Copied to
clipboard

G = C42.6D4order 128 = 27

6th non-split extension by C42 of D4 acting faithfully

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — C42.6D4
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C22×Q8 — C23.32C23 — C42.6D4
 Lower central C1 — C2 — C2×C4 — C42.6D4
 Upper central C1 — C2 — C22×C4 — C42.6D4
 Jennings C1 — C2 — C2 — C22×C4 — C42.6D4

Generators and relations for C42.6D4
G = < a,b,c,d | a4=b4=c4=1, d2=b, ab=ba, cac-1=dad-1=a-1b-1, cbc-1=b-1, bd=db, dcd-1=b-1c-1 >

Subgroups: 300 in 155 conjugacy classes, 54 normal (34 characteristic)
C1, C2, C2 [×4], C4 [×4], C4 [×10], C22 [×3], C22 [×3], C8 [×3], C2×C4 [×6], C2×C4 [×14], D4 [×4], Q8 [×4], Q8 [×8], C23, C23, C42 [×2], C42 [×6], C22⋊C4 [×2], C22⋊C4 [×2], C4⋊C4 [×7], C2×C8 [×2], C2×C8, M4(2) [×4], SD16 [×4], Q16 [×4], C22×C4, C22×C4 [×2], C2×D4, C2×D4, C2×Q8, C2×Q8 [×6], C2×Q8 [×3], C4○D4 [×4], C23⋊C4 [×2], Q8⋊C4 [×2], C4≀C2 [×2], C42⋊C2 [×3], C42⋊C2 [×2], C4×Q8 [×4], C2×M4(2) [×2], C2×SD16 [×2], C2×Q16 [×2], C8.C22 [×4], C22×Q8, C2×C4○D4, C4.9C42, M4(2)⋊4C4, C23.C23, C23.38D4, C42⋊C22, C23.32C23, C2×C8.C22, C42.6D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×8], C23, C22⋊C4 [×4], C22×C4, C2×D4 [×4], C4○D4 [×2], C2×C22⋊C4, C4×D4 [×2], C22≀C2, C4⋊D4 [×2], C22.D4, C23.23D4, C42.6D4

Character table of C42.6D4

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 4M 4N 4O 4P 4Q 4R 4S 8A 8B 8C 8D size 1 1 2 2 2 8 2 2 2 2 4 4 4 4 4 4 4 4 4 4 4 4 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 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 -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 -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 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 -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 -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 -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 -1 -1 -1 -1 -1 -1 linear of order 2 ρ9 1 1 1 -1 -1 -1 1 1 -1 -1 -i i 1 -1 1 i -i i -i i -i -1 1 i -i 1 i -1 -i linear of order 4 ρ10 1 1 1 -1 -1 -1 1 1 -1 -1 i -i 1 -1 1 -i i -i i -i i -1 1 -i i 1 -i -1 i linear of order 4 ρ11 1 1 1 -1 -1 -1 1 1 -1 -1 i i -1 1 -1 -i i -i -i i -i 1 1 i -i -1 -i 1 i linear of order 4 ρ12 1 1 1 -1 -1 -1 1 1 -1 -1 -i -i -1 1 -1 i -i i i -i i 1 1 -i i -1 i 1 -i linear of order 4 ρ13 1 1 1 -1 -1 1 1 1 -1 -1 i i -1 1 -1 -i i -i -i i -i 1 -1 -i i 1 i -1 -i linear of order 4 ρ14 1 1 1 -1 -1 1 1 1 -1 -1 -i -i -1 1 -1 i -i i i -i i 1 -1 i -i 1 -i -1 i linear of order 4 ρ15 1 1 1 -1 -1 1 1 1 -1 -1 -i i 1 -1 1 i -i i -i i -i -1 -1 -i i -1 -i 1 i linear of order 4 ρ16 1 1 1 -1 -1 1 1 1 -1 -1 i -i 1 -1 1 -i i -i i -i i -1 -1 i -i -1 i 1 -i linear of order 4 ρ17 2 2 2 2 2 0 -2 -2 -2 -2 0 -2 0 0 0 0 0 0 2 2 -2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ18 2 2 -2 2 -2 0 2 -2 2 -2 0 0 2 -2 -2 0 0 0 0 0 0 2 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ19 2 2 -2 2 -2 0 2 -2 2 -2 0 0 -2 2 2 0 0 0 0 0 0 -2 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ20 2 2 -2 -2 2 0 -2 2 2 -2 -2 0 0 0 0 -2 2 2 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ21 2 2 2 2 2 0 -2 -2 -2 -2 0 2 0 0 0 0 0 0 -2 -2 2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ22 2 2 -2 -2 2 0 -2 2 2 -2 2 0 0 0 0 2 -2 -2 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ23 2 2 -2 -2 2 0 2 -2 -2 2 0 0 -2 -2 2 0 0 0 0 0 0 2 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ24 2 2 -2 -2 2 0 2 -2 -2 2 0 0 2 2 -2 0 0 0 0 0 0 -2 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ25 2 2 2 -2 -2 0 -2 -2 2 2 0 -2i 0 0 0 0 0 0 -2i 2i 2i 0 0 0 0 0 0 0 0 complex lifted from C4○D4 ρ26 2 2 2 -2 -2 0 -2 -2 2 2 0 2i 0 0 0 0 0 0 2i -2i -2i 0 0 0 0 0 0 0 0 complex lifted from C4○D4 ρ27 2 2 -2 2 -2 0 -2 2 -2 2 -2i 0 0 0 0 2i 2i -2i 0 0 0 0 0 0 0 0 0 0 0 complex lifted from C4○D4 ρ28 2 2 -2 2 -2 0 -2 2 -2 2 2i 0 0 0 0 -2i -2i 2i 0 0 0 0 0 0 0 0 0 0 0 complex lifted from C4○D4 ρ29 8 -8 0 0 0 0 0 0 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

