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G = C22.97C25  order 128 = 27

78th central stem extension by C22 of C25

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

Aliases: C22.97C25, C23.47C24, C42.89C23, C24.138C23, C4.1562+ (1+4), D4⋊5D4⋊21C2, Q8⋊6D4⋊20C2, (C4×D4)⋊48C22, (C2×C4).87C24, C4⋊Q8⋊109C22, (C4×Q8)⋊47C22, C4⋊D4⋊84C22, C4⋊1D4⋊18C22, C4⋊C4.492C23, (C2×C42)⋊62C22, C22≀C2.9C22, (C2×D4).305C23, C4.4D4⋊85C22, (C2×Q8).452C23, C42.C2⋊57C22, C22.11C24⋊19C2, C22.29C24⋊25C2, C22⋊C4.107C23, (C22×C4).368C23, C22.D4⋊9C22, C42⋊C2⋊102C22, C22⋊Q8.228C22, C2.37(C2×2+ (1+4)), C22.26C24⋊40C2, (C22×D4).427C22, C22.34C24⋊10C2, C23.37C23⋊40C2, C22.53C24⋊13C2, (C4×C4○D4)⋊29C2, (C2×C4)⋊7(C4○D4), (C2×C4⋊1D4)⋊27C2, C22⋊C4○(C4⋊1D4), C4.180(C2×C4○D4), (C2×C4○D4)⋊34C22, C22.30(C2×C4○D4), C2.53(C22×C4○D4), (C2×C22⋊C4).383C22, SmallGroup(128,2240)

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

Derived series
C1 — C22 — C22.97C25
Chief series
C1 — C2 — C22 — C2×C4 — C22×C4 — C2×C42 — C4×C4○D4 — C22.97C25
Lower central
C1 — C22 — C22.97C25
Upper central
C1 — C22 — C22.97C25
Jennings
C1 — C22 — C22.97C25

Subgroups: 1052 in 616 conjugacy classes, 392 normal (18 characteristic)
C1, C2, C2 [×2], C2 [×12], C4 [×8], C4 [×16], C22, C22 [×2], C22 [×40], C2×C4 [×2], C2×C4 [×22], C2×C4 [×32], D4 [×62], Q8 [×10], C23, C23 [×10], C23 [×16], C42 [×2], C42 [×12], C22⋊C4 [×42], C4⋊C4 [×22], C22×C4, C22×C4 [×24], C2×D4, C2×D4 [×42], C2×D4 [×16], C2×Q8, C2×Q8 [×4], C4○D4 [×20], C24 [×4], C2×C42, C2×C42 [×2], C2×C22⋊C4 [×8], C42⋊C2, C42⋊C2 [×4], C4×D4 [×30], C4×Q8 [×6], C22≀C2 [×8], C4⋊D4 [×32], C22⋊Q8 [×4], C22.D4 [×16], C4.4D4 [×12], C42.C2 [×2], C4⋊1D4 [×12], C4⋊Q8 [×2], C22×D4 [×6], C2×C4○D4, C2×C4○D4 [×8], C4×C4○D4, C22.11C24 [×2], C2×C4⋊1D4, C22.26C24 [×4], C23.37C23, C22.29C24 [×2], C22.34C24 [×4], D4⋊5D4 [×8], Q8⋊6D4 [×4], C22.53C24 [×4], C22.97C25

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], C22.97C25

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

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

(2 28)(4 26)(5 30)(7 32)(10 16)(12 14)(18 24)(20 22)

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

Matrix representation ►G ⊆ GL6(𝔽5)

100000
010000
004000
000400
000040
000004
,
400000
040000
001000
000100
000010
000001
,
040000
400000
000400
004000
000001
000010
,
300000
030000
000010
000001
001000
000100
,
040000
100000
004000
000400
000040
000004
,
400000
040000
001000
000100
000040
000004
,
400000
040000
000100
004000
000001
000040

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

44 conjugacy classes

class 1 2A2B2C2D2E2F···2O4A···4P4Q···4AB
order1222222···24···44···4
size1111224···42···24···4

44 irreducible representations

dim1111111111124
type++++++++++++
imageC1C2C2C2C2C2C2C2C2C2C2C4○D42+ (1+4)
kernelC22.97C25C4×C4○D4C22.11C24C2×C4⋊1D4C22.26C24C23.37C23C22.29C24C22.34C24D4⋊5D4Q8⋊6D4C22.53C24C2×C4C4
# reps1121412484484

In GAP, Magma, Sage, TeX 

C_2^2._{97}C_2^5
 
% in TeX
 
G:=Group("C2^2.97C2^5");
 
// GroupNames label
 
G:=SmallGroup(128,2240);
 
// by ID
 
G=gap.SmallGroup(128,2240);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,2,-2,2,477,456,1430,352,570,1684,102]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=f^2=1,d^2=e^2=b,g^2=a,a*b=b*a,d*c*d^-1=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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