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G = C22.130C25order 128 = 27

111st central stem extension by C22 of C25

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

Aliases: C23.71C24, C42.113C23, C24.518C23, C22.130C25, C4.892+ 1+4, C22.32- 1+4, (D4×Q8)⋊27C2, C4⋊Q843C22, D46D435C2, D43Q834C2, Q85D428C2, (C4×D4)⋊64C22, (C4×Q8)⋊61C22, C4⋊C4.318C23, C4⋊D438C22, (C2×C4).120C24, C22⋊Q848C22, (C2×D4).322C23, C4.4D439C22, C22⋊C4.46C23, (C2×Q8).463C23, C42.C223C22, (C22×Q8)⋊42C22, C422C217C22, C42⋊C258C22, C22.19C2443C2, C22.32C2417C2, C22≀C2.33C22, (C23×C4).620C22, (C22×C4).390C23, C2.42(C2×2- 1+4), C2.59(C2×2+ 1+4), C2.48(C2.C25), C22.56C246C2, C22.57C247C2, C22.D419C22, C22.50C2432C2, C23.38C2329C2, C22.47C2430C2, C22.33C2416C2, C22.31C2422C2, C22.35C2418C2, C22.46C2431C2, C22.36C2428C2, (C2×C4⋊C4)⋊88C22, (C2×C22⋊Q8)⋊85C2, (C2×C4○D4)⋊47C22, (C2×C22⋊C4).392C22, SmallGroup(128,2273)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C22.130C25
C1C2C22C2×C4C22×C4C23×C4C22.19C24 — C22.130C25
C1C22 — C22.130C25
C1C22 — C22.130C25
C1C22 — C22.130C25

Generators and relations for C22.130C25
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=e2=1, f2=g2=a, ab=ba, dcd=gcg-1=ac=ca, fdf-1=ad=da, ae=ea, af=fa, ag=ga, ece=fcf-1=bc=cb, ede=bd=db, be=eb, bf=fb, bg=gb, dg=gd, ef=fe, eg=ge, fg=gf >

Subgroups: 772 in 516 conjugacy classes, 382 normal (122 characteristic)
C1, C2 [×3], C2 [×8], C4 [×2], C4 [×23], C22, C22 [×2], C22 [×24], C2×C4 [×10], C2×C4 [×14], C2×C4 [×27], D4 [×23], Q8 [×13], C23 [×5], C23 [×2], C23 [×4], C42 [×2], C42 [×10], C22⋊C4 [×10], C22⋊C4 [×34], C4⋊C4 [×10], C4⋊C4 [×38], C22×C4 [×8], C22×C4 [×14], C22×C4 [×2], C2×D4 [×8], C2×D4 [×10], C2×Q8 [×4], C2×Q8 [×6], C2×Q8 [×4], C4○D4 [×6], C24, C2×C22⋊C4 [×2], C2×C4⋊C4 [×2], C2×C4⋊C4 [×4], C42⋊C2 [×3], C42⋊C2 [×2], C4×D4 [×5], C4×D4 [×14], C4×Q8 [×3], C4×Q8 [×2], C22≀C2 [×2], C22≀C2 [×2], C4⋊D4 [×5], C4⋊D4 [×14], C22⋊Q8 [×11], C22⋊Q8 [×20], C22.D4 [×2], C22.D4 [×16], C4.4D4, C4.4D4 [×10], C42.C2, C42.C2 [×8], C422C2 [×14], C4⋊Q8 [×2], C4⋊Q8 [×4], C23×C4, C22×Q8 [×2], C2×C4○D4 [×3], C2×C22⋊Q8, C22.19C24 [×2], C23.38C23, C22.31C24, C22.32C24 [×4], C22.33C24 [×2], C22.35C24, C22.36C24, C22.36C24 [×2], D46D4, Q85D4 [×2], D4×Q8, C22.46C24, C22.47C24, C22.47C24 [×2], D43Q8, D43Q8 [×2], C22.50C24, C22.56C24 [×2], C22.57C24 [×2], C22.130C25
Quotients: C1, C2 [×31], C22 [×155], C23 [×155], C24 [×31], 2+ 1+4 [×2], 2- 1+4 [×2], C25, C2×2+ 1+4, C2×2- 1+4, C2.C25, C22.130C25

Smallest permutation representation of C22.130C25
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 9)(2 10)(3 11)(4 12)(5 31)(6 32)(7 29)(8 30)(13 19)(14 20)(15 17)(16 18)(21 27)(22 28)(23 25)(24 26)
(1 28)(2 23)(3 26)(4 21)(5 17)(6 16)(7 19)(8 14)(9 22)(10 25)(11 24)(12 27)(13 29)(15 31)(18 32)(20 30)
(1 14)(2 13)(3 16)(4 15)(5 25)(6 28)(7 27)(8 26)(9 20)(10 19)(11 18)(12 17)(21 29)(22 32)(23 31)(24 30)
(1 9)(2 10)(3 11)(4 12)(5 31)(6 32)(7 29)(8 30)
(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 4 3 2)(5 30 7 32)(6 31 8 29)(9 12 11 10)(13 14 15 16)(17 18 19 20)(21 28 23 26)(22 25 24 27)

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

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

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

38 conjugacy classes

class 1 2A2B2C2D2E2F···2K4A4B4C4D4E···4Z
order1222222···244444···4
size1111224···422224···4

38 irreducible representations

dim111111111111111111444
type+++++++++++++++++++-
imageC1C2C2C2C2C2C2C2C2C2C2C2C2C2C2C2C2C22+ 1+42- 1+4C2.C25
kernelC22.130C25C2×C22⋊Q8C22.19C24C23.38C23C22.31C24C22.32C24C22.33C24C22.35C24C22.36C24D46D4Q85D4D4×Q8C22.46C24C22.47C24D43Q8C22.50C24C22.56C24C22.57C24C4C22C2
# reps112114213121133122222

Matrix representation of C22.130C25 in GL8(𝔽5)

10000000
01000000
00100000
00010000
00004000
00000400
00000040
00000004
,
40000000
04000000
00400000
00040000
00001000
00000100
00000010
00000001
,
10300000
00110000
00400000
01100000
00000034
00000002
00002100
00000300
,
43000000
01000000
04010000
01100000
00000010
00000001
00001000
00000100
,
40000000
11000000
40100000
00040000
00001000
00000100
00000010
00000001
,
10000000
01000000
10400000
40040000
00004200
00004100
00000013
00000014
,
10000000
01000000
00100000
00010000
00001300
00001400
00000013
00000014

G:=sub<GL(8,GF(5))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,3,1,4,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,1,3,0,0,0,0,3,0,0,0,0,0,0,0,4,2,0,0],[4,0,0,0,0,0,0,0,3,1,4,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[4,1,4,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,1,4,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,4,0,0,0,0,0,0,2,1,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,3,4],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,3,4,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,3,4] >;

C22.130C25 in GAP, Magma, Sage, TeX

C_2^2._{130}C_2^5
% in TeX

G:=Group("C2^2.130C2^5");
// GroupNames label

G:=SmallGroup(128,2273);
// by ID

G=gap.SmallGroup(128,2273);
# by ID

G:=PCGroup([7,-2,2,2,2,2,-2,2,477,1430,723,2019,570,136,1684,102]);
// Polycyclic

G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=e^2=1,f^2=g^2=a,a*b=b*a,d*c*d=g*c*g^-1=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=f*c*f^-1=b*c=c*b,e*d*e=b*d=d*b,b*e=e*b,b*f=f*b,b*g=g*b,d*g=g*d,e*f=f*e,e*g=g*e,f*g=g*f>;
// generators/relations

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