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G = C22.87C25order 128 = 27

68th central stem extension by C22 of C25

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

Aliases: C422+ 1+4, C23.43C24, C22.87C25, C42.575C23, C24.135C23, D4214C2, Q8(C4⋊Q8), C4○D410D4, D416(C2×D4), D4(C41D4), Q814(C2×D4), Q86D418C2, (C4×D4)⋊44C22, (C2×C4).77C24, C4⋊Q8108C22, (C4×Q8)⋊99C22, C2.33(D4×C23), C22≀C28C22, C41D451C22, C4⋊C4.489C23, C4⋊D427C22, (C2×C42)⋊59C22, C4.122(C22×D4), (C2×D4).470C23, C4.4D482C22, (C22×D4)⋊37C22, C22⋊C4.22C23, (C2×Q8).447C23, C22.14(C22×D4), C22.29C2422C2, (C2×2+ 1+4)⋊11C2, C42⋊C2100C22, C2.32(C2×2+ 1+4), C22.26C2436C2, (C22×C4).1208C23, (C2×C4)⋊5(C2×D4), (C4×C4○D4)⋊26C2, (C2×D4)(C41D4), (C2×C41D4)⋊26C2, (C2×C4○D4)⋊30C22, SmallGroup(128,2230)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C22.87C25
C1C2C22C2×C4C22×C4C2×C42C4×C4○D4 — C22.87C25
C1C22 — C22.87C25
C1C22 — C22.87C25
C1C22 — C22.87C25

Generators and relations for C22.87C25
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=f2=1, e2=b, g2=a, ab=ba, dcd=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 >

Subgroups: 1628 in 874 conjugacy classes, 432 normal (10 characteristic)
C1, C2, C2 [×2], C2 [×18], C4 [×12], C4 [×10], C22, C22 [×6], C22 [×66], C2×C4, C2×C4 [×27], C2×C4 [×36], D4 [×12], D4 [×108], Q8 [×4], Q8 [×4], C23 [×15], C23 [×48], C42, C42 [×9], C22⋊C4 [×30], C4⋊C4 [×10], C22×C4 [×21], C2×D4 [×69], C2×D4 [×96], C2×Q8, C2×Q8 [×2], C4○D4 [×8], C4○D4 [×40], C24 [×12], C2×C42 [×3], C42⋊C2 [×3], C4×D4 [×18], C4×Q8 [×2], C22≀C2 [×24], C4⋊D4 [×36], C4.4D4 [×6], C41D4 [×21], C4⋊Q8, C22×D4 [×30], C2×C4○D4, C2×C4○D4 [×10], 2+ 1+4 [×16], C4×C4○D4, C2×C41D4 [×3], C22.26C24 [×3], C22.29C24 [×6], D42 [×12], Q86D4 [×4], C2×2+ 1+4 [×2], C22.87C25
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], C22.87C25

Smallest permutation representation of C22.87C25
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 10)(2 9)(3 12)(4 11)(5 23)(6 22)(7 21)(8 24)(13 28)(14 27)(15 26)(16 25)(17 30)(18 29)(19 32)(20 31)
(1 6)(2 7)(3 8)(4 5)(9 29)(10 30)(11 31)(12 32)(13 21)(14 22)(15 23)(16 24)(17 27)(18 28)(19 25)(20 26)
(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 23)(2 24)(3 21)(4 22)(5 10)(6 11)(7 12)(8 9)(13 19)(14 20)(15 17)(16 18)(25 29)(26 30)(27 31)(28 32)
(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), (1,10)(2,9)(3,12)(4,11)(5,23)(6,22)(7,21)(8,24)(13,28)(14,27)(15,26)(16,25)(17,30)(18,29)(19,32)(20,31), (1,6)(2,7)(3,8)(4,5)(9,29)(10,30)(11,31)(12,32)(13,21)(14,22)(15,23)(16,24)(17,27)(18,28)(19,25)(20,26), (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), (1,23)(2,24)(3,21)(4,22)(5,10)(6,11)(7,12)(8,9)(13,19)(14,20)(15,17)(16,18)(25,29)(26,30)(27,31)(28,32), (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), (1,10)(2,9)(3,12)(4,11)(5,23)(6,22)(7,21)(8,24)(13,28)(14,27)(15,26)(16,25)(17,30)(18,29)(19,32)(20,31), (1,6)(2,7)(3,8)(4,5)(9,29)(10,30)(11,31)(12,32)(13,21)(14,22)(15,23)(16,24)(17,27)(18,28)(19,25)(20,26), (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), (1,23)(2,24)(3,21)(4,22)(5,10)(6,11)(7,12)(8,9)(13,19)(14,20)(15,17)(16,18)(25,29)(26,30)(27,31)(28,32), (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)], [(1,10),(2,9),(3,12),(4,11),(5,23),(6,22),(7,21),(8,24),(13,28),(14,27),(15,26),(16,25),(17,30),(18,29),(19,32),(20,31)], [(1,6),(2,7),(3,8),(4,5),(9,29),(10,30),(11,31),(12,32),(13,21),(14,22),(15,23),(16,24),(17,27),(18,28),(19,25),(20,26)], [(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)], [(1,23),(2,24),(3,21),(4,22),(5,10),(6,11),(7,12),(8,9),(13,19),(14,20),(15,17),(16,18),(25,29),(26,30),(27,31),(28,32)], [(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···2I2J···2U4A···4L4M···4V
order12222···22···24···44···4
size11112···24···42···24···4

44 irreducible representations

dim1111111124
type++++++++++
imageC1C2C2C2C2C2C2C2D42+ 1+4
kernelC22.87C25C4×C4○D4C2×C41D4C22.26C24C22.29C24D42Q86D4C2×2+ 1+4C4○D4C4
# reps11336124284

Matrix representation of C22.87C25 in GL6(ℤ)

100000
010000
00-1000
000-100
0000-10
00000-1
,
-100000
0-10000
001000
000100
000010
000001
,
-1-20000
010000
000-100
00-1000
00000-1
0000-10
,
-100000
0-10000
000001
0000-10
000-100
001000
,
-1-20000
110000
00-1000
000-100
0000-10
00000-1
,
100000
010000
000010
000001
001000
000100
,
100000
010000
000100
00-1000
000001
0000-10

G:=sub<GL(6,Integers())| [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,-1,0,0,0,0,0,0,-1],[-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[-1,0,0,0,0,0,-2,1,0,0,0,0,0,0,0,-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,-1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,-1,0,0,0,0,-1,0,0,0,0,1,0,0,0],[-1,1,0,0,0,0,-2,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,-1,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,1,0] >;

C22.87C25 in GAP, Magma, Sage, TeX

C_2^2._{87}C_2^5
% in TeX

G:=Group("C2^2.87C2^5");
// GroupNames label

G:=SmallGroup(128,2230);
// by ID

G=gap.SmallGroup(128,2230);
# by ID

G:=PCGroup([7,-2,2,2,2,2,-2,2,477,1430,352,570,1684,102]);
// Polycyclic

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