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## G = D5×C8⋊C22order 320 = 26·5

### Direct product of D5 and C8⋊C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — D5×C8⋊C22
 Chief series C1 — C5 — C10 — C20 — C4×D5 — C2×C4×D5 — C2×D4×D5 — D5×C8⋊C22
 Lower central C5 — C10 — C20 — D5×C8⋊C22
 Upper central C1 — C2 — C2×C4 — C8⋊C22

Generators and relations for D5×C8⋊C22
G = < a,b,c,d,e | a5=b2=c8=d2=e2=1, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd=c3, ece=c5, de=ed >

Subgroups: 1358 in 298 conjugacy classes, 101 normal (51 characteristic)
C1, C2, C2 [×10], C4 [×2], C4 [×4], C22, C22 [×24], C5, C8 [×2], C8 [×2], C2×C4, C2×C4 [×10], D4, D4 [×2], D4 [×14], Q8, Q8 [×2], C23 [×12], D5 [×2], D5 [×4], C10, C10 [×4], C2×C8 [×2], M4(2), M4(2) [×3], D8 [×2], D8 [×6], SD16 [×2], SD16 [×6], C22×C4 [×2], C2×D4, C2×D4 [×10], C2×Q8, C4○D4, C4○D4 [×5], C24, Dic5 [×2], Dic5, C20 [×2], C20, D10 [×2], D10 [×17], C2×C10, C2×C10 [×5], C2×M4(2), C2×D8 [×2], C2×SD16 [×2], C8⋊C22, C8⋊C22 [×7], C22×D4, C2×C4○D4, C52C8 [×2], C40 [×2], Dic10, Dic10, C4×D5 [×4], C4×D5 [×3], D20, D20 [×2], D20 [×2], C2×Dic5, C2×Dic5, C5⋊D4 [×7], C2×C20, C2×C20, C5×D4, C5×D4 [×2], C5×D4 [×2], C5×Q8, C22×D5, C22×D5 [×10], C22×C10, C2×C8⋊C22, C8×D5 [×2], C8⋊D5 [×2], C40⋊C2 [×2], D40 [×2], C4.Dic5, D4⋊D5 [×4], D4.D5 [×2], Q8⋊D5 [×2], C5×M4(2), C5×D8 [×2], C5×SD16 [×2], C2×C4×D5, C2×C4×D5, C2×D20, C4○D20, C4○D20, D4×D5, D4×D5 [×4], D4×D5 [×3], D42D5, D42D5, Q8×D5, Q82D5, C2×C5⋊D4, D4×C10, C5×C4○D4, C23×D5, D5×M4(2), C8⋊D10, D5×D8 [×2], D8⋊D5 [×2], D5×SD16 [×2], D40⋊C2 [×2], D4.D10, D4⋊D10, C5×C8⋊C22, C2×D4×D5, D5×C4○D4, D5×C8⋊C22
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D5, C2×D4 [×6], C24, D10 [×7], C8⋊C22 [×2], C22×D4, C22×D5 [×7], C2×C8⋊C22, D4×D5 [×2], C23×D5, C2×D4×D5, D5×C8⋊C22

Smallest permutation representation of D5×C8⋊C22
On 40 points
Generators in S40
(1 27 24 35 16)(2 28 17 36 9)(3 29 18 37 10)(4 30 19 38 11)(5 31 20 39 12)(6 32 21 40 13)(7 25 22 33 14)(8 26 23 34 15)
(1 12)(2 13)(3 14)(4 15)(5 16)(6 9)(7 10)(8 11)(17 21)(18 22)(19 23)(20 24)(25 37)(26 38)(27 39)(28 40)(29 33)(30 34)(31 35)(32 36)
(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)
(1 3)(2 6)(5 7)(9 13)(10 16)(12 14)(17 21)(18 24)(20 22)(25 31)(27 29)(28 32)(33 39)(35 37)(36 40)
(1 5)(3 7)(10 14)(12 16)(18 22)(20 24)(25 29)(27 31)(33 37)(35 39)

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

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

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

44 conjugacy classes

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

44 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 8 type + + + + + + + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 D4 D4 D4 D5 D10 D10 D10 D10 D10 C8⋊C22 D4×D5 D4×D5 D5×C8⋊C22 kernel D5×C8⋊C22 D5×M4(2) C8⋊D10 D5×D8 D8⋊D5 D5×SD16 D40⋊C2 D4.D10 D4⋊D10 C5×C8⋊C22 C2×D4×D5 D5×C4○D4 C4×D5 C2×Dic5 C22×D5 C8⋊C22 M4(2) D8 SD16 C2×D4 C4○D4 D5 C4 C22 C1 # reps 1 1 1 2 2 2 2 1 1 1 1 1 2 1 1 2 2 4 4 2 2 2 2 2 2

Matrix representation of D5×C8⋊C22 in GL8(𝔽41)

 40 0 1 0 0 0 0 0 0 40 0 1 0 0 0 0 5 0 35 0 0 0 0 0 0 5 0 35 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 36 0 40 0 0 0 0 0 0 36 0 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 40
,
 0 40 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 40 40 1 32 0 0 0 0 0 0 40 0 0 0 0 0 40 0 0 0 0 0 0 0 0 23 0 1
,
 40 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 18 1
,
 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 1 0 0 0 0 0 23 23 0 1

G:=sub<GL(8,GF(41))| [40,0,5,0,0,0,0,0,0,40,0,5,0,0,0,0,1,0,35,0,0,0,0,0,0,1,0,35,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,36,0,0,0,0,0,0,1,0,36,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40],[0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,40,0,40,0,0,0,0,0,40,0,0,23,0,0,0,0,1,40,0,0,0,0,0,0,32,0,0,1],[40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,40,18,0,0,0,0,0,0,0,1],[40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,23,0,0,0,0,0,40,0,23,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1] >;

D5×C8⋊C22 in GAP, Magma, Sage, TeX

D_5\times C_8\rtimes C_2^2
% in TeX

G:=Group("D5xC8:C2^2");
// GroupNames label

G:=SmallGroup(320,1444);
// by ID

G=gap.SmallGroup(320,1444);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,570,185,438,235,102,12550]);
// Polycyclic

G:=Group<a,b,c,d,e|a^5=b^2=c^8=d^2=e^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,d*c*d=c^3,e*c*e=c^5,d*e=e*d>;
// generators/relations

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