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G = C22.64C25order 128 = 27

45th central stem extension by C22 of C25

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

Aliases: C22.64C25, C23.127C24, C42.594C23, C24.498C23, C4(D43Q8), C4⋊Q887C22, C42(D45D4), C42(D46D4), D410(C4○D4), D46D447C2, D45D443C2, D43Q847C2, (C4×D4)⋊31C22, (C4×Q8)⋊34C22, D42(C42⋊C2), C4⋊C4.473C23, C4⋊D476C22, (C2×C4).166C24, (C2×C42)⋊54C22, C22⋊Q891C22, (C2×D4).459C23, C4.4D475C22, C22⋊C4.88C23, (C2×Q8).281C23, C42.C250C22, C4(C22.45C24), C42⋊C299C22, C22.19C2418C2, C422C232C22, C22≀C2.24C22, (C23×C4).602C22, C22.45C2427C2, C2.11(C2.C25), C4(C22.50C24), C4(C22.47C24), C4(C22.46C24), C4(C22.49C24), (C22×C4).1200C23, (C22×D4).594C22, C22.D445C22, C22.49C2428C2, C22.46C2444C2, C22.50C2444C2, C23.36C2321C2, C22.47C2443C2, C23.37C2331C2, (C2×C4×D4)⋊87C2, (C4×C4○D4)⋊21C2, C42⋊C2(C4×D4), (C2×C4)⋊18(C4○D4), (C2×C4)(D43Q8), C4.136(C2×C4○D4), (C2×C4⋊C4)⋊139C22, (C2×D4)(C42⋊C2), C2.36(C22×C4○D4), C22.38(C2×C4○D4), (C2×C42⋊C2)⋊64C2, C22⋊C4(C42⋊C2), (C2×C4○D4).324C22, (C2×C4)(C22.45C24), (C2×C22⋊C4).544C22, (C2×C4)(C22.46C24), (C2×C4)(C22.49C24), SmallGroup(128,2207)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C22.64C25
C1C2C22C2×C4C22×C4C23×C4C2×C42⋊C2 — C22.64C25
C1C22 — C22.64C25
C1C2×C4 — C22.64C25
C1C22 — C22.64C25

Generators and relations for C22.64C25
 G = < a,b,c,d,e,f,g | a2=b2=c2=e2=f2=1, d2=b, g2=a, ab=ba, dcd-1=ac=ca, fdf=ad=da, ae=ea, af=fa, ag=ga, ece=bc=cb, bd=db, be=eb, bf=fb, bg=gb, cf=fc, cg=gc, de=ed, dg=gd, ef=fe, eg=ge, fg=gf >

Subgroups: 788 in 566 conjugacy classes, 398 normal (50 characteristic)
C1, C2 [×3], C2 [×10], C4 [×6], C4 [×21], C22, C22 [×6], C22 [×26], C2×C4 [×6], C2×C4 [×22], C2×C4 [×45], D4 [×4], D4 [×24], Q8 [×8], C23, C23 [×6], C23 [×10], C42 [×4], C42 [×18], C22⋊C4 [×42], C4⋊C4 [×4], C4⋊C4 [×38], C22×C4 [×3], C22×C4 [×24], C22×C4 [×8], C2×D4 [×3], C2×D4 [×12], C2×D4 [×4], C2×Q8, C2×Q8 [×4], C4○D4 [×12], C24 [×2], C2×C42, C2×C42 [×6], C2×C22⋊C4 [×6], C2×C4⋊C4, C2×C4⋊C4 [×4], C42⋊C2, C42⋊C2 [×22], C4×D4 [×6], C4×D4 [×22], C4×Q8 [×2], C4×Q8 [×6], C22≀C2 [×4], C4⋊D4 [×10], C22⋊Q8 [×14], C22.D4 [×16], C4.4D4 [×6], C42.C2 [×8], C422C2 [×12], C4⋊Q8 [×2], C23×C4 [×2], C22×D4, C2×C4○D4, C2×C4○D4 [×2], C2×C42⋊C2 [×2], C2×C4×D4, C4×C4○D4, C4×C4○D4 [×2], C22.19C24 [×2], C23.36C23 [×6], C23.37C23, D45D4 [×2], D46D4, C22.45C24 [×4], C22.46C24 [×4], C22.47C24 [×2], D43Q8, C22.49C24, C22.50C24, C22.64C25
Quotients: C1, C2 [×31], C22 [×155], C23 [×155], C4○D4 [×8], C24 [×31], C2×C4○D4 [×12], C25, C22×C4○D4 [×2], C2.C25, C22.64C25

Smallest permutation representation of C22.64C25
On 32 points
Generators in S32
(1 15)(2 16)(3 13)(4 14)(5 18)(6 19)(7 20)(8 17)(9 24)(10 21)(11 22)(12 23)(25 30)(26 31)(27 32)(28 29)
(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 31)(2 27)(3 29)(4 25)(5 21)(6 11)(7 23)(8 9)(10 18)(12 20)(13 28)(14 30)(15 26)(16 32)(17 24)(19 22)
(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)
(9 11)(10 12)(21 23)(22 24)(25 27)(26 28)(29 31)(30 32)
(1 3)(2 14)(4 16)(5 20)(6 8)(7 18)(9 11)(10 23)(12 21)(13 15)(17 19)(22 24)(25 32)(26 28)(27 30)(29 31)
(1 17 15 8)(2 18 16 5)(3 19 13 6)(4 20 14 7)(9 31 24 26)(10 32 21 27)(11 29 22 28)(12 30 23 25)

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

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

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

50 conjugacy classes

class 1 2A2B2C2D···2I2J2K2L2M4A4B4C4D4E···4V4W···4AJ
order12222···2222244444···44···4
size11112···2444411112···24···4

50 irreducible representations

dim111111111111111224
type+++++++++++++++
imageC1C2C2C2C2C2C2C2C2C2C2C2C2C2C2C4○D4C4○D4C2.C25
kernelC22.64C25C2×C42⋊C2C2×C4×D4C4×C4○D4C22.19C24C23.36C23C23.37C23D45D4D46D4C22.45C24C22.46C24C22.47C24D43Q8C22.49C24C22.50C24C2×C4D4C2
# reps121326121442111882

Matrix representation of C22.64C25 in GL4(𝔽5) generated by

1000
0100
0040
0004
,
4000
0400
0010
0001
,
0100
1000
0040
0001
,
2000
0200
0001
0010
,
1000
0400
0010
0001
,
4000
0400
0010
0004
,
4000
0400
0030
0003
G:=sub<GL(4,GF(5))| [1,0,0,0,0,1,0,0,0,0,4,0,0,0,0,4],[4,0,0,0,0,4,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,4,0,0,0,0,1],[2,0,0,0,0,2,0,0,0,0,0,1,0,0,1,0],[1,0,0,0,0,4,0,0,0,0,1,0,0,0,0,1],[4,0,0,0,0,4,0,0,0,0,1,0,0,0,0,4],[4,0,0,0,0,4,0,0,0,0,3,0,0,0,0,3] >;

C22.64C25 in GAP, Magma, Sage, TeX

C_2^2._{64}C_2^5
% in TeX

G:=Group("C2^2.64C2^5");
// GroupNames label

G:=SmallGroup(128,2207);
// by ID

G=gap.SmallGroup(128,2207);
# by ID

G:=PCGroup([7,-2,2,2,2,2,-2,2,477,456,1430,570,102]);
// Polycyclic

G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=e^2=f^2=1,d^2=b,g^2=a,a*b=b*a,d*c*d^-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=b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,b*g=g*b,c*f=f*c,c*g=g*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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