Copied to
clipboard

## G = D5×M4(2)  order 160 = 25·5

### Direct product of D5 and M4(2)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — D5×M4(2)
 Chief series C1 — C5 — C10 — C20 — C4×D5 — C2×C4×D5 — D5×M4(2)
 Lower central C5 — C10 — D5×M4(2)
 Upper central C1 — C4 — M4(2)

Generators and relations for D5×M4(2)
G = < a,b,c,d | a5=b2=c8=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c5 >

Subgroups: 184 in 68 conjugacy classes, 39 normal (27 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C8, C2×C4, C2×C4, C23, D5, D5, C10, C10, C2×C8, M4(2), M4(2), C22×C4, Dic5, C20, D10, D10, C2×C10, C2×M4(2), C52C8, C40, C4×D5, C2×Dic5, C2×C20, C22×D5, C8×D5, C8⋊D5, C4.Dic5, C5×M4(2), C2×C4×D5, D5×M4(2)
Quotients: C1, C2, C4, C22, C2×C4, C23, D5, M4(2), C22×C4, D10, C2×M4(2), C4×D5, C22×D5, C2×C4×D5, D5×M4(2)

Smallest permutation representation of D5×M4(2)
On 40 points
Generators in S40
(1 26 37 18 10)(2 27 38 19 11)(3 28 39 20 12)(4 29 40 21 13)(5 30 33 22 14)(6 31 34 23 15)(7 32 35 24 16)(8 25 36 17 9)
(1 14)(2 15)(3 16)(4 9)(5 10)(6 11)(7 12)(8 13)(17 29)(18 30)(19 31)(20 32)(21 25)(22 26)(23 27)(24 28)(33 37)(34 38)(35 39)(36 40)
(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)(33 34 35 36 37 38 39 40)
(2 6)(4 8)(9 13)(11 15)(17 21)(19 23)(25 29)(27 31)(34 38)(36 40)

G:=sub<Sym(40)| (1,26,37,18,10)(2,27,38,19,11)(3,28,39,20,12)(4,29,40,21,13)(5,30,33,22,14)(6,31,34,23,15)(7,32,35,24,16)(8,25,36,17,9), (1,14)(2,15)(3,16)(4,9)(5,10)(6,11)(7,12)(8,13)(17,29)(18,30)(19,31)(20,32)(21,25)(22,26)(23,27)(24,28)(33,37)(34,38)(35,39)(36,40), (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)(33,34,35,36,37,38,39,40), (2,6)(4,8)(9,13)(11,15)(17,21)(19,23)(25,29)(27,31)(34,38)(36,40)>;

G:=Group( (1,26,37,18,10)(2,27,38,19,11)(3,28,39,20,12)(4,29,40,21,13)(5,30,33,22,14)(6,31,34,23,15)(7,32,35,24,16)(8,25,36,17,9), (1,14)(2,15)(3,16)(4,9)(5,10)(6,11)(7,12)(8,13)(17,29)(18,30)(19,31)(20,32)(21,25)(22,26)(23,27)(24,28)(33,37)(34,38)(35,39)(36,40), (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)(33,34,35,36,37,38,39,40), (2,6)(4,8)(9,13)(11,15)(17,21)(19,23)(25,29)(27,31)(34,38)(36,40) );

G=PermutationGroup([[(1,26,37,18,10),(2,27,38,19,11),(3,28,39,20,12),(4,29,40,21,13),(5,30,33,22,14),(6,31,34,23,15),(7,32,35,24,16),(8,25,36,17,9)], [(1,14),(2,15),(3,16),(4,9),(5,10),(6,11),(7,12),(8,13),(17,29),(18,30),(19,31),(20,32),(21,25),(22,26),(23,27),(24,28),(33,37),(34,38),(35,39),(36,40)], [(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),(33,34,35,36,37,38,39,40)], [(2,6),(4,8),(9,13),(11,15),(17,21),(19,23),(25,29),(27,31),(34,38),(36,40)]])

40 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 5A 5B 8A 8B 8C 8D 8E 8F 8G 8H 10A 10B 10C 10D 20A 20B 20C 20D 20E 20F 40A ··· 40H order 1 2 2 2 2 2 4 4 4 4 4 4 5 5 8 8 8 8 8 8 8 8 10 10 10 10 20 20 20 20 20 20 40 ··· 40 size 1 1 2 5 5 10 1 1 2 5 5 10 2 2 2 2 2 2 10 10 10 10 2 2 4 4 2 2 2 2 4 4 4 ··· 4

40 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 4 type + + + + + + + + + image C1 C2 C2 C2 C2 C2 C4 C4 C4 D5 M4(2) D10 D10 C4×D5 C4×D5 D5×M4(2) kernel D5×M4(2) C8×D5 C8⋊D5 C4.Dic5 C5×M4(2) C2×C4×D5 C4×D5 C2×Dic5 C22×D5 M4(2) D5 C8 C2×C4 C4 C22 C1 # reps 1 2 2 1 1 1 4 2 2 2 4 4 2 4 4 4

Matrix representation of D5×M4(2) in GL4(𝔽41) generated by

 1 0 0 0 0 1 0 0 0 0 40 1 0 0 33 7
,
 40 0 0 0 0 40 0 0 0 0 40 0 0 0 33 1
,
 0 1 0 0 9 0 0 0 0 0 40 0 0 0 0 40
,
 1 0 0 0 0 40 0 0 0 0 1 0 0 0 0 1
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,40,33,0,0,1,7],[40,0,0,0,0,40,0,0,0,0,40,33,0,0,0,1],[0,9,0,0,1,0,0,0,0,0,40,0,0,0,0,40],[1,0,0,0,0,40,0,0,0,0,1,0,0,0,0,1] >;

D5×M4(2) in GAP, Magma, Sage, TeX

D_5\times M_4(2)
% in TeX

G:=Group("D5xM4(2)");
// GroupNames label

G:=SmallGroup(160,127);
// by ID

G=gap.SmallGroup(160,127);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-5,188,50,69,4613]);
// Polycyclic

G:=Group<a,b,c,d|a^5=b^2=c^8=d^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^5>;
// generators/relations

׿
×
𝔽