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G = C24.176C23order 128 = 27

16th non-split extension by C24 of C23 acting via C23/C2=C22

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

Aliases: C24.176C23, C22⋊C49Q8, (C22×Q8)⋊7C4, C23.6(C2×Q8), C22.13(C4×Q8), C4.15(C23⋊C4), C23.9(C4○D4), C23.576(C2×D4), (C22×C4).306D4, C22.12(C4⋊Q8), C23.9D4.6C2, C23.200(C22×C4), (C23×C4).266C22, C22.30(C22⋊Q8), C23.7Q8.17C2, C22.22(C4.4D4), C2.29(C23.C23), C2.18(C23.67C23), (C2×C4⋊C4)⋊15C4, C2.29(C2×C23⋊C4), (C4×C22⋊C4).17C2, (C22×C4).26(C2×C4), (C2×C22⋊Q8).11C2, (C2×C4).209(C22⋊C4), (C2×C22⋊C4).16C22, C22.305(C2×C22⋊C4), SmallGroup(128,728)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C24.176C23
C1C2C22C23C24C23×C4C4×C22⋊C4 — C24.176C23
C1C2C23 — C24.176C23
C1C22C23×C4 — C24.176C23
C1C2C24 — C24.176C23

Generators and relations for C24.176C23
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=1, e2=c, f2=abc, g2=b, ab=ba, ac=ca, eae-1=faf-1=ad=da, ag=ga, bc=cb, bd=db, geg-1=be=eb, bf=fb, bg=gb, fcf-1=cd=dc, ce=ec, cg=gc, de=ed, df=fd, dg=gd, fef-1=ae, fg=gf >

Subgroups: 364 in 162 conjugacy classes, 58 normal (28 characteristic)
C1, C2 [×3], C2 [×6], C4 [×2], C4 [×13], C22 [×3], C22 [×4], C22 [×10], C2×C4 [×4], C2×C4 [×31], Q8 [×4], C23 [×3], C23 [×4], C23 [×2], C42 [×2], C22⋊C4 [×4], C22⋊C4 [×10], C4⋊C4 [×8], C22×C4 [×6], C22×C4 [×4], C22×C4 [×6], C2×Q8 [×4], C24, C2.C42 [×2], C2×C42, C2×C22⋊C4 [×6], C2×C4⋊C4, C2×C4⋊C4 [×2], C2×C4⋊C4, C22⋊Q8 [×4], C23×C4, C22×Q8, C23.9D4 [×4], C4×C22⋊C4, C23.7Q8, C2×C22⋊Q8, C24.176C23
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], Q8 [×4], C23, C22⋊C4 [×4], C22×C4, C2×D4 [×2], C2×Q8 [×2], C4○D4 [×2], C23⋊C4 [×2], C2×C22⋊C4, C4×Q8 [×2], C22⋊Q8 [×2], C4.4D4, C4⋊Q8, C23.67C23, C2×C23⋊C4, C23.C23, C24.176C23

Smallest permutation representation of C24.176C23
On 32 points
Generators in S32
(1 7)(2 15)(3 9)(4 13)(5 8)(6 16)(10 11)(12 14)(17 28)(18 27)(19 26)(20 25)(21 31)(22 30)(23 29)(24 32)
(1 4)(2 3)(5 12)(6 11)(7 13)(8 14)(9 15)(10 16)(17 22)(18 23)(19 24)(20 21)(25 31)(26 32)(27 29)(28 30)
(17 19)(18 20)(21 23)(22 24)(25 27)(26 28)(29 31)(30 32)
(1 6)(2 5)(3 12)(4 11)(7 16)(8 15)(9 14)(10 13)(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 25 13 23)(2 19 9 30)(3 24 15 28)(4 31 7 18)(5 17 14 32)(6 27 10 21)(8 26 12 22)(11 29 16 20)
(1 3 4 2)(5 6 12 11)(7 9 13 15)(8 16 14 10)(17 27 22 29)(18 30 23 28)(19 25 24 31)(20 32 21 26)

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

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

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

32 conjugacy classes

class 1 2A2B2C2D···2I4A4B4C4D4E···4N4O···4V
order12222···244444···44···4
size11112···222224···48···8

32 irreducible representations

dim111111122244
type+++++-++
imageC1C2C2C2C2C4C4Q8D4C4○D4C23⋊C4C23.C23
kernelC24.176C23C23.9D4C4×C22⋊C4C23.7Q8C2×C22⋊Q8C2×C4⋊C4C22×Q8C22⋊C4C22×C4C23C4C2
# reps141116244422

Matrix representation of C24.176C23 in GL6(𝔽5)

100000
010000
000123
001023
000010
000024
,
400000
040000
004000
000400
000040
000004
,
400000
040000
001003
000103
000040
000004
,
100000
010000
004000
000400
000040
000004
,
310000
020000
000300
002001
000023
000003
,
100000
010000
002010
002000
001400
002003
,
120000
440000
000340
003040
000020
000043

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

C24.176C23 in GAP, Magma, Sage, TeX

C_2^4._{176}C_2^3
% in TeX

G:=Group("C2^4.176C2^3");
// GroupNames label

G:=SmallGroup(128,728);
// by ID

G=gap.SmallGroup(128,728);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,400,422,100,2019,1018,2028]);
// Polycyclic

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

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