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G = C25.85C22  order 128 = 27

6th non-split extension by C25 of C22 acting via C22/C2=C2

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

Aliases: C25.85C22, C24.528C23, C23.162C24, (C23×C4)⋊17C4, C4○(C24⋊3C4), (C24×C4).10C2, (C2×C42)⋊2C22, C24.122(C2×C4), C23.597(C2×D4), (C22×C4).768D4, C4○(C23.7Q8), C24⋊3C4.18C2, C4○(C23.34D4), C22.53(C23×C4), C22.62(C22×D4), C22⋊4(C42⋊C2), C23.213(C4○D4), C23.34D4⋊65C2, (C22×C4).440C23, (C23×C4).643C22, C23.207(C22×C4), C23.7Q8⋊118C2, C2.C42⋊50C22, C2.1(C22.19C24), (C4×C22⋊C4)⋊4C2, (C2×C4⋊C4)⋊97C22, C4.96(C2×C22⋊C4), (C2×C4)⋊13(C22⋊C4), (C2×C4)○(C24⋊3C4), (C2×C42⋊C2)⋊4C2, (C2×C4).1554(C2×D4), C2.8(C2×C42⋊C2), C22.55(C2×C4○D4), C2.7(C22×C22⋊C4), (C22×C4).489(C2×C4), (C2×C4).622(C22×C4), C22.73(C2×C22⋊C4), (C2×C4)○(C23.7Q8), (C22×C4)○(C24⋊3C4), (C2×C22⋊C4).411C22, (C22×C4)○(C23.7Q8), SmallGroup(128,1012)

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

Derived series
C1 — C22 — C25.85C22
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C23×C4 — C24×C4 — C25.85C22
Lower central
C1 — C22 — C25.85C22
Upper central
C1 — C22×C4 — C25.85C22
Jennings
C1 — C23 — C25.85C22

Subgroups: 908 in 544 conjugacy classes, 196 normal (12 characteristic)
C1, C2, C2 [×6], C2 [×12], C4 [×8], C4 [×12], C22, C22 [×18], C22 [×68], C2×C4 [×32], C2×C4 [×68], C23, C23 [×18], C23 [×68], C42 [×8], C22⋊C4 [×24], C4⋊C4 [×8], C22×C4 [×2], C22×C4 [×42], C22×C4 [×52], C24, C24 [×6], C24 [×12], C2.C42 [×8], C2×C42 [×4], C2×C22⋊C4 [×12], C2×C4⋊C4 [×4], C42⋊C2 [×8], C23×C4 [×2], C23×C4 [×12], C23×C4 [×8], C25, C4×C22⋊C4 [×4], C24⋊3C4 [×2], C23.7Q8 [×4], C23.34D4 [×2], C2×C42⋊C2 [×2], C24×C4, C25.85C22

Quotients:
C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], D4 [×8], C23 [×15], C22⋊C4 [×16], C22×C4 [×14], C2×D4 [×12], C4○D4 [×8], C24, C2×C22⋊C4 [×12], C42⋊C2 [×8], C23×C4, C22×D4 [×2], C2×C4○D4 [×4], C22×C22⋊C4, C2×C42⋊C2 [×2], C22.19C24 [×4], C25.85C22

Generators and relations
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=e2=1, f2=e, g2=c, ab=ba, faf-1=ac=ca, ad=da, ae=ea, ag=ga, bc=cb, fbf-1=bd=db, be=eb, bg=gb, cd=dc, ce=ec, cf=fc, cg=gc, de=ed, df=fd, dg=gd, ef=fe, eg=ge, fg=gf >

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

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

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

Matrix representation ►G ⊆ GL5(𝔽5)

40000
04400
00100
00040
00004
,
10000
04400
00100
00010
00004
,
10000
04000
00400
00010
00001
,
10000
04000
00400
00040
00004
,
40000
01000
00100
00010
00001
,
20000
01000
03400
00001
00010
,
40000
02000
00200
00010
00001

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

56 conjugacy classes

class 1 2A···2G2H···2S4A···4H4I···4T4U···4AJ
order12···22···24···44···44···4
size11···12···21···12···24···4

56 irreducible representations

dim1111111122
type++++++++
imageC1C2C2C2C2C2C2C4D4C4○D4
kernelC25.85C22C4×C22⋊C4C24⋊3C4C23.7Q8C23.34D4C2×C42⋊C2C24×C4C23×C4C22×C4C23
# reps142422116816

In GAP, Magma, Sage, TeX 

C_2^5._{85}C_2^2
 
% in TeX
 
G:=Group("C2^5.85C2^2");
 
// GroupNames label
 
G:=SmallGroup(128,1012);
 
// by ID
 
G=gap.SmallGroup(128,1012);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,758,80]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=e^2=1,f^2=e,g^2=c,a*b=b*a,f*a*f^-1=a*c=c*a,a*d=d*a,a*e=e*a,a*g=g*a,b*c=c*b,f*b*f^-1=b*d=d*b,b*e=e*b,b*g=g*b,c*d=d*c,c*e=e*c,c*f=f*c,c*g=g*c,d*e=e*d,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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