Copied to
clipboard

G = C24.162C23order 128 = 27

2nd non-split extension by C24 of C23 acting via C23/C2=C22

p-group, metabelian, nilpotent (class 3), monomial

Aliases: C24.162C23, (C2×C42)⋊5C4, (C2×C4).68C42, C42⋊C217C4, C23.51(C2×Q8), (C22×C4).38Q8, C4(C23.9D4), C23.539(C2×D4), (C22×C4).755D4, C22.5(C2×C42), C23.9D4.9C2, (C23×C4).220C22, C23.175(C22×C4), C4.21(C2.C42), C2.4(C23.C23), (C2×C4⋊C4)⋊24C4, C22.13(C2×C4⋊C4), (C2×C4).125(C4⋊C4), C22⋊C4.47(C2×C4), (C22×C4).46(C2×C4), (C2×C42⋊C2).9C2, (C2×C4).113(C22⋊C4), C22.108(C2×C22⋊C4), C2.17(C2×C2.C42), (C2×C22⋊C4).407C22, SmallGroup(128,472)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C24.162C23
C1C2C22C23C24C23×C4C2×C42⋊C2 — C24.162C23
C1C2C22 — C24.162C23
C1C2×C4C23×C4 — C24.162C23
C1C2C24 — C24.162C23

Generators and relations for C24.162C23
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=1, e2=b, f2=c, g2=d, ab=ba, ac=ca, eae-1=ad=da, af=fa, ag=ga, bc=cb, fbf-1=bd=db, be=eb, bg=gb, cd=dc, ce=ec, cf=fc, cg=gc, de=ed, df=fd, dg=gd, fef-1=ade, eg=ge, fg=gf >

Subgroups: 372 in 210 conjugacy classes, 108 normal (16 characteristic)
C1, C2, C2 [×2], C2 [×6], C4 [×8], C4 [×12], C22 [×3], C22 [×4], C22 [×10], C2×C4 [×2], C2×C4 [×26], C2×C4 [×24], C23 [×3], C23 [×4], C23 [×2], C42 [×12], C22⋊C4 [×8], C22⋊C4 [×8], C4⋊C4 [×12], C22×C4 [×2], C22×C4 [×16], C22×C4 [×4], C24, C2×C42 [×2], C2×C42 [×2], C2×C22⋊C4 [×6], C2×C4⋊C4 [×2], C2×C4⋊C4 [×2], C42⋊C2 [×8], C42⋊C2 [×8], C23×C4, C23.9D4 [×4], C2×C42⋊C2, C2×C42⋊C2 [×2], C24.162C23
Quotients: C1, C2 [×7], C4 [×12], C22 [×7], C2×C4 [×18], D4 [×6], Q8 [×2], C23, C42 [×4], C22⋊C4 [×12], C4⋊C4 [×12], C22×C4 [×3], C2×D4 [×3], C2×Q8, C2.C42 [×8], C2×C42, C2×C22⋊C4 [×3], C2×C4⋊C4 [×3], C2×C2.C42, C23.C23 [×2], C24.162C23

Smallest permutation representation of C24.162C23
On 32 points
Generators in S32
(1 6)(2 25)(3 8)(4 27)(5 16)(7 14)(9 23)(10 32)(11 21)(12 30)(13 28)(15 26)(17 24)(18 29)(19 22)(20 31)
(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 26)(2 27)(3 28)(4 25)(5 14)(6 15)(7 16)(8 13)(9 29)(10 30)(11 31)(12 32)(17 22)(18 23)(19 24)(20 21)
(1 13)(2 14)(3 15)(4 16)(5 27)(6 28)(7 25)(8 26)(9 20)(10 17)(11 18)(12 19)(21 29)(22 30)(23 31)(24 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 32 26 12)(2 18 27 23)(3 22 28 17)(4 9 25 29)(5 31 14 11)(6 10 15 30)(7 21 16 20)(8 19 13 24)
(1 11 13 18)(2 12 14 19)(3 9 15 20)(4 10 16 17)(5 24 27 32)(6 21 28 29)(7 22 25 30)(8 23 26 31)

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

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

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

44 conjugacy classes

class 1 2A2B2C2D···2I4A4B4C4D4E···4J4K···4AH
order12222···244444···44···4
size11112···211112···24···4

44 irreducible representations

dim111111224
type++++-
imageC1C2C2C4C4C4D4Q8C23.C23
kernelC24.162C23C23.9D4C2×C42⋊C2C2×C42C2×C4⋊C4C42⋊C2C22×C4C22×C4C2
# reps1434416624

Matrix representation of C24.162C23 in GL6(𝔽5)

400000
040000
000400
004000
000004
000040
,
400000
040000
000100
001000
000004
000040
,
400000
040000
001000
000100
000010
000001
,
100000
010000
004000
000400
000040
000004
,
430000
110000
000002
000030
002000
000300
,
240000
030000
000010
000001
001000
000100
,
100000
010000
003000
000300
000030
000003

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

C24.162C23 in GAP, Magma, Sage, TeX

C_2^4._{162}C_2^3
% in TeX

G:=Group("C2^4.162C2^3");
// GroupNames label

G:=SmallGroup(128,472);
// by ID

G=gap.SmallGroup(128,472);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,112,141,232,352,2019,1411]);
// Polycyclic

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

׿
×
𝔽