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G = 2+ (1+6)  order 128 = 27

Extraspecial group

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

Aliases: 2+ (1+6), C2.7C26, C4.17C25, C24⋊5C23, Q8○2- (1+4), D4○2+ (1+4), D4.13C24, C22.3C25, Q8.13C24, C23.99C24, 2+ (1+4)⋊14C22, 2- (1+4)⋊13C22, C4○D4⋊9C23, (C2×D4)⋊24C23, (C22×C4)⋊6C23, (C2×Q8)⋊25C23, (C2×C4).151C24, C2.C25⋊8C2, (C22×D4)⋊52C22, (C2×2+ (1+4))⋊16C2, (C2×C4○D4)⋊61C22, SmallGroup(128,2326)

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

Derived series
C1 — C2 — 2+ (1+6)
Chief series
C1 — C2 — C22 — C23 — C24 — C22×D4 — C2×2+ (1+4) — 2+ (1+6)
Lower central
C1 — C2 — 2+ (1+6)
Upper central
C1 — C2 — 2+ (1+6)
Jennings
C1 — C2 — 2+ (1+6)

Subgroups: 3556 in 2996 conjugacy classes, 2826 normal (3 characteristic)
C1, C2, C2 [×35], C4 [×28], C22 [×35], C22 [×105], C2×C4 [×210], D4 [×280], Q8 [×56], C23 [×105], C23 [×30], C22×C4 [×105], C2×D4 [×630], C2×Q8 [×70], C4○D4 [×560], C24 [×30], C22×D4 [×105], C2×C4○D4 [×210], 2+ (1+4) [×280], 2- (1+4) [×56], C2×2+ (1+4) [×35], C2.C25 [×28], 2+ (1+6)

Quotients:
C1, C2 [×63], C22 [×651], C23 [×1395], C24 [×651], C25 [×63], C26, 2+ (1+6)

Generators and relations
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=e2=f2=g2=1, cbc=gbg=ab=ba, fcf=ac=ca, ede=ad=da, ae=ea, af=fa, ag=ga, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cg=gc, df=fd, dg=gd, ef=fe, eg=ge, fg=gf >

Permutation representations
►On 16 points - transitive group 16T197
Generators in S16
(1 2)(3 4)(5 6)(7 8)(9 10)(11 12)(13 14)(15 16)

(1 15)(2 16)(3 14)(4 13)(5 11)(6 12)(7 10)(8 9)

(1 9)(2 10)(3 11)(4 12)(5 13)(6 14)(7 15)(8 16)

(1 5)(2 6)(3 7)(4 8)(9 13)(10 14)(11 15)(12 16)

(1 3)(2 4)(5 8)(6 7)(9 11)(10 12)(13 16)(14 15)

(1 3)(2 4)(5 7)(6 8)(9 12)(10 11)(13 16)(14 15)

(1 3)(2 4)(5 7)(6 8)(9 11)(10 12)(13 15)(14 16)
 
G:=sub<Sym(16)| (1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16), (1,15)(2,16)(3,14)(4,13)(5,11)(6,12)(7,10)(8,9), (1,9)(2,10)(3,11)(4,12)(5,13)(6,14)(7,15)(8,16), (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16), (1,3)(2,4)(5,8)(6,7)(9,11)(10,12)(13,16)(14,15), (1,3)(2,4)(5,7)(6,8)(9,12)(10,11)(13,16)(14,15), (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,15)(14,16)>;
 
G:=Group( (1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16), (1,15)(2,16)(3,14)(4,13)(5,11)(6,12)(7,10)(8,9), (1,9)(2,10)(3,11)(4,12)(5,13)(6,14)(7,15)(8,16), (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16), (1,3)(2,4)(5,8)(6,7)(9,11)(10,12)(13,16)(14,15), (1,3)(2,4)(5,7)(6,8)(9,12)(10,11)(13,16)(14,15), (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,15)(14,16) );
 
G=PermutationGroup([(1,2),(3,4),(5,6),(7,8),(9,10),(11,12),(13,14),(15,16)], [(1,15),(2,16),(3,14),(4,13),(5,11),(6,12),(7,10),(8,9)], [(1,9),(2,10),(3,11),(4,12),(5,13),(6,14),(7,15),(8,16)], [(1,5),(2,6),(3,7),(4,8),(9,13),(10,14),(11,15),(12,16)], [(1,3),(2,4),(5,8),(6,7),(9,11),(10,12),(13,16),(14,15)], [(1,3),(2,4),(5,7),(6,8),(9,12),(10,11),(13,16),(14,15)], [(1,3),(2,4),(5,7),(6,8),(9,11),(10,12),(13,15),(14,16)])
 
G:=TransitiveGroup(16,197);
 

Matrix representation ►G ⊆ GL8(ℤ)

-10000000
0-1000000
00-100000
000-10000
0000-1000
00000-100
000000-10
0000000-1
,
00000001
000000-10
00000100
0000-1000
000-10000
00100000
0-1000000
10000000
,
00001000
00000100
00000010
00000001
10000000
01000000
00100000
00010000
,
01000000
10000000
000-10000
00-100000
00000100
00001000
0000000-1
000000-10
,
00100000
00010000
10000000
01000000
00000010
00000001
00001000
00000100
,
01000000
10000000
00010000
00100000
00000-100
0000-1000
0000000-1
000000-10
,
01000000
10000000
00010000
00100000
00000100
00001000
00000001
00000010

G:=sub<GL(8,Integers())| [-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1],[0,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,0],[0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0],[0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0] >;
 

65 conjugacy classes

class 1 2A2B···2AJ4A···4AB
order122···24···4
size112···22···2

65 irreducible representations

dim1118
type++++
imageC1C2C22+ (1+6)
kernel2+ (1+6)C2×2+ (1+4)C2.C25C1
# reps135281

In GAP, Magma, Sage, TeX 

2_+^{(1+6)}
 
% in TeX
 
G:=Group("ES+(2,3)");
 
// GroupNames label
 
G:=SmallGroup(128,2326);
 
// by ID
 
G=gap.SmallGroup(128,2326);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,2,2,-2,925,521,1411,4037]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=e^2=f^2=g^2=1,c*b*c=g*b*g=a*b=b*a,f*c*f=a*c=c*a,e*d*e=a*d=d*a,a*e=e*a,a*f=f*a,a*g=g*a,b*d=d*b,b*e=e*b,b*f=f*b,c*d=d*c,c*e=e*c,c*g=g*c,d*f=f*d,d*g=g*d,e*f=f*e,e*g=g*e,f*g=g*f>;
 
// generators/relations
 

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