Copied to
clipboard

## G = (C2×C4)⋊Q16order 128 = 27

### The semidirect product of C2×C4 and Q16 acting via Q16/C2=D4

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C22×C4 — (C2×C4)⋊Q16
 Chief series C1 — C2 — C22 — C23 — C22×C4 — C2×C4⋊C4 — C23.41C23 — (C2×C4)⋊Q16
 Lower central C1 — C22 — C22×C4 — (C2×C4)⋊Q16
 Upper central C1 — C22 — C22×C4 — (C2×C4)⋊Q16
 Jennings C1 — C2 — C22 — C22×C4 — (C2×C4)⋊Q16

Generators and relations for (C2×C4)⋊Q16
G = < a,b,c,d | a2=b4=c8=1, d2=c4, cbc-1=ab=ba, cac-1=dad-1=ab2, bd=db, dcd-1=c-1 >

Subgroups: 268 in 119 conjugacy classes, 34 normal (18 characteristic)
C1, C2, C2, C4, C22, C22, C8, C2×C4, C2×C4, Q8, C23, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, Q16, C22×C4, C22×C4, C2×Q8, C2×Q8, C2.C42, C2.C42, C22⋊C8, Q8⋊C4, C2×C4⋊C4, C2×C4⋊C4, C42⋊C2, C22⋊Q8, C22⋊Q8, C42.C2, C4⋊Q8, C2×Q16, C22×Q8, C22.M4(2), C23.31D4, C23.78C23, C22⋊Q16, C23.41C23, (C2×C4)⋊Q16
Quotients: C1, C2, C22, D4, C23, Q16, C2×D4, C22≀C2, C2×Q16, C8.C22, C22⋊Q16, D4.10D4, C2≀C22, (C2×C4)⋊Q16

Character table of (C2×C4)⋊Q16

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 4M 8A 8B 8C 8D size 1 1 1 1 2 2 4 4 4 4 8 8 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 2 2 2 2 -2 -2 0 0 -2 2 0 0 -2 0 2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ10 2 2 2 2 2 2 2 2 -2 -2 0 0 0 0 0 0 -2 0 0 0 0 0 0 orthogonal lifted from D4 ρ11 2 2 2 2 -2 -2 0 0 2 -2 0 2 0 -2 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ12 2 2 2 2 -2 -2 0 0 2 -2 0 -2 0 2 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ13 2 2 2 2 2 2 -2 -2 -2 -2 0 0 0 0 0 0 2 0 0 0 0 0 0 orthogonal lifted from D4 ρ14 2 2 2 2 -2 -2 0 0 -2 2 0 0 2 0 -2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ15 2 2 -2 -2 -2 2 -2 2 0 0 0 0 0 0 0 0 0 0 0 √2 -√2 -√2 √2 symplectic lifted from Q16, Schur index 2 ρ16 2 2 -2 -2 -2 2 2 -2 0 0 0 0 0 0 0 0 0 0 0 √2 √2 -√2 -√2 symplectic lifted from Q16, Schur index 2 ρ17 2 2 -2 -2 -2 2 -2 2 0 0 0 0 0 0 0 0 0 0 0 -√2 √2 √2 -√2 symplectic lifted from Q16, Schur index 2 ρ18 2 2 -2 -2 -2 2 2 -2 0 0 0 0 0 0 0 0 0 0 0 -√2 -√2 √2 √2 symplectic lifted from Q16, Schur index 2 ρ19 4 -4 4 -4 0 0 0 0 0 0 0 0 0 0 0 2 0 -2 0 0 0 0 0 orthogonal lifted from C2≀C22 ρ20 4 -4 4 -4 0 0 0 0 0 0 0 0 0 0 0 -2 0 2 0 0 0 0 0 orthogonal lifted from C2≀C22 ρ21 4 -4 -4 4 0 0 0 0 0 0 2 0 0 0 0 0 0 0 -2 0 0 0 0 symplectic lifted from D4.10D4, Schur index 2 ρ22 4 4 -4 -4 4 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from C8.C22, Schur index 2 ρ23 4 -4 -4 4 0 0 0 0 0 0 -2 0 0 0 0 0 0 0 2 0 0 0 0 symplectic lifted from D4.10D4, Schur index 2

Smallest permutation representation of (C2×C4)⋊Q16
On 32 points
Generators in S32
(1 28)(2 6)(3 30)(4 8)(5 32)(7 26)(9 13)(10 19)(11 15)(12 21)(14 23)(16 17)(18 22)(20 24)(25 29)(27 31)
(1 21 32 16)(2 13 25 18)(3 10 26 23)(4 20 27 15)(5 17 28 12)(6 9 29 22)(7 14 30 19)(8 24 31 11)
(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 13 5 9)(2 12 6 16)(3 11 7 15)(4 10 8 14)(17 29 21 25)(18 28 22 32)(19 27 23 31)(20 26 24 30)

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

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

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

Matrix representation of (C2×C4)⋊Q16 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 10 1 0 0 0 0 1 7 0 0 0 0 0 0 10 1 0 0 0 0 1 7
,
 6 4 0 0 0 0 4 0 0 0 0 0 0 0 0 0 1 7 0 0 0 0 7 16 0 0 10 1 0 0 0 0 1 7 0 0
,
 4 11 0 0 0 0 0 13 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 16 0 0 0 0 0 0 16 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],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,10,1,0,0,0,0,1,7,0,0,0,0,0,0,10,1,0,0,0,0,1,7],[6,4,0,0,0,0,4,0,0,0,0,0,0,0,0,0,10,1,0,0,0,0,1,7,0,0,1,7,0,0,0,0,7,16,0,0],[4,0,0,0,0,0,11,13,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,1,0,0,0,0,0,0,1,0,0] >;

(C2×C4)⋊Q16 in GAP, Magma, Sage, TeX

(C_2\times C_4)\rtimes Q_{16}
% in TeX

G:=Group("(C2xC4):Q16");
// GroupNames label

G:=SmallGroup(128,337);
// by ID

G=gap.SmallGroup(128,337);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,224,141,232,422,352,1123,570,521,136,1411]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^4=c^8=1,d^2=c^4,c*b*c^-1=a*b=b*a,c*a*c^-1=d*a*d^-1=a*b^2,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

Export

׿
×
𝔽