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G = C22.103C25  order 128 = 27

84th central stem extension by C22 of C25

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

Aliases: C42.94C23, C24.140C23, C22.103C25, C23.141C24, C4.1582+ (1+4), (D42)⋊16C2, (D4×Q8)⋊21C2, C4⋊Q8⋊94C22, D4⋊14(C4○D4), D4⋊5D4⋊23C2, Q8⋊5D4⋊20C2, D4○2(C4.4D4), (C4×D4)⋊51C22, (C2×C4).93C24, (C4×Q8)⋊50C22, C4⋊C4.497C23, C4⋊D4⋊29C22, (C2×C42)⋊65C22, C22⋊Q8⋊37C22, C22≀C2⋊11C22, C22.32C24⋊7C2, (C2×D4).476C23, C4.4D4⋊30C22, C22⋊C4.27C23, (C2×Q8).455C23, C42.C2⋊76C22, (C22×Q8)⋊35C22, C22.45C24⋊9C2, C42⋊C2⋊44C22, C22.11C24⋊22C2, C42⋊2C2⋊39C22, C4⋊1D4.188C22, (C22×C4).373C23, C2.40(C2×2+ (1+4)), C2.34(C2.C25), C22.26C24⋊43C2, (C22×D4).430C22, C22.D4⋊54C22, C23.36C23⋊34C2, C22.49C24⋊15C2, C22.53C24⋊16C2, C22.50C24⋊25C2, C22.36C24⋊17C2, (C4×C4○D4)⋊33C2, C4⋊C4○(C42.C2), C4.276(C2×C4○D4), (C2×D4)○(C4.4D4), (C2×C4.4D4)⋊56C2, (C2×C4○D4)⋊36C22, C22⋊C4○(C4.4D4), C2.59(C22×C4○D4), C22.42(C2×C4○D4), (C2×C22⋊C4)⋊52C22, SmallGroup(128,2246)

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

Derived series
C1 — C22 — C22.103C25
Chief series
C1 — C2 — C22 — C2×C4 — C22×C4 — C2×C42 — C4×C4○D4 — C22.103C25
Lower central
C1 — C22 — C22.103C25
Upper central
C1 — C22 — C22.103C25
Jennings
C1 — C22 — C22.103C25

Subgroups: 916 in 571 conjugacy classes, 390 normal (50 characteristic)
C1, C2 [×3], C2 [×11], C4 [×4], C4 [×20], C22, C22 [×4], C22 [×33], C2×C4 [×6], C2×C4 [×16], C2×C4 [×31], D4 [×4], D4 [×34], Q8 [×12], C23, C23 [×8], C23 [×16], C42 [×4], C42 [×12], C22⋊C4 [×52], C4⋊C4 [×4], C4⋊C4 [×24], C22×C4 [×3], C22×C4 [×20], C2×D4 [×2], C2×D4 [×22], C2×D4 [×12], C2×Q8 [×2], C2×Q8 [×6], C2×Q8 [×4], C4○D4 [×8], C24 [×4], C2×C42, C2×C42 [×2], C2×C22⋊C4 [×12], C42⋊C2, C42⋊C2 [×10], C4×D4 [×3], C4×D4 [×24], C4×Q8 [×3], C4×Q8 [×2], C22≀C2 [×8], C4⋊D4, C4⋊D4 [×16], C22⋊Q8, C22⋊Q8 [×12], C22.D4 [×14], C4.4D4, C4.4D4 [×20], C42.C2, C42⋊2C2 [×10], C4⋊1D4, C4⋊Q8, C4⋊Q8 [×2], C22×D4 [×4], C22×Q8 [×2], C2×C4○D4, C2×C4○D4 [×2], C4×C4○D4, C22.11C24 [×2], C2×C4.4D4 [×2], C23.36C23, C23.36C23 [×2], C22.26C24, C22.32C24 [×4], C22.36C24 [×2], D42, D4⋊5D4 [×4], Q8⋊5D4 [×2], D4×Q8, C22.45C24 [×4], C22.49C24 [×2], C22.50C24, C22.53C24, C22.103C25

Quotients:
C1, C2 [×31], C22 [×155], C23 [×155], C4○D4 [×4], C24 [×31], C2×C4○D4 [×6], 2+ (1+4) [×2], C25, C22×C4○D4, C2×2+ (1+4), C2.C25, C22.103C25

Generators and relations
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=f2=1, e2=a, g2=b, ab=ba, dcd=gcg-1=ac=ca, fdf=ad=da, ae=ea, af=fa, ag=ga, ece-1=bc=cb, bd=db, be=eb, bf=fb, bg=gb, cf=fc, de=ed, 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 15)(2 16)(3 13)(4 14)(5 19)(6 20)(7 17)(8 18)(9 25)(10 26)(11 27)(12 28)(21 29)(22 30)(23 31)(24 32)

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

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

(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)

(9 11)(10 12)(21 23)(22 24)(25 27)(26 28)(29 31)(30 32)

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

Matrix representation ►G ⊆ GL6(𝔽5)

100000
010000
004000
000400
000040
000004
,
400000
040000
001000
000100
000010
000001
,
410000
010000
000220
003021
000034
000032
,
400000
040000
000110
000100
001400
000404
,
230000
430000
002000
000200
000020
000002
,
400000
040000
001002
000102
000040
000004
,
300000
030000
000110
001010
000040
000041

G:=sub<GL(6,GF(5))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[4,0,0,0,0,0,1,1,0,0,0,0,0,0,0,3,0,0,0,0,2,0,0,0,0,0,2,2,3,3,0,0,0,1,4,2],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,1,1,4,4,0,0,1,0,0,0,0,0,0,0,0,4],[2,4,0,0,0,0,3,3,0,0,0,0,0,0,2,0,0,0,0,0,0,2,0,0,0,0,0,0,2,0,0,0,0,0,0,2],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,2,2,0,4],[3,0,0,0,0,0,0,3,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,1,1,4,4,0,0,0,0,0,1] >;
 

44 conjugacy classes

class 1 2A2B2C2D2E2F2G2H···2N4A···4N4O···4AC
order122222222···24···44···4
size111122224···42···24···4

44 irreducible representations

dim1111111111111111244
type+++++++++++++++++
imageC1C2C2C2C2C2C2C2C2C2C2C2C2C2C2C2C4○D42+ (1+4)C2.C25
kernelC22.103C25C4×C4○D4C22.11C24C2×C4.4D4C23.36C23C22.26C24C22.32C24C22.36C24D42D4⋊5D4Q8⋊5D4D4×Q8C22.45C24C22.49C24C22.50C24C22.53C24D4C4C2
# reps1122314214214211822

In GAP, Magma, Sage, TeX 

C_2^2._{103}C_2^5
 
% in TeX
 
G:=Group("C2^2.103C2^5");
 
// GroupNames label
 
G:=SmallGroup(128,2246);
 
// by ID
 
G=gap.SmallGroup(128,2246);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,2,-2,2,477,1430,184,570,1684,172]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=f^2=1,e^2=a,g^2=b,a*b=b*a,d*c*d=g*c*g^-1=a*c=c*a,f*d*f=a*d=d*a,a*e=e*a,a*f=f*a,a*g=g*a,e*c*e^-1=b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,b*g=g*b,c*f=f*c,d*e=e*d,d*g=g*d,e*f=f*e,e*g=g*e,f*g=g*f>;
 
// generators/relations
 

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