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G = C22.157C25order 128 = 27

138th central stem extension by C22 of C25

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

Aliases: C23.97C24, C24.160C23, C42.139C23, C22.157C25, C4⋊Q849C22, (C4×D4)⋊77C22, (C4×Q8)⋊73C22, C4⋊C4.510C23, (C2×C4).147C24, (C2×C42)⋊80C22, C22⋊Q858C22, C22≀C222C22, C24⋊C229C2, (C2×D4).345C23, C4.4D448C22, C22⋊C4.65C23, (C2×Q8).322C23, C42.C271C22, C422C250C22, C42⋊C271C22, C22.32C2427C2, C4⋊D4.127C22, (C22×C4).416C23, C22.45C2426C2, C2.68(C2.C25), C22.D465C22, C23.36C2365C2, C22.50C2443C2, C22.57C2423C2, C22.36C2444C2, (C2×C22⋊C4)⋊67C22, SmallGroup(128,2300)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C22.157C25
C1C2C22C2×C4C22×C4C2×C42C23.36C23 — C22.157C25
C1C22 — C22.157C25
C1C22 — C22.157C25
C1C22 — C22.157C25

Generators and relations for C22.157C25
 G = < a,b,c,d,e,f,g | a2=b2=d2=1, c2=e2=a, f2=g2=ba=ab, dcd=gcg-1=ac=ca, fdf-1=ad=da, ae=ea, af=fa, ag=ga, ece-1=fcf-1=bc=cb, ede-1=bd=db, be=eb, bf=fb, bg=gb, dg=gd, ef=fe, eg=ge, fg=gf >

Subgroups: 708 in 479 conjugacy classes, 378 normal (7 characteristic)
C1, C2 [×3], C2 [×7], C4 [×24], C22, C22 [×25], C2×C4 [×24], C2×C4 [×15], D4 [×15], Q8 [×9], C23, C23 [×6], C23 [×6], C42, C42 [×17], C22⋊C4 [×54], C4⋊C4 [×42], C22×C4 [×15], C2×D4 [×15], C2×Q8 [×9], C24 [×2], C2×C42, C2×C22⋊C4 [×6], C42⋊C2 [×12], C4×D4 [×15], C4×Q8 [×9], C22≀C2 [×8], C4⋊D4 [×9], C22⋊Q8 [×21], C22.D4 [×18], C4.4D4 [×21], C42.C2 [×3], C422C2 [×26], C4⋊Q8 [×6], C23.36C23 [×3], C22.32C24 [×6], C22.36C24 [×6], C22.45C24 [×6], C22.50C24 [×6], C24⋊C22, C22.57C24 [×3], C22.157C25
Quotients: C1, C2 [×31], C22 [×155], C23 [×155], C24 [×31], C25, C2.C25 [×3], C22.157C25

Smallest permutation representation of C22.157C25
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 10)(2 11)(3 12)(4 9)(5 30)(6 31)(7 32)(8 29)(13 25)(14 26)(15 27)(16 28)(17 21)(18 22)(19 23)(20 24)
(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 2)(3 4)(5 29)(6 32)(7 31)(8 30)(9 12)(10 11)(13 16)(14 15)(17 22)(18 21)(19 24)(20 23)(25 28)(26 27)
(1 16 3 14)(2 25 4 27)(5 22 7 24)(6 19 8 17)(9 15 11 13)(10 28 12 26)(18 32 20 30)(21 31 23 29)
(1 20 12 22)(2 21 9 19)(3 18 10 24)(4 23 11 17)(5 14 32 28)(6 27 29 13)(7 16 30 26)(8 25 31 15)
(1 15 12 25)(2 14 9 28)(3 13 10 27)(4 16 11 26)(5 21 32 19)(6 24 29 18)(7 23 30 17)(8 22 31 20)

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

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

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

38 conjugacy classes

class 1 2A2B2C2D···2J4A···4F4G···4AA
order12222···24···44···4
size11114···42···24···4

38 irreducible representations

dim111111114
type++++++++
imageC1C2C2C2C2C2C2C2C2.C25
kernelC22.157C25C23.36C23C22.32C24C22.36C24C22.45C24C22.50C24C24⋊C22C22.57C24C2
# reps136666136

Matrix representation of C22.157C25 in GL8(𝔽5)

40000000
04000000
00400000
00040000
00001000
00000100
00000010
00000001
,
10000000
01000000
00100000
00010000
00004000
00000400
00000040
00000004
,
31000000
02000000
02020000
03200000
00000020
00002224
00003000
00000333
,
24000000
33000000
33020000
02300000
00000020
00003331
00003000
00002202
,
30000000
03000000
00300000
00030000
00000010
00004443
00001000
00000001
,
40300000
00110000
10100000
44400000
00000100
00004000
00004443
00001011
,
43000000
11000000
01010000
44400000
00002000
00000200
00000020
00000002

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

C22.157C25 in GAP, Magma, Sage, TeX

C_2^2._{157}C_2^5
% in TeX

G:=Group("C2^2.157C2^5");
// GroupNames label

G:=SmallGroup(128,2300);
// by ID

G=gap.SmallGroup(128,2300);
# by ID

G:=PCGroup([7,-2,2,2,2,2,-2,2,224,477,1430,723,184,2019,570,360,1684,242]);
// Polycyclic

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