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G = C23.434C24  order 128 = 27

151st central stem extension by C23 of C24

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

Aliases: C25.47C22, C24.317C23, C23.434C24, C22.2242+ (1+4), (C22×C4)⋊28D4, (C2×D4).211D4, C23.45(C2×D4), C24⋊3C4⋊18C2, (C2×C42)⋊25C22, (C23×C4)⋊11C22, C2.65(D4⋊5D4), C23.7Q8⋊62C2, C23.Q8⋊26C2, C23.150(C4○D4), C23.10D4⋊38C2, C23.34D4⋊35C2, C23.23D4⋊52C2, C2.11(C23⋊3D4), C22.76(C4⋊D4), (C22×C4).535C23, C22.285(C22×D4), C2.C42⋊26C22, C24.C22⋊76C2, (C22×D4).160C22, C2.57(C22.19C24), C2.45(C22.45C24), (C2×C4×D4)⋊41C2, (C2×C4⋊C4)⋊21C22, (C2×C4).350(C2×D4), C2.29(C2×C4⋊D4), (C2×C22≀C2).11C2, (C22×C22⋊C4)⋊23C2, (C2×C22⋊C4)⋊20C22, C22.311(C2×C4○D4), (C2×C22.D4)⋊19C2, SmallGroup(128,1266)

Series: Derived ►Chief ►Lower central ►Upper central ►Jennings

Derived series
C1 — C23 — C23.434C24
Chief series
C1 — C2 — C22 — C23 — C24 — C25 — C22×C22⋊C4 — C23.434C24
Lower central
C1 — C23 — C23.434C24
Upper central
C1 — C23 — C23.434C24
Jennings
C1 — C23 — C23.434C24

Subgroups: 884 in 400 conjugacy classes, 112 normal (34 characteristic)
C1, C2 [×3], C2 [×4], C2 [×10], C4 [×14], C22 [×3], C22 [×8], C22 [×58], C2×C4 [×6], C2×C4 [×42], D4 [×16], C23, C23 [×12], C23 [×54], C42 [×2], C22⋊C4 [×30], C4⋊C4 [×8], C22×C4 [×5], C22×C4 [×10], C22×C4 [×12], C2×D4 [×4], C2×D4 [×14], C24 [×2], C24 [×2], C24 [×12], C2.C42 [×6], C2×C42, C2×C22⋊C4 [×4], C2×C22⋊C4 [×12], C2×C22⋊C4 [×4], C2×C4⋊C4, C2×C4⋊C4 [×4], C4×D4 [×4], C22≀C2 [×4], C22.D4 [×4], C23×C4 [×3], C22×D4, C22×D4 [×2], C25, C24⋊3C4, C23.7Q8, C23.34D4, C23.23D4 [×2], C24.C22 [×2], C23.10D4 [×2], C23.Q8 [×2], C22×C22⋊C4, C2×C4×D4, C2×C22≀C2, C2×C22.D4, C23.434C24

Quotients:
C1, C2 [×15], C22 [×35], D4 [×8], C23 [×15], C2×D4 [×12], C4○D4 [×6], C24, C4⋊D4 [×4], C22×D4 [×2], C2×C4○D4 [×3], 2+ (1+4) [×2], C2×C4⋊D4, C22.19C24, C23⋊3D4, D4⋊5D4 [×2], C22.45C24 [×2], C23.434C24

Generators and relations
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=f2=g2=1, e2=ca=ac, ab=ba, ede-1=gdg=ad=da, ae=ea, af=fa, ag=ga, bc=cb, fdf=bd=db, be=eb, bf=fb, bg=gb, cd=dc, fef=ce=ec, cf=fc, cg=gc, eg=ge, fg=gf >

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

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

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

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

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

Matrix representation ►G ⊆ GL6(𝔽5)

100000
010000
004000
000400
000010
000001
,
400000
040000
001000
000100
000040
000004
,
100000
010000
001000
000100
000040
000004
,
010000
100000
000100
001000
000021
000023
,
100000
010000
003000
000200
000013
000014
,
400000
010000
001000
000100
000040
000041
,
400000
040000
001000
000400
000010
000001

G:=sub<GL(6,GF(5))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,2,2,0,0,0,0,1,3],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,3,0,0,0,0,0,0,2,0,0,0,0,0,0,1,1,0,0,0,0,3,4],[4,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,4,0,0,0,0,0,1],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;
 

38 conjugacy classes

class 1 2A···2G2H2I2J2K2L···2Q4A4B4C4D4E···4P4Q4R4S4T
order12···222222···244444···44444
size11···122224···422224···48888

38 irreducible representations

dim1111111111112224
type+++++++++++++++
imageC1C2C2C2C2C2C2C2C2C2C2C2D4D4C4○D42+ (1+4)
kernelC23.434C24C24⋊3C4C23.7Q8C23.34D4C23.23D4C24.C22C23.10D4C23.Q8C22×C22⋊C4C2×C4×D4C2×C22≀C2C2×C22.D4C22×C4C2×D4C23C22
# reps11112222111144122

In GAP, Magma, Sage, TeX 

C_2^3._{434}C_2^4
 
% in TeX
 
G:=Group("C2^3.434C2^4");
 
// GroupNames label
 
G:=SmallGroup(128,1266);
 
// by ID
 
G=gap.SmallGroup(128,1266);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,253,568,758,723,675]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=f^2=g^2=1,e^2=c*a=a*c,a*b=b*a,e*d*e^-1=g*d*g=a*d=d*a,a*e=e*a,a*f=f*a,a*g=g*a,b*c=c*b,f*d*f=b*d=d*b,b*e=e*b,b*f=f*b,b*g=g*b,c*d=d*c,f*e*f=c*e=e*c,c*f=f*c,c*g=g*c,e*g=g*e,f*g=g*f>;
 
// generators/relations
 

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