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G = C22.47C25  order 128 = 27

28th central stem extension by C22 of C25

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

Aliases: C22.47C25, C23.119C24, C24.487C23, C42.548C23, C4⋊Q8⋊79C22, (C22×C4)⋊18Q8, C2.9(Q8×C23), (C2×C4).49C24, (C4×Q8)⋊32C22, C4○(C23⋊2Q8), C23.74(C2×Q8), C4.20(C22×Q8), C4⋊C4.287C23, C23⋊2Q8.6C2, C22.7(C22×Q8), C22⋊C4.77C23, (C2×Q8).277C23, C42.C2⋊44C22, C2.6(C2.C25), (C2×C42).922C22, (C23×C4).591C22, C22⋊Q8.223C22, C4○(C23.41C23), (C22×C4).1186C23, C23.41C23⋊28C2, C23.37C23⋊29C2, C42⋊C2.340C22, (C2×C4).145(C2×Q8), (C2×C4⋊C4).951C22, (C2×C42⋊C2).65C2, (C2×C22⋊C4).536C22, SmallGroup(128,2190)

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

Derived series
C1 — C22 — C22.47C25
Chief series
C1 — C2 — C22 — C2×C4 — C22×C4 — C2×C42 — C2×C42⋊C2 — C22.47C25
Lower central
C1 — C22 — C22.47C25
Upper central
C1 — C2×C4 — C22.47C25
Jennings
C1 — C22 — C22.47C25

Generators and relations for C22.47C25
 G = < a,b,c,d,e,f,g | a2=b2=d2=f2=1, c2=e2=b, g2=a, ab=ba, dcd=fcf=ac=ca, ede-1=ad=da, ae=ea, af=fa, ag=ga, ece-1=bc=cb, bd=db, be=eb, bf=fb, bg=gb, cg=gc, df=fd, dg=gd, ef=fe, eg=ge, fg=gf >

Subgroups: 636 in 498 conjugacy classes, 428 normal (7 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C2×C4, C2×C4, C2×C4, Q8, C23, C23, C42, C22⋊C4, C4⋊C4, C22×C4, C2×Q8, C24, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C42⋊C2, C4×Q8, C22⋊Q8, C42.C2, C4⋊Q8, C23×C4, C2×C42⋊C2, C23.37C23, C23⋊2Q8, C23.41C23, C22.47C25
Quotients: C1, C2, C22, Q8, C23, C2×Q8, C24, C22×Q8, C25, Q8×C23, C2.C25, C22.47C25

Smallest permutation representation of C22.47C25
►On 32 points
Generators in S32
(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)

(2 28)(4 26)(6 17)(8 19)(10 14)(12 16)(21 29)(23 31)

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

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

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

44 conjugacy classes

class 1 2A2B2C2D···2I4A4B4C4D4E···4J4K···4AH
order12222···244444···44···4
size11112···211112···24···4

44 irreducible representations

dim1111124
type+++++-
imageC1C2C2C2C2Q8C2.C25
kernelC22.47C25C2×C42⋊C2C23.37C23C23⋊2Q8C23.41C23C22×C4C2
# reps131241284

Matrix representation of C22.47C25 ►in GL6(𝔽5)

100000
010000
004000
000400
000040
000004
,
400000
040000
001000
000100
000010
000001
,
020000
200000
000320
003002
001002
000120
,
400000
040000
001000
000400
000040
000001
,
040000
100000
000100
001000
000001
000010
,
100000
010000
001000
000100
000240
002004
,
100000
010000
003000
000300
000030
000003

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

C22.47C25 in GAP, Magma, Sage, TeX 

C_2^2._{47}C_2^5
 
% in TeX
 
G:=Group("C2^2.47C2^5");
 
// GroupNames label
 
G:=SmallGroup(128,2190);
 
// by ID
 
G=gap.SmallGroup(128,2190);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,2,-2,2,448,477,1430,387,352,1123,102]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=d^2=f^2=1,c^2=e^2=b,g^2=a,a*b=b*a,d*c*d=f*c*f=a*c=c*a,e*d*e^-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*g=g*c,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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