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G = C23.584C24order 128 = 27

301st central stem extension by C23 of C24

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

Aliases: C25.58C22, C23.584C24, C24.391C23, C22.3582+ 1+4, (C2×D4).138D4, C23.63(C2×D4), C243C424C2, (C23×C4)⋊12C22, (C2×C42)⋊31C22, C23⋊Q841C2, C2.89(D45D4), (C22×Q8)⋊8C22, C23.168(C4○D4), C23.23D483C2, C23.10D479C2, C23.11D477C2, C2.42(C233D4), (C22×C4).179C23, C22.393(C22×D4), C2.C4237C22, C2.5(C24⋊C22), (C22×D4).223C22, C24.C22122C2, C2.61(C22.32C24), C2.75(C22.45C24), (C2×C4⋊C4)⋊33C22, (C2×C4).416(C2×D4), (C2×C4.4D4)⋊27C2, (C2×C22≀C2).13C2, (C2×C22⋊C4)⋊30C22, C22.446(C2×C4○D4), SmallGroup(128,1416)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C23.584C24
C1C2C22C23C24C25C243C4 — C23.584C24
C1C23 — C23.584C24
C1C23 — C23.584C24
C1C23 — C23.584C24

Generators and relations for C23.584C24
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=f2=g2=1, e2=b, ab=ba, ac=ca, ede-1=ad=da, geg=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, gdg=abd, fg=gf >

Subgroups: 836 in 344 conjugacy classes, 96 normal (22 characteristic)
C1, C2 [×3], C2 [×4], C2 [×8], C4 [×12], C22 [×3], C22 [×4], C22 [×56], C2×C4 [×2], C2×C4 [×36], D4 [×12], Q8 [×4], C23, C23 [×8], C23 [×52], C42 [×2], C22⋊C4 [×27], C4⋊C4 [×2], C22×C4, C22×C4 [×10], C22×C4 [×3], C2×D4 [×4], C2×D4 [×12], C2×Q8 [×4], C24 [×2], C24 [×2], C24 [×10], C2.C42 [×8], C2×C42, C2×C22⋊C4 [×2], C2×C22⋊C4 [×16], C2×C4⋊C4 [×2], C22≀C2 [×4], C4.4D4 [×4], C23×C4, C22×D4, C22×D4 [×2], C22×Q8, C25, C243C4 [×2], C23.23D4, C23.23D4 [×2], C24.C22 [×2], C23⋊Q8 [×2], C23.10D4 [×2], C23.11D4 [×2], C2×C22≀C2, C2×C4.4D4, C23.584C24
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×4], C24, C22×D4, C2×C4○D4 [×2], 2+ 1+4 [×4], C233D4, C22.32C24 [×2], D45D4 [×2], C22.45C24, C24⋊C22, C23.584C24

Smallest permutation representation of C23.584C24
On 32 points
Generators in S32
(1 7)(2 8)(3 5)(4 6)(9 30)(10 31)(11 32)(12 29)(13 21)(14 22)(15 23)(16 24)(17 27)(18 28)(19 25)(20 26)
(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 15)(2 16)(3 13)(4 14)(5 21)(6 22)(7 23)(8 24)(9 27)(10 28)(11 25)(12 26)(17 30)(18 31)(19 32)(20 29)
(1 28)(2 19)(3 26)(4 17)(5 20)(6 27)(7 18)(8 25)(9 22)(10 15)(11 24)(12 13)(14 30)(16 32)(21 29)(23 31)
(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 13)(2 4)(3 15)(5 23)(6 8)(7 21)(10 28)(12 26)(14 16)(18 31)(20 29)(22 24)
(1 15)(2 24)(3 13)(4 22)(5 21)(6 14)(7 23)(8 16)(9 25)(10 20)(11 27)(12 18)(17 32)(19 30)(26 31)(28 29)

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

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

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

32 conjugacy classes

class 1 2A···2G2H···2O4A···4J4K···4P
order12···22···24···44···4
size11···14···44···48···8

32 irreducible representations

dim111111111224
type+++++++++++
imageC1C2C2C2C2C2C2C2C2D4C4○D42+ 1+4
kernelC23.584C24C243C4C23.23D4C24.C22C23⋊Q8C23.10D4C23.11D4C2×C22≀C2C2×C4.4D4C2×D4C23C22
# reps123222211484

Matrix representation of C23.584C24 in GL6(𝔽5)

100000
010000
001000
000100
000040
000004
,
400000
040000
001000
000100
000010
000001
,
100000
010000
004000
000400
000010
000001
,
010000
100000
001000
000100
000014
000004
,
300000
030000
000100
001000
000032
000012
,
400000
010000
004000
000100
000010
000001
,
100000
040000
004000
000400
000010
000024

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

C23.584C24 in GAP, Magma, Sage, TeX

C_2^3._{584}C_2^4
% in TeX

G:=Group("C2^3.584C2^4");
// GroupNames label

G:=SmallGroup(128,1416);
// by ID

G=gap.SmallGroup(128,1416);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,253,232,758,723,1571,346]);
// 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=b,a*b=b*a,a*c=c*a,e*d*e^-1=a*d=d*a,g*e*g=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,g*d*g=a*b*d,f*g=g*f>;
// generators/relations

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