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G = C22.95C25  order 128 = 27

76th central stem extension by C22 of C25

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

Aliases: C42.87C23, C23.46C24, C22.95C25, C24.508C23, C4.812+ (1+4), C4⋊Q8⋊93C22, D4⋊6D4⋊23C2, Q8⋊6D4⋊19C2, (C4×D4)⋊47C22, (C2×C4).85C24, (C4×Q8)⋊46C22, C4⋊1D4⋊53C22, C4⋊D4⋊83C22, C4⋊C4.491C23, C22⋊Q8⋊96C22, C42⋊2C2⋊5C22, C22.32C24⋊5C2, (C2×D4).304C23, C4.4D4⋊27C22, (C2×Q8).290C23, C42.C2⋊56C22, C42⋊C2⋊41C22, C22.19C24⋊31C2, C22.29C24⋊24C2, C22≀C2.29C22, C22⋊C4.105C23, (C22×C4).366C23, (C2×C42).948C22, (C23×C4).612C22, C2.36(C2×2+ (1+4)), C2.29(C2.C25), C22.26C24⋊39C2, (C22×D4).600C22, C22.D4⋊52C22, C23.33C23⋊21C2, C22.49C24⋊13C2, C22.53C24⋊11C2, C23.37C23⋊38C2, C22.50C24⋊20C2, C22.47C24⋊16C2, (C2×C4×D4)⋊92C2, (C2×C4)⋊6(C4○D4), (C2×C4⋊C4)⋊76C22, C4.178(C2×C4○D4), (C2×C4○D4)⋊33C22, C2.51(C22×C4○D4), C22.16(C2×C4○D4), (C2×C22⋊C4).547C22, SmallGroup(128,2238)

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

Derived series
C1 — C22 — C22.95C25
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C2×C42 — C2×C4×D4 — C22.95C25
Lower central
C1 — C22 — C22.95C25
Upper central
C1 — C22 — C22.95C25
Jennings
C1 — C22 — C22.95C25

Subgroups: 884 in 570 conjugacy classes, 390 normal (30 characteristic)
C1, C2 [×3], C2 [×10], C4 [×6], C4 [×19], C22, C22 [×2], C22 [×34], C2×C4 [×6], C2×C4 [×20], C2×C4 [×33], D4 [×42], Q8 [×10], C23, C23 [×8], C23 [×8], C42 [×4], C42 [×12], C22⋊C4 [×44], C4⋊C4 [×32], C22×C4 [×3], C22×C4 [×22], C22×C4 [×4], C2×D4 [×30], C2×D4 [×4], C2×Q8 [×6], C4○D4 [×16], C24 [×2], C2×C42, C2×C22⋊C4 [×2], C2×C4⋊C4, C2×C4⋊C4 [×4], C42⋊C2 [×10], C4×D4 [×32], C4×Q8 [×8], C22≀C2 [×8], C4⋊D4 [×24], C22⋊Q8 [×8], C22.D4 [×16], C4.4D4 [×14], C42.C2 [×2], C42⋊2C2 [×8], C4⋊1D4, C4⋊1D4 [×4], C4⋊Q8 [×3], C23×C4 [×2], C22×D4, C2×C4○D4 [×6], C2×C4×D4, C23.33C23 [×2], C22.19C24 [×4], C22.26C24, C23.37C23, C22.29C24 [×2], C22.32C24 [×4], D4⋊6D4 [×2], Q8⋊6D4 [×2], C22.47C24 [×4], C22.49C24 [×2], C22.50C24 [×2], C22.53C24 [×4], C22.95C25

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, C22.95C25

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

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

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

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

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

Matrix representation ►G ⊆ GL6(𝔽5)

100000
010000
004000
000400
000040
000004
,
400000
040000
001000
000100
000010
000001
,
040000
400000
000010
000001
001000
000100
,
200000
020000
000100
001000
000004
000040
,
300000
020000
001000
000100
000010
000001
,
100000
010000
000100
004000
000001
000040
,
100000
010000
001000
000100
000040
000004

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

44 conjugacy classes

class 1 2A2B2C2D2E2F···2M4A···4P4Q···4AD
order1222222···24···44···4
size1111224···42···24···4

44 irreducible representations

dim11111111111111244
type+++++++++++++++
imageC1C2C2C2C2C2C2C2C2C2C2C2C2C2C4○D42+ (1+4)C2.C25
kernelC22.95C25C2×C4×D4C23.33C23C22.19C24C22.26C24C23.37C23C22.29C24C22.32C24D4⋊6D4Q8⋊6D4C22.47C24C22.49C24C22.50C24C22.53C24C2×C4C4C2
# reps11241124224224822

In GAP, Magma, Sage, TeX 

C_2^2._{95}C_2^5
 
% in TeX
 
G:=Group("C2^2.95C2^5");
 
// GroupNames label
 
G:=SmallGroup(128,2238);
 
// by ID
 
G=gap.SmallGroup(128,2238);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,2,-2,2,477,456,1430,352,570,136,1684]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=g^2=1,d^2=e^2=b,f^2=a,a*b=b*a,d*c*d^-1=g*c*g=a*c=c*a,f*d*f^-1=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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