p-group, metabelian, nilpotent (class 4), monomial
Aliases: C4⋊Q8.5C4, C42.12(C2×C4), (C2×Q8).119D4, C4.4D4.1C4, C4⋊Q8.95C22, C42.3C4⋊6C2, C42.C4⋊6C2, (C22×C4).100D4, (C2×Q8).13C23, C42⋊C2.10C4, C23.85(C22⋊C4), C22.10(C23⋊C4), C4.10D4.7C22, C4.4D4.16C22, (C22×Q8).86C22, C23.38C23.8C2, (C2×C4).10(C2×D4), (C2×C4○D4).10C4, (C2×D4).41(C2×C4), C2.45(C2×C23⋊C4), (C22×C4).34(C2×C4), (C2×Q8).109(C2×C4), (C2×C4.10D4)⋊27C2, (C2×C4).31(C22⋊C4), (C2×C4).102(C22×C4), C22.69(C2×C22⋊C4), SmallGroup(128,865)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C4⋊Q8.C4
G = < a,b,c,d | a4=b4=1, c2=d4=b2, ab=ba, cac-1=a-1, dad-1=ab-1, cbc-1=b-1, dbd-1=a2b, cd=dc >
Subgroups: 252 in 115 conjugacy classes, 42 normal (22 characteristic)
C1, C2, C2, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C42, C22⋊C4, C4⋊C4, C2×C8, M4(2), C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C2×Q8, C4○D4, C4.10D4, C4.10D4, C42⋊C2, C22⋊Q8, C22.D4, C4.4D4, C4⋊Q8, C2×M4(2), C22×Q8, C2×C4○D4, C42.C4, C42.3C4, C2×C4.10D4, C23.38C23, C4⋊Q8.C4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, C23⋊C4, C2×C22⋊C4, C2×C23⋊C4, C4⋊Q8.C4
Character table of C4⋊Q8.C4
class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | |
size | 1 | 1 | 2 | 2 | 2 | 8 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | 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 | 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 | 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 | 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 | linear of order 2 |
ρ5 | 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 |
ρ6 | 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 |
ρ7 | 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 |
ρ8 | 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 |
ρ9 | 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 |
ρ10 | 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 | i | -i | -i | -i | -i | i | i | i | linear of order 4 |
ρ12 | 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 |
ρ13 | 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 |
ρ14 | 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 |
ρ15 | 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 |
ρ16 | 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 |
ρ17 | 2 | 2 | -2 | 2 | -2 | 0 | 2 | 2 | -2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
ρ18 | 2 | 2 | 2 | 2 | 2 | 0 | 2 | -2 | -2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
ρ19 | 2 | 2 | 2 | 2 | 2 | 0 | -2 | 2 | -2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
ρ20 | 2 | 2 | -2 | 2 | -2 | 0 | -2 | -2 | -2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
ρ21 | 4 | 4 | 4 | -4 | -4 | 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 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C23⋊C4 |
ρ23 | 8 | -8 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic faithful, Schur index 2 |
(1 25)(2 8 6 4)(3 31)(5 29)(7 27)(9 11 13 15)(10 23)(12 21)(14 19)(16 17)(18 24 22 20)(26 28 30 32)
(1 31 5 27)(2 32 6 28)(3 29 7 25)(4 30 8 26)(9 20 13 24)(10 17 14 21)(11 18 15 22)(12 23 16 19)
