p-group, metabelian, nilpotent (class 4), monomial
Aliases: (C2×Q8).2Q8, (C2×C4).2C42, (C2×Q8).40D4, C4.10D4⋊5C4, (C2×C42).13C4, (C22×C4).38D4, (C22×Q8).1C22, C22.15(C23⋊C4), C2.3(C42.C4), C2.3(C42.3C4), C23.162(C22⋊C4), C2.16(C23.9D4), C23.67C23.3C2, C22.5(C2.C42), (C2×C4⋊C4).10C4, (C2×C4).3(C4⋊C4), (C2×Q8).41(C2×C4), (C2×C4).3(C22⋊C4), (C22×C4).67(C2×C4), (C2×C4.10D4).5C2, SmallGroup(128,126)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for (C2×Q8).Q8
G = < a,b,c,d,e | a2=b4=d4=1, c2=b2, e2=ab-1d2, dbd-1=ab=ba, ece-1=ac=ca, ad=da, eae-1=ab2, cbc-1=b-1, be=eb, cd=dc, ede-1=cd-1 >
Subgroups: 192 in 86 conjugacy classes, 32 normal (18 characteristic)
C1, C2 [×3], C2 [×2], C4 [×9], C22 [×3], C22 [×2], C8 [×4], C2×C4 [×2], C2×C4 [×4], C2×C4 [×13], Q8 [×4], C23, C42, C4⋊C4, C2×C8 [×2], M4(2) [×6], C22×C4, C22×C4 [×2], C22×C4 [×2], C2×Q8 [×2], C2×Q8 [×2], C2×Q8 [×2], C2.C42 [×2], C4.10D4 [×4], C4.10D4 [×2], C2×C42, C2×C4⋊C4, C2×M4(2) [×2], C22×Q8, C23.67C23, C2×C4.10D4 [×2], (C2×Q8).Q8
Quotients: C1, C2 [×3], C4 [×6], C22, C2×C4 [×3], D4 [×3], Q8, C42, C22⋊C4 [×3], C4⋊C4 [×3], C2.C42, C23⋊C4 [×2], C23.9D4, C42.C4, C42.3C4, (C2×Q8).Q8
Character table of (C2×Q8).Q8
class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | |
size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | |
ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
ρ2 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | linear of order 2 |
ρ3 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | linear of order 2 |
ρ4 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | linear of order 2 |
ρ5 | 1 | -1 | 1 | -1 | -1 | 1 | i | 1 | -1 | -1 | 1 | i | -1 | -i | -i | 1 | i | -i | 1 | 1 | -i | -i | i | i | -1 | -1 | linear of order 4 |
ρ6 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | -i | i | i | -i | i | -i | -i | i | linear of order 4 |
ρ7 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | 1 | -1 | -1 | -1 | i | -i | i | -i | i | -i | i | -i | linear of order 4 |
ρ8 | 1 | -1 | 1 | -1 | -1 | 1 | -i | 1 | -1 | 1 | -1 | -i | 1 | i | i | -1 | i | -i | -i | i | 1 | -1 | -1 | 1 | i | -i | linear of order 4 |
ρ9 | 1 | -1 | 1 | -1 | -1 | 1 | i | 1 | -1 | -1 | 1 | i | -1 | -i | -i | 1 | i | -i | -1 | -1 | i | i | -i | -i | 1 | 1 | linear of order 4 |
ρ10 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | i | -i | -i | i | -i | i | i | -i | linear of order 4 |
ρ11 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | 1 | -1 | -1 | -1 | -i | i | -i | i | -i | i | -i | i | linear of order 4 |
ρ12 | 1 | -1 | 1 | -1 | -1 | 1 | -i | 1 | -1 | 1 | -1 | -i | 1 | i | i | -1 | i | -i | i | -i | -1 | 1 | 1 | -1 | -i | i | linear of order 4 |
ρ13 | 1 | -1 | 1 | -1 | -1 | 1 | -i | 1 | -1 | -1 | 1 | -i | -1 | i | i | 1 | -i | i | -1 | -1 | -i | -i | i | i | 1 | 1 | linear of order 4 |
ρ14 | 1 | -1 | 1 | -1 | -1 | 1 | i | 1 | -1 | 1 | -1 | i | 1 | -i | -i | -1 | -i | i | i | -i | 1 | -1 | -1 | 1 | -i | i | linear of order 4 |
ρ15 | 1 | -1 | 1 | -1 | -1 | 1 | -i | 1 | -1 | -1 | 1 | -i | -1 | i | i | 1 | -i | i | 1 | 1 | i | i | -i | -i | -1 | -1 | linear of order 4 |
ρ16 | 1 | -1 | 1 | -1 | -1 | 1 | i | 1 | -1 | 1 | -1 | i | 1 | -i | -i | -1 | -i | i | -i | i | -1 | 1 | 1 | -1 | i | -i | linear of order 4 |
ρ17 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | -2 | -2 | 2 | 2 | 0 | -2 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
ρ18 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | -2 | -2 | -2 | -2 | 0 | 2 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
ρ19 | 2 | -2 | 2 | -2 | -2 | 2 | 0 | -2 | 2 | -2 | 2 | 0 | 2 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
ρ20 | 2 | -2 | 2 | -2 | -2 | 2 | 0 | -2 | 2 | 2 | -2 | 0 | -2 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
ρ21 | 4 | -4 | 4 | -4 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C23⋊C4 |
ρ22 | 4 | 4 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C23⋊C4 |
ρ23 | 4 | 4 | -4 | -4 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 2 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from C42.3C4, Schur index 2 |
