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## G = D5×C4.D4order 320 = 26·5

### Direct product of D5 and C4.D4

Series: Derived Chief Lower central Upper central

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

Generators and relations for D5×C4.D4
G = < a,b,c,d,e | a5=b2=c4=1, d4=c2, e2=c, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd-1=c-1, ce=ec, ede-1=c-1d3 >

Subgroups: 942 in 186 conjugacy classes, 53 normal (21 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, C23, C23, D5, D5, C10, C10, C2×C8, M4(2), M4(2), C22×C4, C2×D4, C2×D4, C24, Dic5, C20, D10, D10, C2×C10, C2×C10, C4.D4, C4.D4, C2×M4(2), C22×D4, C52C8, C40, C4×D5, D20, C2×Dic5, C5⋊D4, C2×C20, C5×D4, C22×D5, C22×D5, C22×D5, C22×C10, C2×C4.D4, C8×D5, C8⋊D5, C4.Dic5, C5×M4(2), C2×C4×D5, C2×D20, D4×D5, C2×C5⋊D4, D4×C10, C23×D5, C20.46D4, C20.D4, C5×C4.D4, D5×M4(2), C2×D4×D5, D5×C4.D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D5, C22⋊C4, C22×C4, C2×D4, D10, C4.D4, C2×C22⋊C4, C4×D5, C22×D5, C2×C4.D4, C2×C4×D5, D4×D5, D5×C22⋊C4, D5×C4.D4

Smallest permutation representation of D5×C4.D4
On 40 points
Generators in S40
(1 31 11 33 22)(2 32 12 34 23)(3 25 13 35 24)(4 26 14 36 17)(5 27 15 37 18)(6 28 16 38 19)(7 29 9 39 20)(8 30 10 40 21)
(1 22)(2 23)(3 24)(4 17)(5 18)(6 19)(7 20)(8 21)(25 35)(26 36)(27 37)(28 38)(29 39)(30 40)(31 33)(32 34)
(1 3 5 7)(2 8 6 4)(9 11 13 15)(10 16 14 12)(17 23 21 19)(18 20 22 24)(25 27 29 31)(26 32 30 28)(33 35 37 39)(34 40 38 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 4 3 2 5 8 7 6)(9 16 11 14 13 12 15 10)(17 24 23 18 21 20 19 22)(25 32 27 30 29 28 31 26)(33 36 35 34 37 40 39 38)

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

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

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

44 conjugacy classes

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

44 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 4 4 8 type + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C4 C4 D4 D5 D10 D10 C4×D5 C4.D4 D4×D5 D5×C4.D4 kernel D5×C4.D4 C20.46D4 C20.D4 C5×C4.D4 D5×M4(2) C2×D4×D5 C2×C5⋊D4 C23×D5 C4×D5 C4.D4 M4(2) C2×D4 C23 D5 C4 C1 # reps 1 2 1 1 2 1 4 4 4 2 4 2 8 2 4 2

Matrix representation of D5×C4.D4 in GL8(𝔽41)

 40 0 1 0 0 0 0 0 0 40 0 1 0 0 0 0 33 0 7 0 0 0 0 0 0 33 0 7 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 8 0 40 0 0 0 0 0 0 8 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
,
 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 0 1 0 0 0 0 0 0 40 0 0 0 0 0 0 0 36 0 0 1 0 0 0 0 0 5 40 0
,
 0 1 0 0 0 0 0 0 1 0 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 5 37 0 39 0 0 0 0 37 5 39 0 0 0 0 0 1 0 36 4 0 0 0 0 0 0 4 36
,
 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 37 5 39 0 0 0 0 0 5 37 0 39 0 0 0 0 0 0 4 36 0 0 0 0 1 0 36 4

G:=sub<GL(8,GF(41))| [40,0,33,0,0,0,0,0,0,40,0,33,0,0,0,0,1,0,7,0,0,0,0,0,0,1,0,7,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,8,0,0,0,0,0,0,1,0,8,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40],[40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,40,36,0,0,0,0,0,1,0,0,5,0,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,5,37,1,0,0,0,0,0,37,5,0,0,0,0,0,0,0,39,36,4,0,0,0,0,39,0,4,36],[0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,37,5,0,1,0,0,0,0,5,37,0,0,0,0,0,0,39,0,4,36,0,0,0,0,0,39,36,4] >;

D5×C4.D4 in GAP, Magma, Sage, TeX

D_5\times C_4.D_4
% in TeX

G:=Group("D5xC4.D4");
// GroupNames label

G:=SmallGroup(320,371);
// by ID

G=gap.SmallGroup(320,371);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,219,58,570,136,438,12550]);
// Polycyclic

G:=Group<a,b,c,d,e|a^5=b^2=c^4=1,d^4=c^2,e^2=c,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^-1=c^-1,c*e=e*c,e*d*e^-1=c^-1*d^3>;
// generators/relations

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