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G = C23.224C24order 128 = 27

77th central extension by C23 of C24

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

Aliases: C25.23C22, C23.224C24, C24.551C23, C22.612+ 1+4, C24.70(C2×C4), (C2×C42)⋊4C22, C22.65(C4×D4), C2.3(D45D4), C243C4.5C2, C22⋊C4.182D4, C23.413(C2×D4), C23.8Q812C2, C23.7Q822C2, C222(C42⋊C2), C23.287(C4○D4), C23.34D415C2, (C22×C4).753C23, (C23×C4).298C22, C22.115(C23×C4), C23.215(C22×C4), C22.103(C22×D4), C2.C4252C22, C24.C2212C2, C2.1(C22.45C24), C2.24(C22.11C24), C2.29(C2×C4×D4), (C4×C22⋊C4)⋊8C2, (C2×C4⋊C4)⋊8C22, C22⋊C439(C2×C4), (C2×C22⋊C4)⋊23C4, (C22×C4)⋊30(C2×C4), (C2×C4).884(C2×D4), C22⋊C43(C22⋊C4), (C2×C42⋊C2)⋊11C2, (C2×C4).229(C22×C4), C2.25(C2×C42⋊C2), C22.109(C2×C4○D4), (C22×C22⋊C4).12C2, (C2×C22⋊C4).435C22, C22⋊C42(C2×C22⋊C4), SmallGroup(128,1074)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C23.224C24
C1C2C22C23C24C25C22×C22⋊C4 — C23.224C24
C1C22 — C23.224C24
C1C23 — C23.224C24
C1C23 — C23.224C24

Generators and relations for C23.224C24
 G = < a,b,c,d,e,f,g | a2=b2=c2=f2=g2=1, d2=c, 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, ce=ec, cf=fc, cg=gc, dg=gd, ef=fe, fg=gf >

Subgroups: 780 in 396 conjugacy classes, 156 normal (34 characteristic)
C1, C2 [×3], C2 [×4], C2 [×10], C4 [×18], C22 [×3], C22 [×12], C22 [×50], C2×C4 [×12], C2×C4 [×50], C23, C23 [×14], C23 [×50], C42 [×6], C22⋊C4 [×20], C22⋊C4 [×16], C4⋊C4 [×6], C22×C4 [×4], C22×C4 [×16], C22×C4 [×16], C24 [×3], C24 [×4], C24 [×10], C2.C42 [×8], C2×C42 [×4], C2×C22⋊C4 [×8], C2×C22⋊C4 [×14], C2×C22⋊C4 [×4], C2×C4⋊C4 [×4], C42⋊C2 [×4], C23×C4 [×2], C23×C4 [×2], C25, C4×C22⋊C4, C4×C22⋊C4 [×2], C243C4, C23.7Q8, C23.34D4, C23.8Q8 [×2], C24.C22 [×4], C22×C22⋊C4 [×2], C2×C42⋊C2, C23.224C24
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], D4 [×4], C23 [×15], C22×C4 [×14], C2×D4 [×6], C4○D4 [×6], C24, C42⋊C2 [×4], C4×D4 [×4], C23×C4, C22×D4, C2×C4○D4 [×3], 2+ 1+4 [×2], C2×C42⋊C2, C2×C4×D4, C22.11C24, D45D4 [×2], C22.45C24 [×2], C23.224C24

Smallest permutation representation of C23.224C24
On 32 points
Generators in S32
(1 11)(2 12)(3 9)(4 10)(5 22)(6 23)(7 24)(8 21)(13 25)(14 26)(15 27)(16 28)(17 31)(18 32)(19 29)(20 30)
(1 27)(2 28)(3 25)(4 26)(5 20)(6 17)(7 18)(8 19)(9 13)(10 14)(11 15)(12 16)(21 29)(22 30)(23 31)(24 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 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 31 27 23)(2 18 28 7)(3 29 25 21)(4 20 26 5)(6 11 17 15)(8 9 19 13)(10 30 14 22)(12 32 16 24)
(1 3)(2 26)(4 28)(5 18)(6 8)(7 20)(9 11)(10 16)(12 14)(13 15)(17 19)(21 23)(22 32)(24 30)(25 27)(29 31)
(1 25)(2 26)(3 27)(4 28)(5 32)(6 29)(7 30)(8 31)(9 15)(10 16)(11 13)(12 14)(17 21)(18 22)(19 23)(20 24)

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

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

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

50 conjugacy classes

class 1 2A···2G2H···2O2P2Q4A···4P4Q···4AF
order12···22···2224···44···4
size11···12···2442···24···4

50 irreducible representations

dim1111111111224
type+++++++++++
imageC1C2C2C2C2C2C2C2C2C4D4C4○D42+ 1+4
kernelC23.224C24C4×C22⋊C4C243C4C23.7Q8C23.34D4C23.8Q8C24.C22C22×C22⋊C4C2×C42⋊C2C2×C22⋊C4C22⋊C4C23C22
# reps131112421164122

Matrix representation of C23.224C24 in GL6(𝔽5)

400000
040000
004000
000400
000010
000001
,
100000
010000
001000
000100
000040
000004
,
100000
010000
004000
000400
000010
000001
,
100000
340000
003000
000200
000001
000010
,
110000
040000
000100
001000
000020
000002
,
100000
010000
004000
000400
000010
000004
,
100000
340000
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],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,3,0,0,0,0,0,4,0,0,0,0,0,0,3,0,0,0,0,0,0,2,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[1,0,0,0,0,0,1,4,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,2,0,0,0,0,0,0,2],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,4],[1,3,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4] >;

C23.224C24 in GAP, Magma, Sage, TeX

C_2^3._{224}C_2^4
% in TeX

G:=Group("C2^3.224C2^4");
// GroupNames label

G:=SmallGroup(128,1074);
// by ID

G=gap.SmallGroup(128,1074);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,232,758,346]);
// Polycyclic

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

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