ρ24 | 4 | 4 | -4 | -4 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | -2 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from C42.3C4, Schur index 2 |
ρ25 | 4 | -4 | -4 | 4 | 0 | 0 | 2i | 0 | 0 | 0 | 0 | -2i | 0 | 2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C42.C4 |
ρ26 | 4 | -4 | -4 | 4 | 0 | 0 | -2i | 0 | 0 | 0 | 0 | 2i | 0 | -2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C42.C4 |
(1 27)(2 32)(3 29)(4 26)(5 31)(6 28)(7 25)(8 30)(9 24)(10 21)(11 18)(12 23)(13 20)(14 17)(15 22)(16 19)
(1 25 5 29)(2 26 6 30)(3 27 7 31)(4 28 8 32)(9 18 13 22)(10 19 14 23)(11 20 15 24)(12 21 16 17)
(1 21 5 17)(2 11 6 15)(3 19 7 23)(4 9 8 13)(10 31 14 27)(12 29 16 25)(18 28 22 32)(20 26 24 30)
(2 22 6 18)(3 29)(4 9 8 13)(7 25)(11 32 15 28)(12 23)(16 19)(20 26 24 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)
G:=sub<Sym(32)| (1,27)(2,32)(3,29)(4,26)(5,31)(6,28)(7,25)(8,30)(9,24)(10,21)(11,18)(12,23)(13,20)(14,17)(15,22)(16,19), (1,25,5,29)(2,26,6,30)(3,27,7,31)(4,28,8,32)(9,18,13,22)(10,19,14,23)(11,20,15,24)(12,21,16,17), (1,21,5,17)(2,11,6,15)(3,19,7,23)(4,9,8,13)(10,31,14,27)(12,29,16,25)(18,28,22,32)(20,26,24,30), (2,22,6,18)(3,29)(4,9,8,13)(7,25)(11,32,15,28)(12,23)(16,19)(20,26,24,30), (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)>;
G:=Group( (1,27)(2,32)(3,29)(4,26)(5,31)(6,28)(7,25)(8,30)(9,24)(10,21)(11,18)(12,23)(13,20)(14,17)(15,22)(16,19), (1,25,5,29)(2,26,6,30)(3,27,7,31)(4,28,8,32)(9,18,13,22)(10,19,14,23)(11,20,15,24)(12,21,16,17), (1,21,5,17)(2,11,6,15)(3,19,7,23)(4,9,8,13)(10,31,14,27)(12,29,16,25)(18,28,22,32)(20,26,24,30), (2,22,6,18)(3,29)(4,9,8,13)(7,25)(11,32,15,28)(12,23)(16,19)(20,26,24,30), (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) );
G=PermutationGroup([(1,27),(2,32),(3,29),(4,26),(5,31),(6,28),(7,25),(8,30),(9,24),(10,21),(11,18),(12,23),(13,20),(14,17),(15,22),(16,19)], [(1,25,5,29),(2,26,6,30),(3,27,7,31),(4,28,8,32),(9,18,13,22),(10,19,14,23),(11,20,15,24),(12,21,16,17)], [(1,21,5,17),(2,11,6,15),(3,19,7,23),(4,9,8,13),(10,31,14,27),(12,29,16,25),(18,28,22,32),(20,26,24,30)], [(2,22,6,18),(3,29),(4,9,8,13),(7,25),(11,32,15,28),(12,23),(16,19),(20,26,24,30)], [(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)])
Matrix representation of (C2×Q8).Q8 ►in GL6(𝔽17)
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 | 16 | 0 |
0 | 0 | 0 | 0 | 0 | 16 |
16 | 0 | 0 | 0 | 0 | 0 |
0 | 16 | 0 | 0 | 0 | 0 |
0 | 0 | 4 | 0 | 0 | 0 |
0 | 0 | 0 | 13 | 0 | 0 |
0 | 0 | 0 | 0 | 4 | 0 |
0 | 0 | 0 | 0 | 0 | 13 |
1 | 0 | 0 | 0 | 0 | 0 |
0 | 1 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 16 | 0 | 0 |
0 | 0 | 1 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 1 |
0 | 0 | 0 | 0 | 16 | 0 |
4 | 0 | 0 | 0 | 0 | 0 |
0 | 13 | 0 | 0 | 0 | 0 |
0 | 0 | 1 | 0 | 0 | 0 |
0 | 0 | 0 | 1 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 1 |
0 | 0 | 0 | 0 | 16 | 0 |
0 | 16 | 0 | 0 | 0 | 0 |
16 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 1 | 0 |
0 | 0 | 0 | 0 | 0 | 1 |
0 | 0 | 13 | 0 | 0 | 0 |
0 | 0 | 0 | 4 | 0 | 0 |
G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,4,0,0,0,0,0,0,13,0,0,0,0,0,0,4,0,0,0,0,0,0,13],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0],[4,0,0,0,0,0,0,13,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,16,0,0,0,0,1,0],[0,16,0,0,0,0,16,0,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,4,0,0,1,0,0,0,0,0,0,1,0,0] >;
(C2×Q8).Q8 in GAP, Magma, Sage, TeX
(C_2\times Q_8).Q_8
% in TeX
G:=Group("(C2xQ8).Q8");
// GroupNames label
G:=SmallGroup(128,126);
// by ID
G=gap.SmallGroup(128,126);
# by ID
G:=PCGroup([7,-2,2,-2,2,2,-2,-2,56,85,120,758,352,1466,521,248,2804,4037]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^4=d^4=1,c^2=b^2,e^2=a*b^-1*d^2,d*b*d^-1=a*b=b*a,e*c*e^-1=a*c=c*a,a*d=d*a,e*a*e^-1=a*b^2,c*b*c^-1=b^-1,b*e=e*b,c*d=d*c,e*d*e^-1=c*d^-1>;
// generators/relations
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