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G = C23.335C24  order 128 = 27

52nd central stem extension by C23 of C24

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

Aliases: C25.41C22, C23.335C24, C24.266C23, C22.1452+ (1+4), C4.31C22≀C2, (C2×D4).285D4, C24⋊3C4⋊16C2, (C2×C42)⋊23C22, (D4×C23).12C2, C23.163(C2×D4), (C22×C4).378D4, C2.31(D4⋊5D4), (C22×Q8)⋊2C22, C22⋊3(C4.4D4), (C22×C4).56C23, C23.302(C4○D4), C23.10D4⋊22C2, (C23×C4).348C22, C22.215(C22×D4), C2.C42⋊66C22, C24.3C22⋊38C2, (C22×D4).508C22, C2.14(C22.29C24), (C2×C22⋊Q8)⋊9C2, (C4×C22⋊C4)⋊58C2, (C2×C4⋊C4)⋊16C22, (C2×C4.4D4)⋊8C2, (C2×C4).319(C2×D4), C2.23(C2×C22≀C2), C2.11(C2×C4.4D4), (C2×C22⋊C4)⋊18C22, C22.212(C2×C4○D4), SmallGroup(128,1167)

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

Derived series
C1 — C23 — C23.335C24
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C2×C42 — C4×C22⋊C4 — C23.335C24
Lower central
C1 — C23 — C23.335C24
Upper central
C1 — C23 — C23.335C24
Jennings
C1 — C23 — C23.335C24

Subgroups: 1220 in 544 conjugacy classes, 124 normal (16 characteristic)
C1, C2 [×3], C2 [×4], C2 [×12], C4 [×4], C4 [×10], C22, C22 [×10], C22 [×84], C2×C4 [×8], C2×C4 [×34], D4 [×32], Q8 [×4], C23, C23 [×14], C23 [×92], C42 [×4], C22⋊C4 [×32], C4⋊C4 [×6], C22×C4 [×2], C22×C4 [×12], C22×C4 [×4], C2×D4 [×8], C2×D4 [×52], C2×Q8 [×6], C24, C24 [×4], C24 [×24], C2.C42 [×2], C2×C42 [×2], C2×C22⋊C4 [×20], C2×C4⋊C4, C2×C4⋊C4 [×2], C22⋊Q8 [×4], C4.4D4 [×8], C23×C4, C22×D4 [×4], C22×D4 [×12], C22×Q8, C25 [×2], C4×C22⋊C4, C24⋊3C4 [×4], C24.3C22 [×2], C23.10D4 [×4], C2×C22⋊Q8, C2×C4.4D4 [×2], D4×C23, C23.335C24

Quotients:
C1, C2 [×15], C22 [×35], D4 [×12], C23 [×15], C2×D4 [×18], C4○D4 [×4], C24, C22≀C2 [×4], C4.4D4 [×4], C22×D4 [×3], C2×C4○D4 [×2], 2+ (1+4) [×2], C2×C22≀C2, C2×C4.4D4, C22.29C24, D4⋊5D4 [×4], C23.335C24

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

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

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

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

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

Matrix representation ►G ⊆ GL6(𝔽5)

400000
040000
001000
000100
000010
000001
,
100000
010000
004000
000400
000040
000004
,
100000
010000
004000
000400
000010
000001
,
010000
100000
004000
001100
000040
000001
,
010000
400000
004000
001100
000010
000001
,
200000
020000
004300
001100
000001
000040
,
010000
400000
001000
000100
000040
000004

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

38 conjugacy classes

class 1 2A···2G2H2I2J2K2L···2S4A4B4C4D4E···4N4O4P4Q4R
order12···222222···244444···44444
size11···122224···422224···48888

38 irreducible representations

dim111111112224
type+++++++++++
imageC1C2C2C2C2C2C2C2D4D4C4○D42+ (1+4)
kernelC23.335C24C4×C22⋊C4C24⋊3C4C24.3C22C23.10D4C2×C22⋊Q8C2×C4.4D4D4×C23C22×C4C2×D4C23C22
# reps114241214882

In GAP, Magma, Sage, TeX 

C_2^3._{335}C_2^4
 
% in TeX
 
G:=Group("C2^3.335C2^4");
 
// GroupNames label
 
G:=SmallGroup(128,1167);
 
// by ID
 
G=gap.SmallGroup(128,1167);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,253,120,758,723,268,675,80]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=1,e^2=g^2=a,f^2=b*a=a*b,a*c=c*a,e*d*e^-1=g*d*g^-1=a*d=d*a,a*e=e*a,a*f=f*a,a*g=g*a,b*c=c*b,f*d*f^-1=b*d=d*b,b*e=e*b,b*f=f*b,b*g=g*b,c*d=d*c,f*e*f^-1=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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