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G = C22.108C25  order 128 = 27

89th central stem extension by C22 of C25

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

Aliases: C42.99C23, C22.108C25, C23.143C24, C24.510C23, C4.822+ (1+4), C22.192+ (1+4), (D42)⋊17C2, D4○3(C4⋊D4), C4⋊Q8⋊95C22, D4⋊15(C4○D4), D4⋊5D4⋊24C2, D4⋊3Q8⋊28C2, Q8⋊6D4⋊22C2, (C4×D4)⋊52C22, (C2×C4).98C24, (C4×Q8)⋊51C22, C4⋊1D4⋊19C22, C4⋊C4.302C23, C4⋊D4⋊30C22, (C23×C4)⋊45C22, (C2×C42)⋊66C22, C22≀C2⋊35C22, C42⋊2C2⋊6C22, C22.32C24⋊8C2, (C2×D4).480C23, C4.4D4⋊86C22, (C22×D4)⋊39C22, C22⋊C4.31C23, C22⋊Q8⋊105C22, (C2×Q8).294C23, C42.C2⋊59C22, C22.19C24⋊33C2, C42⋊C2⋊45C22, C22.11C24⋊23C2, (C22×C4).377C23, C2.42(C2×2+ (1+4)), C22.26C24⋊45C2, C22.D4⋊55C22, C22.47C24⋊23C2, C22.34C24⋊14C2, C23.36C23⋊39C2, C22.49C24⋊17C2, (C2×C4×D4)⋊94C2, (C2×C4⋊D4)⋊69C2, (C2×C4⋊C4)⋊78C22, C4.281(C2×C4○D4), (C2×C4○D4)⋊37C22, C2.64(C22×C4○D4), C22.44(C2×C4○D4), (C2×C22⋊C4)⋊53C22, SmallGroup(128,2251)

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

Derived series
C1 — C22 — C22.108C25
Chief series
C1 — C2 — C22 — C2×C4 — C22×C4 — C23×C4 — C2×C4×D4 — C22.108C25
Lower central
C1 — C22 — C22.108C25
Upper central
C1 — C22 — C22.108C25
Jennings
C1 — C22 — C22.108C25

Subgroups: 1084 in 626 conjugacy classes, 392 normal (50 characteristic)
C1, C2 [×3], C2 [×14], C4 [×4], C4 [×18], C22, C22 [×6], C22 [×46], C2×C4 [×6], C2×C4 [×14], C2×C4 [×36], D4 [×4], D4 [×54], Q8 [×4], C23, C23 [×10], C23 [×26], C42 [×4], C42 [×6], C22⋊C4 [×46], C4⋊C4 [×4], C4⋊C4 [×22], C22×C4 [×3], C22×C4 [×24], C22×C4 [×4], C2×D4 [×3], C2×D4 [×34], C2×D4 [×24], C2×Q8, C2×Q8 [×2], C4○D4 [×8], C24 [×6], C2×C42, C2×C22⋊C4 [×10], C2×C4⋊C4, C2×C4⋊C4 [×2], C42⋊C2 [×6], C4×D4 [×5], C4×D4 [×26], C4×Q8, C22≀C2 [×12], C4⋊D4, C4⋊D4 [×34], C22⋊Q8, C22⋊Q8 [×6], C22.D4 [×14], C4.4D4, C4.4D4 [×6], C42.C2, C42.C2 [×2], C42⋊2C2 [×6], C4⋊1D4, C4⋊1D4 [×2], C4⋊Q8, C23×C4 [×2], C22×D4, C22×D4 [×8], C2×C4○D4 [×4], C2×C4×D4, C22.11C24 [×2], C2×C4⋊D4 [×2], C22.19C24 [×2], C23.36C23, C22.26C24, C22.32C24 [×4], C22.34C24 [×2], D42, D42 [×2], D4⋊5D4 [×6], Q8⋊6D4, C22.47C24 [×4], D4⋊3Q8, C22.49C24, C22.108C25

Quotients:
C1, C2 [×31], C22 [×155], C23 [×155], C4○D4 [×4], C24 [×31], C2×C4○D4 [×6], 2+ (1+4) [×4], C25, C22×C4○D4, C2×2+ (1+4) [×2], C22.108C25

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

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

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

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

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

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

Matrix representation ►G ⊆ GL6(𝔽5)

100000
010000
004000
000400
000040
000004
,
400000
040000
004000
000400
000040
000004
,
020000
300000
004000
000100
000010
000424
,
100000
010000
000010
002423
001000
000001
,
040000
400000
000400
004000
003132
001012
,
400000
040000
001000
000100
000040
002404
,
200000
020000
000100
001000
002423
004043

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],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[0,3,0,0,0,0,2,0,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,4,0,0,0,0,1,2,0,0,0,0,0,4],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,2,1,0,0,0,0,4,0,0,0,0,1,2,0,0,0,0,0,3,0,1],[0,4,0,0,0,0,4,0,0,0,0,0,0,0,0,4,3,1,0,0,4,0,1,0,0,0,0,0,3,1,0,0,0,0,2,2],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,2,0,0,0,1,0,4,0,0,0,0,4,0,0,0,0,0,0,4],[2,0,0,0,0,0,0,2,0,0,0,0,0,0,0,1,2,4,0,0,1,0,4,0,0,0,0,0,2,4,0,0,0,0,3,3] >;
 

44 conjugacy classes

class 1 2A2B2C2D···2I2J···2Q4A···4L4M···4Z
order12222···22···24···44···4
size11112···24···42···24···4

44 irreducible representations

dim111111111111111244
type+++++++++++++++++
imageC1C2C2C2C2C2C2C2C2C2C2C2C2C2C2C4○D42+ (1+4)2+ (1+4)
kernelC22.108C25C2×C4×D4C22.11C24C2×C4⋊D4C22.19C24C23.36C23C22.26C24C22.32C24C22.34C24D42D4⋊5D4Q8⋊6D4C22.47C24D4⋊3Q8C22.49C24D4C4C22
# reps112221142361411822

In GAP, Magma, Sage, TeX 

C_2^2._{108}C_2^5
 
% in TeX
 
G:=Group("C2^2.108C2^5");
 
// GroupNames label
 
G:=SmallGroup(128,2251);
 
// by ID
 
G=gap.SmallGroup(128,2251);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,2,-2,2,477,1430,570,1684,242]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=e^2=f^2=1,g^2=b*a=a*b,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=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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