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G = C23.372C24order 128 = 27

89th central stem extension by C23 of C24

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

Aliases: C25.43C22, C24.571C23, C23.372C24, C22.1772+ 1+4, C2.26D42, C22⋊C442D4, C243C417C2, (C2×C42)⋊24C22, C23⋊Q813C2, C23.177(C2×D4), C2.54(D45D4), (C22×Q8)⋊4C22, C222(C4.4D4), (C23×C4).93C22, C23.235(C4○D4), C23.23D449C2, C23.10D433C2, C23.11D419C2, (C22×C4).516C23, C22.252(C22×D4), C24.3C2245C2, C24.C2254C2, C2.C4267C22, (C22×D4).139C22, C2.17(C22.32C24), C2.44(C22.19C24), C2.21(C22.45C24), (C2×C4).57(C2×D4), (C2×C4⋊C4)⋊18C22, (C4×C22⋊C4)⋊69C2, (C2×C22⋊Q8)⋊16C2, (C2×C4.4D4)⋊12C2, C2.13(C2×C4.4D4), (C2×C22≀C2).10C2, (C22×C22⋊C4)⋊22C2, (C2×C22⋊C4)⋊19C22, C22.249(C2×C4○D4), SmallGroup(128,1204)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C23.372C24
C1C2C22C23C22×C4C2×C42C4×C22⋊C4 — C23.372C24
C1C23 — C23.372C24
C1C23 — C23.372C24
C1C23 — C23.372C24

Generators and relations for C23.372C24
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=1, e2=g2=b, f2=a, ab=ba, 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 >

Subgroups: 884 in 393 conjugacy classes, 112 normal (82 characteristic)
C1, C2 [×7], C2 [×9], C4 [×15], C22 [×7], C22 [×4], C22 [×55], C2×C4 [×8], C2×C4 [×37], D4 [×12], Q8 [×4], C23, C23 [×10], C23 [×55], C42 [×3], C22⋊C4 [×8], C22⋊C4 [×26], C4⋊C4 [×6], C22×C4 [×11], C22×C4 [×10], C2×D4 [×16], C2×Q8 [×5], C24 [×4], C24 [×12], C2.C42 [×6], C2×C42 [×2], C2×C22⋊C4 [×17], C2×C22⋊C4 [×4], C2×C4⋊C4 [×3], C22≀C2 [×4], C22⋊Q8 [×4], C4.4D4 [×4], C23×C4 [×2], C22×D4 [×3], C22×Q8, C25, C4×C22⋊C4, C243C4, C23.23D4 [×2], C24.C22 [×2], C24.3C22, C23⋊Q8, C23.10D4 [×2], C23.11D4, C22×C22⋊C4, C2×C22≀C2, C2×C22⋊Q8, C2×C4.4D4, C23.372C24
Quotients: C1, C2 [×15], C22 [×35], D4 [×8], C23 [×15], C2×D4 [×12], C4○D4 [×6], C24, C4.4D4 [×4], C22×D4 [×2], C2×C4○D4 [×3], 2+ 1+4 [×2], C22.19C24, C2×C4.4D4, C22.32C24, D42, D45D4 [×2], C22.45C24, C23.372C24

Smallest permutation representation of C23.372C24
On 32 points
Generators in S32
(1 6)(2 7)(3 8)(4 5)(9 24)(10 21)(11 22)(12 23)(13 30)(14 31)(15 32)(16 29)(17 26)(18 27)(19 28)(20 25)
(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 22)(2 23)(3 24)(4 21)(5 10)(6 11)(7 12)(8 9)(13 25)(14 26)(15 27)(16 28)(17 31)(18 32)(19 29)(20 30)
(1 22)(2 12)(3 24)(4 10)(5 21)(6 11)(7 23)(8 9)(13 18)(14 28)(15 20)(16 26)(17 29)(19 31)(25 32)(27 30)
(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 6 14)(2 18 7 27)(3 29 8 16)(4 20 5 25)(9 28 24 19)(10 13 21 30)(11 26 22 17)(12 15 23 32)
(1 7 3 5)(2 8 4 6)(9 21 11 23)(10 22 12 24)(13 17 15 19)(14 18 16 20)(25 31 27 29)(26 32 28 30)

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

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

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

38 conjugacy classes

class 1 2A···2G2H2I2J2K2L2M2N2O2P4A4B4C4D4E···4R4S4T4U
order12···222222222244444···4444
size11···122224444822224···4888

38 irreducible representations

dim1111111111111224
type+++++++++++++++
imageC1C2C2C2C2C2C2C2C2C2C2C2C2D4C4○D42+ 1+4
kernelC23.372C24C4×C22⋊C4C243C4C23.23D4C24.C22C24.3C22C23⋊Q8C23.10D4C23.11D4C22×C22⋊C4C2×C22≀C2C2×C22⋊Q8C2×C4.4D4C22⋊C4C23C22
# reps11122112111118122

Matrix representation of C23.372C24 in GL6(𝔽5)

100000
010000
004000
000400
000010
000001
,
100000
010000
001000
000100
000040
000004
,
400000
040000
001000
000100
000040
000004
,
400000
040000
001000
000400
000040
000021
,
100000
040000
000400
004000
000020
000013
,
010000
100000
003000
000300
000033
000042
,
100000
010000
000100
001000
000020
000002

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

C23.372C24 in GAP, Magma, Sage, TeX

C_2^3._{372}C_2^4
% in TeX

G:=Group("C2^3.372C2^4");
// GroupNames label

G:=SmallGroup(128,1204);
// by ID

G=gap.SmallGroup(128,1204);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,253,232,758,723,100,675,136]);
// Polycyclic

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