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G = C24.19Q8order 128 = 27

3rd non-split extension by C24 of Q8 acting via Q8/C4=C2

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

Aliases: C24.19Q8, (C2×C8).356D4, (C23×C8).17C2, (C22×C8).35C4, C23.79(C2×Q8), (C22×C4).78Q8, C8.44(C22⋊C4), C23.69(C4⋊C4), (C22×C4).546D4, C4.182(C4⋊D4), C4.C4218C2, C221(C8.C4), C4.55(C42⋊C2), C24.4C4.16C2, (C23×C4).672C22, (C22×C8).549C22, C22.22(C22⋊Q8), (C22×C4).1332C23, C2.19(C23.7Q8), (C2×M4(2)).154C22, (C2×C8.C4)⋊1C2, (C2×C4).85(C4⋊C4), (C2×C8).210(C2×C4), C4.89(C2×C22⋊C4), C22.93(C2×C4⋊C4), C2.11(C2×C8.C4), (C2×C4).1515(C2×D4), (C2×C4).554(C4○D4), (C22×C4).482(C2×C4), (C2×C4).530(C22×C4), SmallGroup(128,542)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C24.19Q8
C1C2C4C2×C4C22×C4C23×C4C23×C8 — C24.19Q8
C1C2C2×C4 — C24.19Q8
C1C2×C4C23×C4 — C24.19Q8
C1C2C2C22×C4 — C24.19Q8

Generators and relations for C24.19Q8
 G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e4=d, f2=bde2, ab=ba, faf-1=ac=ca, ad=da, ae=ea, bc=cb, fbf-1=bd=db, be=eb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=e3 >

Subgroups: 236 in 146 conjugacy classes, 64 normal (18 characteristic)
C1, C2, C2 [×2], C2 [×6], C4 [×2], C4 [×2], C4 [×2], C22, C22 [×6], C22 [×14], C8 [×4], C8 [×6], C2×C4 [×2], C2×C4 [×6], C2×C4 [×10], C23, C23 [×2], C23 [×6], C2×C8 [×2], C2×C8 [×6], C2×C8 [×14], M4(2) [×8], C22×C4 [×2], C22×C4 [×4], C22×C4 [×4], C24, C22⋊C8 [×4], C8.C4 [×4], C22×C8 [×2], C22×C8 [×4], C22×C8 [×4], C2×M4(2) [×4], C23×C4, C4.C42 [×2], C24.4C4 [×2], C2×C8.C4 [×2], C23×C8, C24.19Q8
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×6], Q8 [×2], C23, C22⋊C4 [×4], C4⋊C4 [×4], C22×C4, C2×D4 [×3], C2×Q8, C4○D4 [×2], C8.C4 [×4], C2×C22⋊C4, C2×C4⋊C4, C42⋊C2, C4⋊D4 [×2], C22⋊Q8 [×2], C23.7Q8, C2×C8.C4 [×2], C24.19Q8

Smallest permutation representation of C24.19Q8
On 32 points
Generators in S32
(9 28)(10 29)(11 30)(12 31)(13 32)(14 25)(15 26)(16 27)
(1 17)(2 18)(3 19)(4 20)(5 21)(6 22)(7 23)(8 24)(9 28)(10 29)(11 30)(12 31)(13 32)(14 25)(15 26)(16 27)
(1 21)(2 22)(3 23)(4 24)(5 17)(6 18)(7 19)(8 20)(9 28)(10 29)(11 30)(12 31)(13 32)(14 25)(15 26)(16 27)
(1 5)(2 6)(3 7)(4 8)(9 13)(10 14)(11 15)(12 16)(17 21)(18 22)(19 23)(20 24)(25 29)(26 30)(27 31)(28 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 30 23 9 5 26 19 13)(2 25 24 12 6 29 20 16)(3 28 17 15 7 32 21 11)(4 31 18 10 8 27 22 14)

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

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

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

44 conjugacy classes

class 1 2A2B2C2D···2I4A4B4C4D4E···4J8A···8P8Q···8X
order12222···244444···48···88···8
size11112···211112···22···28···8

44 irreducible representations

dim111111222222
type+++++++--
imageC1C2C2C2C2C4D4D4Q8Q8C4○D4C8.C4
kernelC24.19Q8C4.C42C24.4C4C2×C8.C4C23×C8C22×C8C2×C8C22×C4C22×C4C24C2×C4C22
# reps1222184211416

Matrix representation of C24.19Q8 in GL4(𝔽17) generated by

1000
01600
0010
001516
,
1000
01600
00160
00016
,
16000
01600
00160
00016
,
16000
01600
0010
0001
,
8000
0200
00160
00016
,
0100
4000
001616
0021
G:=sub<GL(4,GF(17))| [1,0,0,0,0,16,0,0,0,0,1,15,0,0,0,16],[1,0,0,0,0,16,0,0,0,0,16,0,0,0,0,16],[16,0,0,0,0,16,0,0,0,0,16,0,0,0,0,16],[16,0,0,0,0,16,0,0,0,0,1,0,0,0,0,1],[8,0,0,0,0,2,0,0,0,0,16,0,0,0,0,16],[0,4,0,0,1,0,0,0,0,0,16,2,0,0,16,1] >;

C24.19Q8 in GAP, Magma, Sage, TeX

C_2^4._{19}Q_8
% in TeX

G:=Group("C2^4.19Q8");
// GroupNames label

G:=SmallGroup(128,542);
// by ID

G=gap.SmallGroup(128,542);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,64,422,2019,248,124]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^4=d,f^2=b*d*e^2,a*b=b*a,f*a*f^-1=a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,f*b*f^-1=b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=e^3>;
// generators/relations

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