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G = 2- 1+4:5C4  order 128 = 27

4th semidirect product of 2- 1+4 and C4 acting via C4/C2=C2

p-group, metabelian, nilpotent (class 3), monomial

Aliases: 2- 1+4:5C4, 2+ 1+4:6C4, C42.671C23, M4(2).26C23, D4oC4wrC2, Q8oC4wrC2, C4oD4.57D4, C4wrC2:16C22, (C2xD4).291D4, C4.12(C23xC4), (C2xQ8).226D4, Q8oM4(2):13C2, (C2xC4).182C24, (C2xC42):35C22, C4oD4.21C23, D4.23(C22xC4), C4.182(C22xD4), C23.229(C2xD4), Q8.23(C22xC4), D4.21(C22:C4), C2.C25.3C2, Q8.21(C22:C4), C22.29(C22xD4), C42:C2:77C22, C42:C22:19C2, (C2xM4(2)):43C22, (C22xC4).904C23, C4oD4oC4wrC2, C4oD4:3(C2xC4), (C4xC4oD4):2C2, (C2xC4wrC2):30C2, (C2xD4):25(C2xC4), (C2xQ8):19(C2xC4), (C2xC4).448(C2xD4), C4.35(C2xC22:C4), (C2xC4).64(C22xC4), C22.8(C2xC22:C4), (C2xC4oD4).86C22, C2.44(C22xC22:C4), SmallGroup(128,1633)

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

Derived series
C1 — C4 — 2- 1+4:5C4
Chief series
C1 — C2 — C4 — C2xC4 — C22xC4 — C2xC4oD4 — C2.C25 — 2- 1+4:5C4
Lower central
C1 — C2 — C4 — 2- 1+4:5C4
Upper central
C1 — C4 — C2xC4oD4 — 2- 1+4:5C4
Jennings
C1 — C2 — C2 — C2xC4 — 2- 1+4:5C4

Generators and relations for 2- 1+4:5C4
 G = < a,b,c,d,e | a4=b2=d2=e4=1, c2=a2, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd=a2c, ce=ec, ede-1=a2cd >

Subgroups: 636 in 369 conjugacy classes, 170 normal (13 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C22, C8, C2xC4, C2xC4, C2xC4, D4, D4, Q8, Q8, C23, C23, C42, C42, C42, C22:C4, C4:C4, C2xC8, M4(2), M4(2), C22xC4, C22xC4, C2xD4, C2xD4, C2xQ8, C2xQ8, C2xQ8, C4oD4, C4oD4, C4wrC2, C2xC42, C42:C2, C4xD4, C4xQ8, C2xM4(2), C8oD4, C2xC4oD4, C2xC4oD4, C2xC4oD4, 2+ 1+4, 2+ 1+4, 2- 1+4, 2- 1+4, C2xC4wrC2, C42:C22, C4xC4oD4, Q8oM4(2), C2.C25, 2- 1+4:5C4
Quotients: C1, C2, C4, C22, C2xC4, D4, C23, C22:C4, C22xC4, C2xD4, C24, C2xC22:C4, C23xC4, C22xD4, C22xC22:C4, 2- 1+4:5C4

Permutation representations of 2- 1+4:5C4
►On 16 points - transitive group 16T207
Generators in S16
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)

(1 3)(5 7)(10 12)(13 15)

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

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

(1 10 3 12)(2 11 4 9)(5 7)(6 8)(13 15)(14 16)
 
G:=sub<Sym(16)| (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,3)(5,7)(10,12)(13,15), (1,10,3,12)(2,11,4,9)(5,15,7,13)(6,16,8,14), (1,5)(2,6)(3,7)(4,8)(9,16)(10,13)(11,14)(12,15), (1,10,3,12)(2,11,4,9)(5,7)(6,8)(13,15)(14,16)>;
 
G:=Group( (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,3)(5,7)(10,12)(13,15), (1,10,3,12)(2,11,4,9)(5,15,7,13)(6,16,8,14), (1,5)(2,6)(3,7)(4,8)(9,16)(10,13)(11,14)(12,15), (1,10,3,12)(2,11,4,9)(5,7)(6,8)(13,15)(14,16) );
 
G=PermutationGroup([[(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16)], [(1,3),(5,7),(10,12),(13,15)], [(1,10,3,12),(2,11,4,9),(5,15,7,13),(6,16,8,14)], [(1,5),(2,6),(3,7),(4,8),(9,16),(10,13),(11,14),(12,15)], [(1,10,3,12),(2,11,4,9),(5,7),(6,8),(13,15),(14,16)]])
 
G:=TransitiveGroup(16,207);
 

44 conjugacy classes

class 1 2A2B···2H2I2J2K2L4A4B4C···4M4N···4W8A···8H
order122···22222444···44···48···8
size112···24444112···24···44···4

44 irreducible representations

dim111111112224
type+++++++++
imageC1C2C2C2C2C2C4C4D4D4D42- 1+4:5C4
kernel2- 1+4:5C4C2xC4wrC2C42:C22C4xC4oD4Q8oM4(2)C2.C252+ 1+42- 1+4C2xD4C2xQ8C4oD4C1
# reps166111883144

Matrix representation of 2- 1+4:5C4 ►in GL4(F5) generated by

2000
0300
0020
0003
,
0400
4000
0002
0030
,
2000
0200
0030
0003
,
0020
0001
3000
0100
,
3000
0300
0010
0001
G:=sub<GL(4,GF(5))| [2,0,0,0,0,3,0,0,0,0,2,0,0,0,0,3],[0,4,0,0,4,0,0,0,0,0,0,3,0,0,2,0],[2,0,0,0,0,2,0,0,0,0,3,0,0,0,0,3],[0,0,3,0,0,0,0,1,2,0,0,0,0,1,0,0],[3,0,0,0,0,3,0,0,0,0,1,0,0,0,0,1] >;
 

2- 1+4:5C4 in GAP, Magma, Sage, TeX 

2_-^{1+4}\rtimes_5C_4
 
% in TeX
 
G:=Group("ES-(2,2):5C4");
 
// GroupNames label
 
G:=SmallGroup(128,1633);
 
// by ID
 
G=gap.SmallGroup(128,1633);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,521,2804,1411,172,124]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^4=b^2=d^2=e^4=1,c^2=a^2,b*a*b=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,d*c*d=a^2*c,c*e=e*c,e*d*e^-1=a^2*c*d>;
 
// generators/relations
 

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