Smallest permutation representation of C42.6D4
On 32 points
Generators in S32
(1 32 27 2)(3 26 29 4)(5 28 31 6)(7 30 25 8)(9 22 17 10)(11 24 19 12)(13 18 21 14)(15 20 23 16)
(1 3 5 7)(2 4 6 8)(9 11 13 15)(10 12 14 16)(17 19 21 23)(18 20 22 24)(25 27 29 31)(26 28 30 32)
(1 13 31 21)(2 24 32 16)(3 11 25 19)(4 22 26 14)(5 9 27 17)(6 20 28 12)(7 15 29 23)(8 18 30 10)
(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)| (1,32,27,2)(3,26,29,4)(5,28,31,6)(7,30,25,8)(9,22,17,10)(11,24,19,12)(13,18,21,14)(15,20,23,16), (1,3,5,7)(2,4,6,8)(9,11,13,15)(10,12,14,16)(17,19,21,23)(18,20,22,24)(25,27,29,31)(26,28,30,32), (1,13,31,21)(2,24,32,16)(3,11,25,19)(4,22,26,14)(5,9,27,17)(6,20,28,12)(7,15,29,23)(8,18,30,10), (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( (1,32,27,2)(3,26,29,4)(5,28,31,6)(7,30,25,8)(9,22,17,10)(11,24,19,12)(13,18,21,14)(15,20,23,16), (1,3,5,7)(2,4,6,8)(9,11,13,15)(10,12,14,16)(17,19,21,23)(18,20,22,24)(25,27,29,31)(26,28,30,32), (1,13,31,21)(2,24,32,16)(3,11,25,19)(4,22,26,14)(5,9,27,17)(6,20,28,12)(7,15,29,23)(8,18,30,10), (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([(1,32,27,2),(3,26,29,4),(5,28,31,6),(7,30,25,8),(9,22,17,10),(11,24,19,12),(13,18,21,14),(15,20,23,16)], [(1,3,5,7),(2,4,6,8),(9,11,13,15),(10,12,14,16),(17,19,21,23),(18,20,22,24),(25,27,29,31),(26,28,30,32)], [(1,13,31,21),(2,24,32,16),(3,11,25,19),(4,22,26,14),(5,9,27,17),(6,20,28,12),(7,15,29,23),(8,18,30,10)], [(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 C42.6D4 in GL8(𝔽17)

 0 0 0 0 1 0 0 0 6 11 16 1 1 15 0 0 5 1 9 8 1 0 15 0 3 3 14 3 1 0 0 15 0 0 0 16 0 0 0 0 15 2 5 11 0 6 1 16 12 13 11 14 3 16 8 9 15 11 6 2 3 14 3 14
,
 0 1 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 16 0 0 0 0 0 6 11 16 1 1 15 0 0 6 0 16 0 1 16 0 0 1 15 14 12 0 16 0 1 4 16 11 3 1 16 16 0
,
 8 5 11 6 10 0 7 0 6 15 3 14 7 10 0 10 8 5 11 6 0 0 7 0 15 6 10 7 0 7 0 10 0 12 0 12 0 0 0 0 12 2 9 8 9 16 9 9 0 2 8 9 3 16 15 11 12 12 5 12 0 0 0 13
,
 6 11 16 1 1 15 0 0 0 0 0 0 16 0 0 0 14 14 3 14 16 0 0 2 5 1 9 8 1 0 15 0 1 0 0 0 0 0 0 0 14 9 0 0 7 11 16 16 5 1 5 4 10 12 9 9 13 14 16 10 3 14 14 14

G:=sub<GL(8,GF(17))| [0,6,5,3,0,15,12,15,0,11,1,3,0,2,13,11,0,16,9,14,0,5,11,6,0,1,8,3,16,11,14,2,1,1,1,1,0,0,3,3,0,15,0,0,0,6,16,14,0,0,15,0,0,1,8,3,0,0,0,15,0,16,9,14],[0,16,0,0,6,6,1,4,1,0,0,0,11,0,15,16,0,0,0,16,16,16,14,11,0,0,1,0,1,0,12,3,0,0,0,0,1,1,0,1,0,0,0,0,15,16,16,16,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0],[8,6,8,15,0,12,0,12,5,15,5,6,12,2,2,12,11,3,11,10,0,9,8,5,6,14,6,7,12,8,9,12,10,7,0,0,0,9,3,0,0,10,0,7,0,16,16,0,7,0,7,0,0,9,15,0,0,10,0,10,0,9,11,13],[6,0,14,5,1,14,5,13,11,0,14,1,0,9,1,14,16,0,3,9,0,0,5,16,1,0,14,8,0,0,4,10,1,16,16,1,0,7,10,3,15,0,0,0,0,11,12,14,0,0,0,15,0,16,9,14,0,0,2,0,0,16,9,14] >;

C42.6D4 in GAP, Magma, Sage, TeX

C_4^2._6D_4
% in TeX

G:=Group("C4^2.6D4");
// GroupNames label

G:=SmallGroup(128,637);
// by ID

G=gap.SmallGroup(128,637);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,288,422,352,2019,521,248,1411,718,172,1027,124]);
// Polycyclic

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

Export

׿
×
𝔽