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G = 3+ 1+4  order 243 = 35

Extraspecial group

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

Aliases: 3+ 1+4, He3○He3, C3.4C34, He3⋊4C32, C33⋊3C32, C32.13C33, (C3×He3)⋊7C3, SmallGroup(243,65)

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 — 3+ 1+4

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

Subgroups: 693 in 293 conjugacy classes, 213 normal (3 characteristic)
C1, C3, C3, C32, C32, He3, C33, C3×He3, 3+ 1+4
Quotients: C1, C3, C32, C33, C34, 3+ 1+4

Permutation representations of 3+ 1+4
►On 27 points - transitive group 27T101
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 15 10)(2 13 11)(3 14 12)(4 9 26)(5 7 27)(6 8 25)(16 20 23)(17 21 24)(18 19 22)

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

(4 9 26)(5 7 27)(6 8 25)(16 23 20)(17 24 21)(18 22 19)

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

(1 9 17)(2 7 18)(3 8 16)(4 24 10)(5 22 11)(6 23 12)(13 27 19)(14 25 20)(15 26 21)
 
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), (1,15,10)(2,13,11)(3,14,12)(4,9,26)(5,7,27)(6,8,25)(16,20,23)(17,21,24)(18,19,22), (1,3,11)(2,15,14)(4,6,27)(5,9,8)(7,26,25)(10,12,13)(16,22,17)(18,21,20)(19,24,23), (4,9,26)(5,7,27)(6,8,25)(16,23,20)(17,24,21)(18,22,19), (1,10,15)(2,11,13)(3,12,14)(4,26,9)(5,27,7)(6,25,8)(16,23,20)(17,24,21)(18,22,19), (1,9,17)(2,7,18)(3,8,16)(4,24,10)(5,22,11)(6,23,12)(13,27,19)(14,25,20)(15,26,21)>;
 
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), (1,15,10)(2,13,11)(3,14,12)(4,9,26)(5,7,27)(6,8,25)(16,20,23)(17,21,24)(18,19,22), (1,3,11)(2,15,14)(4,6,27)(5,9,8)(7,26,25)(10,12,13)(16,22,17)(18,21,20)(19,24,23), (4,9,26)(5,7,27)(6,8,25)(16,23,20)(17,24,21)(18,22,19), (1,10,15)(2,11,13)(3,12,14)(4,26,9)(5,27,7)(6,25,8)(16,23,20)(17,24,21)(18,22,19), (1,9,17)(2,7,18)(3,8,16)(4,24,10)(5,22,11)(6,23,12)(13,27,19)(14,25,20)(15,26,21) );
 
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)], [(1,15,10),(2,13,11),(3,14,12),(4,9,26),(5,7,27),(6,8,25),(16,20,23),(17,21,24),(18,19,22)], [(1,3,11),(2,15,14),(4,6,27),(5,9,8),(7,26,25),(10,12,13),(16,22,17),(18,21,20),(19,24,23)], [(4,9,26),(5,7,27),(6,8,25),(16,23,20),(17,24,21),(18,22,19)], [(1,10,15),(2,11,13),(3,12,14),(4,26,9),(5,27,7),(6,25,8),(16,23,20),(17,24,21),(18,22,19)], [(1,9,17),(2,7,18),(3,8,16),(4,24,10),(5,22,11),(6,23,12),(13,27,19),(14,25,20),(15,26,21)]])
 
G:=TransitiveGroup(27,101);
 

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

83 conjugacy classes

class 1 3A3B3C···3CD
order1333···3
size1113···3

83 irreducible representations

dim119
type+
imageC1C33+ 1+4
kernel3+ 1+4C3×He3C1
# reps1802

Matrix representation of 3+ 1+4 ►in GL9(𝔽7)

000000002
000000100
000000040
001000000
400000000
020000000
000004000
000200000
000010000
,
200000000
020000000
002000000
000200000
000020000
000002000
000000200
000000020
000000002
,
001000000
100000000
010000000
000001000
000100000
000010000
000000001
000000100
000000010
,
000001000
000100000
000010000
000000001
000000100
000000010
001000000
100000000
010000000
,
400000000
040000000
004000000
000400000
000040000
000004000
000000400
000000040
000000004
,
000000004
000000400
000000040
001000000
100000000
010000000
000002000
000200000
000020000

G:=sub<GL(9,GF(7))| [0,0,0,0,4,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,1,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0],[2,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,2],[0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0],[4,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,4],[0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,2,0,0,0,4,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0] >;
 

3+ 1+4 in GAP, Magma, Sage, TeX 

3_+^{1+4}
 
% in TeX
 
G:=Group("ES+(3,2)");
 
// GroupNames label
 
G:=SmallGroup(243,65);
 
// by ID
 
G=gap.SmallGroup(243,65);
 
# by ID
 
G:=PCGroup([5,-3,3,3,3,-3,841,457,2163]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f|a^3=b^3=c^3=d^3=f^3=1,e^1=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,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,d*e=e*d,e*f=f*e>;
 
// generators/relations
 

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