p-group, metabelian, nilpotent (class 2), monomial
Aliases: C24⋊8Q8, C25.90C22, C23.748C24, C24.595C23, (C24×C4).11C2, C4.105C22≀C2, (C22×C4).770D4, C23.630(C2×D4), C22⋊4(C22⋊Q8), C23.103(C2×Q8), C24⋊3C4.17C2, (C22×Q8)⋊11C22, C23.248(C4○D4), (C22×C4).258C23, (C23×C4).682C22, C23.8Q8⋊145C2, C23.7Q8⋊116C2, C22.458(C22×D4), C22.178(C22×Q8), C2.C42⋊46C22, 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
Generators and relations for C24⋊8Q8
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, C24⋊3C4, C23.7Q8, C23.8Q8, C2×C22⋊Q8, C24×C4, C24⋊8Q8
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, C24⋊8Q8
(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 | ··· | 2G | 2H | ··· | 2S | 4A | ··· | 4P | 4Q | ··· | 4X |
order | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
size | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 8 | ··· | 8 |
44 irreducible representations
dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 |
type | + | + | + | + | + | + | + | - | |
image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | Q8 | C4○D4 |
kernel | C24⋊8Q8 | C24⋊3C4 | C23.7Q8 | C23.8Q8 | C2×C22⋊Q8 | C24×C4 | C22×C4 | C24 | C23 |
# reps | 1 | 2 | 3 | 6 | 3 | 1 | 12 | 4 | 12 |
Matrix representation of C24⋊8Q8 ►in GL6(𝔽5)
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 | 0 | 0 | 0 |
0 | 0 | 3 | 2 | 0 | 0 |
0 | 0 | 0 | 0 | 4 | 0 |
0 | 0 | 0 | 0 | 0 | 4 |
0 | 1 | 0 | 0 | 0 | 0 |
4 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 1 | 3 | 0 | 0 |
0 | 0 | 1 | 4 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 1 |
0 | 0 | 0 | 0 | 1 | 0 |
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] >;
C24⋊8Q8 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