Copied to
clipboard

## G = (C2×C4)⋊D8order 128 = 27

### The semidirect product of C2×C4 and D8 acting via D8/C2=D4

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C22×C4 — (C2×C4)⋊D8
 Chief series C1 — C2 — C22 — C23 — C22×C4 — C2×C4⋊C4 — C22.31C24 — (C2×C4)⋊D8
 Lower central C1 — C22 — C22×C4 — (C2×C4)⋊D8
 Upper central C1 — C22 — C22×C4 — (C2×C4)⋊D8
 Jennings C1 — C2 — C22 — C22×C4 — (C2×C4)⋊D8

Generators and relations for (C2×C4)⋊D8
G = < a,b,c,d | a2=b4=c8=d2=1, cbc-1=dbd=ab=ba, cac-1=ab2, ad=da, dcd=c-1 >

Subgroups: 452 in 155 conjugacy classes, 34 normal (18 characteristic)
C1, C2 [×3], C2 [×6], C4 [×8], C22 [×3], C22 [×18], C8 [×2], C2×C4 [×4], C2×C4 [×14], D4 [×15], Q8 [×2], C23, C23 [×10], C22⋊C4 [×8], C4⋊C4 [×2], C4⋊C4 [×3], C2×C8 [×2], D8 [×4], C22×C4 [×3], C22×C4 [×3], C2×D4 [×2], C2×D4 [×8], C2×Q8, C4○D4 [×4], C24, C2.C42, C22⋊C8 [×2], D4⋊C4 [×2], C2×C22⋊C4 [×2], C2×C4⋊C4, C4⋊D4 [×2], C4⋊D4 [×3], C22⋊Q8 [×2], C2×D8 [×2], C22×D4, C2×C4○D4, C22.M4(2), C22.SD16 [×2], C23.10D4, C22⋊D8 [×2], C22.31C24, (C2×C4)⋊D8
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, D8 [×2], C2×D4 [×3], C22≀C2, C2×D8, C8⋊C22, C22⋊D8, D4.8D4, C2≀C22, (C2×C4)⋊D8

Character table of (C2×C4)⋊D8

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 4A 4B 4C 4D 4E 4F 4G 4H 4I 8A 8B 8C 8D size 1 1 1 1 2 2 8 8 8 8 4 4 4 4 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 0 0 2 -2 0 0 0 -2 0 0 0 0 0 0 orthogonal lifted from D4 ρ10 2 2 2 2 2 2 0 0 0 0 2 -2 -2 2 0 -2 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ11 2 2 2 2 -2 -2 0 0 -2 0 0 2 -2 0 0 0 2 0 0 0 0 0 0 orthogonal lifted from D4 ρ12 2 2 2 2 -2 -2 0 -2 0 0 0 -2 2 0 0 0 0 2 0 0 0 0 0 orthogonal lifted from D4 ρ13 2 2 2 2 2 2 0 0 0 0 -2 -2 -2 -2 0 2 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ14 2 2 2 2 -2 -2 0 2 0 0 0 -2 2 0 0 0 0 -2 0 0 0 0 0 orthogonal lifted from D4 ρ15 2 2 -2 -2 -2 2 0 0 0 0 2 0 0 -2 0 0 0 0 0 -√2 √2 √2 -√2 orthogonal lifted from D8 ρ16 2 2 -2 -2 -2 2 0 0 0 0 -2 0 0 2 0 0 0 0 0 -√2 -√2 √2 √2 orthogonal lifted from D8 ρ17 2 2 -2 -2 -2 2 0 0 0 0 2 0 0 -2 0 0 0 0 0 √2 -√2 -√2 √2 orthogonal lifted from D8 ρ18 2 2 -2 -2 -2 2 0 0 0 0 -2 0 0 2 0 0 0 0 0 √2 √2 -√2 -√2 orthogonal lifted from D8 ρ19 4 -4 4 -4 0 0 2 0 0 -2 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C2≀C22 ρ20 4 -4 4 -4 0 0 -2 0 0 2 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C2≀C22 ρ21 4 4 -4 -4 4 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C8⋊C22 ρ22 4 -4 -4 4 0 0 0 0 0 0 0 0 0 0 2i 0 0 0 -2i 0 0 0 0 complex lifted from D4.8D4 ρ23 4 -4 -4 4 0 0 0 0 0 0 0 0 0 0 -2i 0 0 0 2i 0 0 0 0 complex lifted from D4.8D4

Smallest permutation representation of (C2×C4)⋊D8
On 32 points
Generators in S32
(1 32)(3 26)(5 28)(7 30)(10 17)(12 19)(14 21)(16 23)
(1 17 32 10)(2 11 25 18)(3 12 26 19)(4 20 27 13)(5 21 28 14)(6 15 29 22)(7 16 30 23)(8 24 31 9)
(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 14)(2 13)(3 12)(4 11)(5 10)(6 9)(7 16)(8 15)(17 28)(18 27)(19 26)(20 25)(21 32)(22 31)(23 30)(24 29)

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

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

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

Matrix representation of (C2×C4)⋊D8 in GL6(𝔽17)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 16 0 6 14 0 0 0 16 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 15 3 14 8 0 0 4 2 5 6 0 0 0 0 13 0 0 0 0 0 15 4
,
 8 11 0 0 0 0 0 15 0 0 0 0 0 0 1 16 10 2 0 0 0 0 8 1 0 0 8 12 10 12 0 0 4 2 5 6
,
 16 0 0 0 0 0 4 1 0 0 0 0 0 0 15 12 8 4 0 0 4 2 5 6 0 0 0 0 16 0 0 0 0 0 8 1

G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,6,0,1,0,0,0,14,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,15,4,0,0,0,0,3,2,0,0,0,0,14,5,13,15,0,0,8,6,0,4],[8,0,0,0,0,0,11,15,0,0,0,0,0,0,1,0,8,4,0,0,16,0,12,2,0,0,10,8,10,5,0,0,2,1,12,6],[16,4,0,0,0,0,0,1,0,0,0,0,0,0,15,4,0,0,0,0,12,2,0,0,0,0,8,5,16,8,0,0,4,6,0,1] >;

(C2×C4)⋊D8 in GAP, Magma, Sage, TeX

(C_2\times C_4)\rtimes D_8
% in TeX

G:=Group("(C2xC4):D8");
// GroupNames label

G:=SmallGroup(128,330);
// by ID

G=gap.SmallGroup(128,330);
# by ID

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

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

Export

׿
×
𝔽