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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), C2.8C26, C4.18C25, D4○2- (1+4), Q8○2+ (1+4), D4.14C24, C22.4C25, Q8.14C24, C23.100C24, 2- (1+4)⋊14C22, 2+ (1+4)⋊15C22, C2.C25⋊9C2, (C2×C4).152C24, C4○D4.34C23, (C2×D4).493C23, (C2×Q8).476C23, (C22×Q8)⋊50C22, (C22×C4).419C23, (C2×2- (1+4))⋊14C2, (C2×C4○D4)⋊62C22, SmallGroup(128,2327)

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

Derived series
C1 — C2 — 2- (1+6)
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C22×Q8 — 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: 3060 in 2898 conjugacy classes, 2826 normal (3 characteristic)
C1, C2, C2 [×27], C4 [×36], C22 [×27], C22 [×45], C2×C4 [×270], D4 [×216], Q8 [×120], C23 [×45], C22×C4 [×135], C2×D4 [×270], C2×Q8 [×270], C4○D4 [×720], C22×Q8 [×45], C2×C4○D4 [×270], 2+ (1+4) [×120], 2- (1+4) [×216], C2×2- (1+4) [×27], C2.C25 [×36], 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=f2=g2=1, d2=e2=a, cbc=gbg=ab=ba, fcf=ac=ca, ede-1=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 >

Smallest permutation representation
►On 32 points
Generators in S32
(1 3)(2 4)(5 7)(6 8)(9 11)(10 12)(13 15)(14 16)(17 19)(18 20)(21 23)(22 24)(25 27)(26 28)(29 31)(30 32)

(1 26)(2 27)(3 28)(4 25)(5 31)(6 32)(7 29)(8 30)(9 14)(10 15)(11 16)(12 13)(17 22)(18 23)(19 24)(20 21)

(1 22)(2 23)(3 24)(4 21)(5 11)(6 12)(7 9)(8 10)(13 30)(14 31)(15 32)(16 29)(17 28)(18 25)(19 26)(20 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 28)(29 30 31 32)

(1 8 3 6)(2 7 4 5)(9 21 11 23)(10 24 12 22)(13 17 15 19)(14 20 16 18)(25 31 27 29)(26 30 28 32)

(1 9)(2 10)(3 11)(4 12)(5 22)(6 23)(7 24)(8 21)(13 25)(14 26)(15 27)(16 28)(17 31)(18 32)(19 29)(20 30)

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

Matrix representation ►G ⊆ GL8(𝔽5)

40000000
04000000
00400000
00040000
00004000
00000400
00000040
00000004
,
00304000
00020100
30000040
02000001
30000020
02000003
00302000
00020300
,
00030000
00300000
02000000
20000000
02000003
20000030
00030200
00302000
,
00030400
00304000
03000004
30000040
00000002
00000020
00000200
00002000
,
00103000
00010300
10000030
01000003
10000040
01000004
00104000
00010400
,
00100000
00010000
10000000
01000000
00000010
00000001
00001000
00000100
,
01000000
10000000
00010000
00100000
00000100
00001000
00000001
00000010

G:=sub<GL(8,GF(5))| [4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[0,0,3,0,3,0,0,0,0,0,0,2,0,2,0,0,3,0,0,0,0,0,3,0,0,2,0,0,0,0,0,2,4,0,0,0,0,0,2,0,0,1,0,0,0,0,0,3,0,0,4,0,2,0,0,0,0,0,0,1,0,3,0,0],[0,0,0,2,0,2,0,0,0,0,2,0,2,0,0,0,0,3,0,0,0,0,0,3,3,0,0,0,0,0,3,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,2,0,0,0,0,0,0,3,0,0,0,0,0,0,3,0,0,0],[0,0,0,3,0,0,0,0,0,0,3,0,0,0,0,0,0,3,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,4,0,0,0,0,0,2,4,0,0,0,0,0,2,0,0,0,0,4,0,2,0,0,0,0,4,0,2,0,0,0],[0,0,1,0,1,0,0,0,0,0,0,1,0,1,0,0,1,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,3,0,0,0,0,0,4,0,0,3,0,0,0,0,0,4,0,0,3,0,4,0,0,0,0,0,0,3,0,4,0,0],[0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[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] >;
 

65 conjugacy classes

class 1 2A2B···2AB4A···4AJ
order122···24···4
size112···22···2

65 irreducible representations

dim1118
type+++-
imageC1C2C22- (1+6)
kernel2- (1+6)C2×2- (1+4)C2.C25C1
# reps127361

In GAP, Magma, Sage, TeX 

2_-^{(1+6)}
 
% in TeX
 
G:=Group("ES-(2,3)");
 
// GroupNames label
 
G:=SmallGroup(128,2327);
 
// by ID
 
G=gap.SmallGroup(128,2327);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,2,2,-2,925,352,521,248,1411,4037]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=f^2=g^2=1,d^2=e^2=a,c*b*c=g*b*g=a*b=b*a,f*c*f=a*c=c*a,e*d*e^-1=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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