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G = C22.75C25order 128 = 27

56th central stem extension by C22 of C25

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

Aliases: C42.77C23, C22.75C25, C23.35C24, C24.619C23, C2212- 1+4, Q8C22≀C2, (D4×Q8)⋊15C2, (C2×Q8)⋊40D4, C4⋊Q829C22, Q8.57(C2×D4), Q85D412C2, (C2×C4).69C24, (C4×Q8)⋊35C22, (Q8×C23)⋊12C2, C2.27(D4×C23), C4⋊C4.482C23, C4.116(C22×D4), C22⋊Q823C22, (C4×D4).229C22, (C2×D4).464C23, C4.4D421C22, C22⋊C4.16C23, (C2×2- 1+4)⋊5C2, (C2×Q8).440C23, (C22×Q8)⋊29C22, C22.50(C22×D4), C22.19C2422C2, C22≀C2.35C22, C4⋊D4.223C22, (C23×C4).603C22, C2.18(C2×2- 1+4), (C22×C4).1206C23, C42⋊C2.223C22, C23.38C2319C2, C23.32C2311C2, C22.D4.29C22, (C2×Q8)C22≀C2, (C2×C4).664(C2×D4), (C2×C4○D4)⋊23C22, SmallGroup(128,2218)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C22.75C25
C1C2C22C23C22×C4C23×C4Q8×C23 — C22.75C25
C1C22 — C22.75C25
C1C22 — C22.75C25
C1C22 — C22.75C25

Generators and relations for C22.75C25
 G = < a,b,c,d,e,f,g | a2=b2=c2=e2=g2=1, d2=f2=a, ab=ba, dcd-1=gcg=ac=ca, fdf-1=ad=da, ae=ea, af=fa, ag=ga, ece=bc=cb, bd=db, be=eb, bf=fb, bg=gb, cf=fc, de=ed, dg=gd, ef=fe, eg=ge, fg=gf >

Subgroups: 1068 in 746 conjugacy classes, 432 normal (10 characteristic)
C1, C2, C2 [×2], C2 [×10], C4 [×12], C4 [×18], C22, C22 [×6], C22 [×26], C2×C4 [×36], C2×C4 [×54], D4 [×36], Q8 [×16], Q8 [×36], C23, C23 [×6], C23 [×6], C42 [×12], C22⋊C4 [×36], C4⋊C4 [×36], C22×C4 [×30], C22×C4 [×12], C2×D4 [×18], C2×Q8 [×30], C2×Q8 [×60], C4○D4 [×40], C24, C42⋊C2 [×6], C4×D4 [×24], C4×Q8 [×8], C22≀C2 [×4], C4⋊D4 [×12], C22⋊Q8 [×36], C22.D4 [×12], C4.4D4 [×12], C4⋊Q8 [×12], C23×C4 [×3], C22×Q8, C22×Q8 [×15], C22×Q8 [×8], C2×C4○D4 [×10], 2- 1+4 [×8], C23.32C23, C22.19C24 [×6], C23.38C23 [×6], Q85D4 [×8], D4×Q8 [×8], Q8×C23, C2×2- 1+4, C22.75C25
Quotients: C1, C2 [×31], C22 [×155], D4 [×8], C23 [×155], C2×D4 [×28], C24 [×31], C22×D4 [×14], 2- 1+4 [×4], C25, D4×C23, C2×2- 1+4 [×2], C22.75C25

Smallest permutation representation of C22.75C25
On 32 points
Generators in S32
(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 15)(2 16)(3 13)(4 14)(5 17)(6 18)(7 19)(8 20)(9 25)(10 26)(11 27)(12 28)(21 29)(22 30)(23 31)(24 32)
(1 21)(2 24)(3 23)(4 22)(5 28)(6 27)(7 26)(8 25)(9 20)(10 19)(11 18)(12 17)(13 31)(14 30)(15 29)(16 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 15)(2 16)(3 13)(4 14)(5 17)(6 18)(7 19)(8 20)
(1 7 3 5)(2 6 4 8)(9 32 11 30)(10 31 12 29)(13 17 15 19)(14 20 16 18)(21 26 23 28)(22 25 24 27)
(1 15)(2 16)(3 13)(4 14)(5 17)(6 18)(7 19)(8 20)(9 27)(10 28)(11 25)(12 26)(21 31)(22 32)(23 29)(24 30)

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

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

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

44 conjugacy classes

class 1 2A2B2C2D···2I2J2K2L2M4A···4L4M···4AD
order12222···222224···44···4
size11112···244442···24···4

44 irreducible representations

dim1111111124
type+++++++++-
imageC1C2C2C2C2C2C2C2D42- 1+4
kernelC22.75C25C23.32C23C22.19C24C23.38C23Q85D4D4×Q8Q8×C23C2×2- 1+4C2×Q8C22
# reps1166881184

Matrix representation of C22.75C25 in GL6(𝔽5)

100000
010000
004000
000400
000040
000004
,
400000
040000
004000
000400
000040
000004
,
040000
400000
000010
003313
001000
000432
,
100000
010000
003000
000200
000020
000123
,
400000
010000
004000
000400
000010
002201
,
400000
040000
000100
004000
003313
000314
,
100000
010000
004000
000400
000010
002201

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

C22.75C25 in GAP, Magma, Sage, TeX

C_2^2._{75}C_2^5
% in TeX

G:=Group("C2^2.75C2^5");
// GroupNames label

G:=SmallGroup(128,2218);
// by ID

G=gap.SmallGroup(128,2218);
# by ID

G:=PCGroup([7,-2,2,2,2,2,-2,2,477,232,1430,570,136,1684]);
// Polycyclic

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