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G = C248Q8order 128 = 27

1st semidirect product of C24 and Q8 acting via Q8/C4=C2

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

Aliases: C248Q8, C25.90C22, C23.748C24, C24.595C23, (C24×C4).11C2, C4.105C22≀C2, (C22×C4).770D4, C23.630(C2×D4), C224(C22⋊Q8), C23.103(C2×Q8), C243C4.17C2, (C22×Q8)⋊11C22, C23.248(C4○D4), (C22×C4).258C23, (C23×C4).682C22, C23.8Q8145C2, C23.7Q8116C2, C22.458(C22×D4), C22.178(C22×Q8), C2.C4246C22, C2.91(C22.19C24), (C2×C4⋊C4)⋊41C22, (C2×C22⋊Q8)⋊49C2, C2.45(C2×C22⋊Q8), C2.31(C2×C22≀C2), (C2×C4).1203(C2×D4), C22.589(C2×C4○D4), (C2×C22⋊C4).359C22, SmallGroup(128,1580)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C248Q8
C1C2C22C23C24C25C24×C4 — C248Q8
C1C23 — C248Q8
C1C23 — C248Q8
C1C23 — C248Q8

Generators and relations for C248Q8
 G = < a,b,c,d,e,f | a2=b2=c2=d2=e4=1, f2=e2, 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=e-1 >

Subgroups: 948 in 534 conjugacy classes, 144 normal (9 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C2×C4, C2×C4, Q8, C23, C23, C23, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×Q8, C24, C24, C2.C42, C2×C22⋊C4, C2×C4⋊C4, C22⋊Q8, C23×C4, C23×C4, C22×Q8, C25, C243C4, C23.7Q8, C23.8Q8, C2×C22⋊Q8, C24×C4, C248Q8
Quotients: C1, C2, C22, D4, Q8, C23, C2×D4, C2×Q8, C4○D4, C24, C22≀C2, C22⋊Q8, C22×D4, C22×Q8, C2×C4○D4, C2×C22≀C2, C2×C22⋊Q8, C22.19C24, C248Q8

Smallest permutation representation of C248Q8
On 32 points
Generators in S32
(1 17)(2 18)(3 19)(4 20)(5 11)(6 12)(7 9)(8 10)(13 32)(14 29)(15 30)(16 31)(21 27)(22 28)(23 25)(24 26)
(1 21)(2 22)(3 23)(4 24)(5 32)(6 29)(7 30)(8 31)(9 15)(10 16)(11 13)(12 14)(17 27)(18 28)(19 25)(20 26)
(1 21)(2 22)(3 23)(4 24)(5 15)(6 16)(7 13)(8 14)(9 32)(10 29)(11 30)(12 31)(17 27)(18 28)(19 25)(20 26)
(1 25)(2 26)(3 27)(4 28)(5 9)(6 10)(7 11)(8 12)(13 30)(14 31)(15 32)(16 29)(17 23)(18 24)(19 21)(20 22)
(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 10 3 12)(2 9 4 11)(5 28 7 26)(6 27 8 25)(13 20 15 18)(14 19 16 17)(21 29 23 31)(22 32 24 30)

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

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

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

44 conjugacy classes

class 1 2A···2G2H···2S4A···4P4Q···4X
order12···22···24···44···4
size11···12···22···28···8

44 irreducible representations

dim111111222
type+++++++-
imageC1C2C2C2C2C2D4Q8C4○D4
kernelC248Q8C243C4C23.7Q8C23.8Q8C2×C22⋊Q8C24×C4C22×C4C24C23
# reps12363112412

Matrix representation of C248Q8 in GL6(𝔽5)

100000
010000
004000
000400
000010
000004
,
100000
040000
001000
000100
000040
000001
,
100000
010000
001000
000100
000040
000004
,
400000
040000
001000
000100
000040
000004
,
300000
020000
003000
003200
000040
000004
,
010000
400000
001300
001400
000001
000010

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

C248Q8 in GAP, Magma, Sage, TeX

C_2^4\rtimes_8Q_8
% in TeX

G:=Group("C2^4:8Q8");
// GroupNames label

G:=SmallGroup(128,1580);
// by ID

G=gap.SmallGroup(128,1580);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,112,253,120,758,2019]);
// Polycyclic

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

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