0,0,1,0],[1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,8,0,0,11,0,0,0,0,0,12,0,0,0,11,0,0,0,0,18,0,0,0,0,11,0,0,0,1,0,0,0,0,0,7,0,0,11,0,0,0,0,0,0,7,0,7,0,0,0,0,0,0,0,7] >;
 

3- 1+4 in GAP, Magma, Sage, TeX 

3_-^{1+4}
 
% in TeX
 
G:=Group("ES-(3,2)");
 
// GroupNames label
 
G:=SmallGroup(243,66);
 
// by ID
 
G=gap.SmallGroup(243,66);
 
# by ID
 
G:=PCGroup([5,-3,3,3,3,-3,405,841,457,2163]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^3=c^3=d^3=e^3=1,b^3=a,c*b*c^-1=e*b*e^-1=a*b=b*a,d*c*d^-1=a*c=c*a,a*d=d*a,a*e=e*a,b*d=d*b,c*e=e*c,d*e=e*d>;
 
// generators/relations
 

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