Copied to
clipboard

G = C23.439C24order 128 = 27

156th central stem extension by C23 of C24

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

Aliases: C25.48C22, C24.26C23, C23.439C24, C22.2282+ 1+4, C22⋊C427D4, (C22×C4)⋊29D4, C232D417C2, (C2×C42)⋊26C22, C23.429(C2×D4), (C22×D4)⋊7C22, C2.67(D45D4), C23.4Q819C2, C23.151(C4○D4), C22.14(C41D4), C2.12(C233D4), (C22×C4).832C23, (C23×C4).392C22, C22.290(C22×D4), C24.3C2252C2, C2.C4268C22, C2.60(C22.19C24), C2.9(C2×C41D4), (C2×C22≀C2)⋊8C2, (C2×C4⋊D4)⋊16C2, (C2×C4⋊C4)⋊22C22, (C4×C22⋊C4)⋊83C2, (C2×C4).352(C2×D4), (C22×C22⋊C4)⋊24C2, (C2×C22⋊C4)⋊21C22, C22.316(C2×C4○D4), (C2×C22.D4)⋊20C2, SmallGroup(128,1271)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C23.439C24
C1C2C22C23C24C25C22×C22⋊C4 — C23.439C24
C1C23 — C23.439C24
C1C23 — C23.439C24
C1C23 — C23.439C24

Generators and relations for C23.439C24
 G = < a,b,c,d,e,f,g | a2=b2=c2=f2=g2=1, d2=b, 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 >

Subgroups: 1028 in 450 conjugacy classes, 120 normal (18 characteristic)
C1, C2 [×3], C2 [×4], C2 [×10], C4 [×16], C22, C22 [×10], C22 [×62], C2×C4 [×12], C2×C4 [×32], D4 [×28], C23, C23 [×10], C23 [×62], C42 [×2], C22⋊C4 [×8], C22⋊C4 [×28], C4⋊C4 [×10], C22×C4 [×2], C22×C4 [×12], C22×C4 [×8], C2×D4 [×36], C24, C24 [×4], C24 [×12], C2.C42 [×2], C2×C42 [×2], C2×C22⋊C4 [×16], C2×C22⋊C4 [×4], C2×C4⋊C4, C2×C4⋊C4 [×4], C22≀C2 [×8], C4⋊D4 [×4], C22.D4 [×8], C23×C4 [×2], C22×D4, C22×D4 [×6], C25, C4×C22⋊C4, C24.3C22 [×4], C232D4 [×2], C23.4Q8 [×2], C22×C22⋊C4, C2×C22≀C2 [×2], C2×C4⋊D4, C2×C22.D4 [×2], C23.439C24
Quotients: C1, C2 [×15], C22 [×35], D4 [×12], C23 [×15], C2×D4 [×18], C4○D4 [×4], C24, C41D4 [×4], C22×D4 [×3], C2×C4○D4 [×2], 2+ 1+4 [×2], C22.19C24, C2×C41D4, C233D4, D45D4 [×4], C23.439C24

Smallest permutation representation of C23.439C24
On 32 points
Generators in S32
(1 23)(2 24)(3 21)(4 22)(5 9)(6 10)(7 11)(8 12)(13 25)(14 26)(15 27)(16 28)(17 30)(18 31)(19 32)(20 29)
(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 27)(2 28)(3 25)(4 26)(5 20)(6 17)(7 18)(8 19)(9 29)(10 30)(11 31)(12 32)(13 21)(14 22)(15 23)(16 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 18 15 11)(2 32 16 8)(3 20 13 9)(4 30 14 6)(5 21 29 25)(7 23 31 27)(10 22 17 26)(12 24 19 28)
(1 3)(5 18)(6 17)(7 20)(8 19)(9 31)(10 30)(11 29)(12 32)(13 15)(21 23)(25 27)
(1 25)(2 14)(3 27)(4 16)(5 18)(6 32)(7 20)(8 30)(9 31)(10 19)(11 29)(12 17)(13 23)(15 21)(22 28)(24 26)

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

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

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

38 conjugacy classes

class 1 2A···2G2H2I2J2K2L2M2N2O2P2Q4A4B4C4D4E···4R4S4T
order12···2222222222244444···444
size11···1222244448822224···488

38 irreducible representations

dim1111111112224
type++++++++++++
imageC1C2C2C2C2C2C2C2C2D4D4C4○D42+ 1+4
kernelC23.439C24C4×C22⋊C4C24.3C22C232D4C23.4Q8C22×C22⋊C4C2×C22≀C2C2×C4⋊D4C2×C22.D4C22⋊C4C22×C4C23C22
# reps1142212128482

Matrix representation of C23.439C24 in GL6(𝔽5)

400000
040000
004000
000400
000010
000001
,
100000
010000
004000
000400
000010
000001
,
100000
010000
001000
000100
000040
000004
,
010000
100000
000100
004000
000040
000004
,
200000
030000
003000
000200
000033
000002
,
100000
010000
004000
000100
000010
000034
,
100000
040000
004000
000100
000040
000004

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

C23.439C24 in GAP, Magma, Sage, TeX

C_2^3._{439}C_2^4
% in TeX

G:=Group("C2^3.439C2^4");
// GroupNames label

G:=SmallGroup(128,1271);
// by ID

G=gap.SmallGroup(128,1271);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,224,253,568,758,723,675]);
// Polycyclic

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

׿
×
𝔽