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G = C25.85C22order 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, C23.162C24, C24.528C23, (C23×C4)⋊17C4, C4(C243C4), (C24×C4).10C2, (C2×C42)⋊2C22, C24.122(C2×C4), (C22×C4).768D4, C23.597(C2×D4), C4(C23.7Q8), C243C4.18C2, C4(C23.34D4), C22.53(C23×C4), C22.62(C22×D4), C224(C42⋊C2), C23.213(C4○D4), C23.34D465C2, C23.207(C22×C4), (C23×C4).643C22, (C22×C4).440C23, C23.7Q8118C2, C2.C4250C22, C2.1(C22.19C24), (C4×C22⋊C4)⋊4C2, (C2×C4⋊C4)⋊97C22, C4.96(C2×C22⋊C4), (C2×C4)⋊13(C22⋊C4), (C2×C4)(C243C4), (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)(C243C4), (C2×C22⋊C4).411C22, (C22×C4)(C23.7Q8), SmallGroup(128,1012)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C25.85C22
C1C2C22C23C22×C4C23×C4C24×C4 — C25.85C22
C1C22 — C25.85C22
C1C22×C4 — C25.85C22
C1C23 — C25.85C22

Generators and relations for C25.85C22
 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 >

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], C243C4 [×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

Smallest permutation representation of C25.85C22
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)])

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⋊C4C243C4C23.7Q8C23.34D4C2×C42⋊C2C24×C4C23×C4C22×C4C23
# reps142422116816

Matrix representation of C25.85C22 in 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] >;

C25.85C22 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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