(1 10 5 14)(2 11 6 15)(3 12 7 16)(4 13 8 9)(17 31 21 27)(18 32 22 28)(19 25 23 29)(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)| (1,25)(2,8,6,4)(3,31)(5,29)(7,27)(9,11,13,15)(10,23)(12,21)(14,19)(16,17)(18,24,22,20)(26,28,30,32), (1,31,5,27)(2,32,6,28)(3,29,7,25)(4,30,8,26)(9,20,13,24)(10,17,14,21)(11,18,15,22)(12,23,16,19), (1,10,5,14)(2,11,6,15)(3,12,7,16)(4,13,8,9)(17,31,21,27)(18,32,22,28)(19,25,23,29)(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( (1,25)(2,8,6,4)(3,31)(5,29)(7,27)(9,11,13,15)(10,23)(12,21)(14,19)(16,17)(18,24,22,20)(26,28,30,32), (1,31,5,27)(2,32,6,28)(3,29,7,25)(4,30,8,26)(9,20,13,24)(10,17,14,21)(11,18,15,22)(12,23,16,19), (1,10,5,14)(2,11,6,15)(3,12,7,16)(4,13,8,9)(17,31,21,27)(18,32,22,28)(19,25,23,29)(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([[(1,25),(2,8,6,4),(3,31),(5,29),(7,27),(9,11,13,15),(10,23),(12,21),(14,19),(16,17),(18,24,22,20),(26,28,30,32)], [(1,31,5,27),(2,32,6,28),(3,29,7,25),(4,30,8,26),(9,20,13,24),(10,17,14,21),(11,18,15,22),(12,23,16,19)], [(1,10,5,14),(2,11,6,15),(3,12,7,16),(4,13,8,9),(17,31,21,27),(18,32,22,28),(19,25,23,29),(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 C4⋊Q8.C4 ►in GL8(𝔽17)
0 | 0 | 0 | 16 | 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 | 0 |
0 | 0 | 0 | 0 | 0 | 16 | 0 | 0 |
0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
16 | 16 | 1 | 1 | 16 | 1 | 16 | 2 |
0 | 16 | 1 | 0 | 16 | 0 | 16 | 1 |
0 | 0 | 1 | 0 | 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 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | 16 | 0 |
16 | 16 | 1 | 1 | 16 | 1 | 16 | 2 |
0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
1 | 1 | 0 | 0 | 1 | 16 | 0 | 16 |
0 | 0 | 0 | 13 | 0 | 0 | 0 | 0 |
0 | 0 | 13 | 0 | 0 | 0 | 0 | 0 |
0 | 13 | 0 | 0 | 0 | 0 | 0 | 0 |
13 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
13 | 13 | 4 | 4 | 13 | 4 | 13 | 8 |
0 | 0 | 0 | 0 | 0 | 0 | 13 | 0 |
0 | 0 | 0 | 0 | 0 | 13 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 4 |
0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
1 | 1 | 16 | 16 | 1 | 16 | 1 | 15 |
0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
16 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 16 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 1 | 16 | 1 |
G:=sub<GL(8,GF(17))| [0,0,0,16,0,0,16,0,0,0,1,0,0,0,16,16,0,1,0,0,0,0,1,1,16,0,0,0,0,0,1,0,0,0,0,0,0,1,16,16,0,0,0,0,16,0,1,0,0,0,0,0,0,0,16,16,0,0,0,0,0,0,2,1],[0,0,16,0,0,16,0,1,0,0,0,16,0,16,0,1,1,0,0,0,0,1,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,16,1,1,0,0,0,0,0,1,0,16,0,0,0,0,16,16,0,0,0,0,0,0,0,2,0,16],[0,0,0,13,13,0,0,0,0,0,13,0,13,0,0,0,0,13,0,0,4,0,0,0,13,0,0,0,4,0,0,0,0,0,0,0,13,0,0,0,0,0,0,0,4,0,13,0,0,0,0,0,13,13,0,0,0,0,0,0,8,0,0,4],[0,0,0,1,0,16,0,0,0,0,0,1,1,0,0,0,0,0,0,16,0,0,0,0,0,0,0,16,0,0,16,0,1,0,0,1,0,0,0,0,0,1,0,16,0,0,0,1,0,0,1,1,0,0,0,16,0,0,0,15,0,0,0,1] >;
C4⋊Q8.C4 in GAP, Magma, Sage, TeX
C_4\rtimes Q_8.C_4
% in TeX
G:=Group("C4:Q8.C4");
// GroupNames label
G:=SmallGroup(128,865);
// by ID
G=gap.SmallGroup(128,865);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,-2,112,141,723,1123,1018,248,1971,375,172,4037]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^2=d^4=b^2,a*b=b*a,c*a*c^-1=a^-1,d*a*d^-1=a*b^-1,c*b*c^-1=b^-1,d*b*d^-1=a^2*b,c*d=d*c>;
// generators/relations
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