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## G = D5×C8.C4order 320 = 26·5

### Direct product of D5 and C8.C4

Series: Derived Chief Lower central Upper central

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

Generators and relations for D5×C8.C4
G = < a,b,c,d | a5=b2=c8=1, d4=c4, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 318 in 106 conjugacy classes, 53 normal (41 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C8, C2×C4, C2×C4, C23, D5, D5, C10, C10, C2×C8, C2×C8, M4(2), M4(2), C22×C4, Dic5, C20, D10, D10, C2×C10, C8.C4, C8.C4, C22×C8, C2×M4(2), C52C8, C52C8, C40, C40, C4×D5, C2×Dic5, C2×C20, C22×D5, C2×C8.C4, C8×D5, C8×D5, C8⋊D5, C2×C52C8, C4.Dic5, C2×C40, C5×M4(2), C2×C4×D5, C40.6C4, C20.53D4, C5×C8.C4, D5×C2×C8, D5×M4(2), D5×C8.C4
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, D5, C4⋊C4, C22×C4, C2×D4, C2×Q8, D10, C8.C4, C2×C4⋊C4, C4×D5, C22×D5, C2×C8.C4, C2×C4×D5, D4×D5, Q8×D5, D5×C4⋊C4, D5×C8.C4

Smallest permutation representation of D5×C8.C4
On 80 points
Generators in S80
(1 57 18 37 75)(2 58 19 38 76)(3 59 20 39 77)(4 60 21 40 78)(5 61 22 33 79)(6 62 23 34 80)(7 63 24 35 73)(8 64 17 36 74)(9 66 53 41 28)(10 67 54 42 29)(11 68 55 43 30)(12 69 56 44 31)(13 70 49 45 32)(14 71 50 46 25)(15 72 51 47 26)(16 65 52 48 27)
(1 79)(2 80)(3 73)(4 74)(5 75)(6 76)(7 77)(8 78)(9 70)(10 71)(11 72)(12 65)(13 66)(14 67)(15 68)(16 69)(17 21)(18 22)(19 23)(20 24)(25 54)(26 55)(27 56)(28 49)(29 50)(30 51)(31 52)(32 53)(33 57)(34 58)(35 59)(36 60)(37 61)(38 62)(39 63)(40 64)(41 45)(42 46)(43 47)(44 48)
(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)(41 42 43 44 45 46 47 48)(49 50 51 52 53 54 55 56)(57 58 59 60 61 62 63 64)(65 66 67 68 69 70 71 72)(73 74 75 76 77 78 79 80)
(1 66 7 68 5 70 3 72)(2 65 8 67 6 69 4 71)(9 73 11 79 13 77 15 75)(10 80 12 78 14 76 16 74)(17 42 23 44 21 46 19 48)(18 41 24 43 22 45 20 47)(25 38 27 36 29 34 31 40)(26 37 28 35 30 33 32 39)(49 59 51 57 53 63 55 61)(50 58 52 64 54 62 56 60)

G:=sub<Sym(80)| (1,57,18,37,75)(2,58,19,38,76)(3,59,20,39,77)(4,60,21,40,78)(5,61,22,33,79)(6,62,23,34,80)(7,63,24,35,73)(8,64,17,36,74)(9,66,53,41,28)(10,67,54,42,29)(11,68,55,43,30)(12,69,56,44,31)(13,70,49,45,32)(14,71,50,46,25)(15,72,51,47,26)(16,65,52,48,27), (1,79)(2,80)(3,73)(4,74)(5,75)(6,76)(7,77)(8,78)(9,70)(10,71)(11,72)(12,65)(13,66)(14,67)(15,68)(16,69)(17,21)(18,22)(19,23)(20,24)(25,54)(26,55)(27,56)(28,49)(29,50)(30,51)(31,52)(32,53)(33,57)(34,58)(35,59)(36,60)(37,61)(38,62)(39,63)(40,64)(41,45)(42,46)(43,47)(44,48), (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)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64)(65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80), (1,66,7,68,5,70,3,72)(2,65,8,67,6,69,4,71)(9,73,11,79,13,77,15,75)(10,80,12,78,14,76,16,74)(17,42,23,44,21,46,19,48)(18,41,24,43,22,45,20,47)(25,38,27,36,29,34,31,40)(26,37,28,35,30,33,32,39)(49,59,51,57,53,63,55,61)(50,58,52,64,54,62,56,60)>;

G:=Group( (1,57,18,37,75)(2,58,19,38,76)(3,59,20,39,77)(4,60,21,40,78)(5,61,22,33,79)(6,62,23,34,80)(7,63,24,35,73)(8,64,17,36,74)(9,66,53,41,28)(10,67,54,42,29)(11,68,55,43,30)(12,69,56,44,31)(13,70,49,45,32)(14,71,50,46,25)(15,72,51,47,26)(16,65,52,48,27), (1,79)(2,80)(3,73)(4,74)(5,75)(6,76)(7,77)(8,78)(9,70)(10,71)(11,72)(12,65)(13,66)(14,67)(15,68)(16,69)(17,21)(18,22)(19,23)(20,24)(25,54)(26,55)(27,56)(28,49)(29,50)(30,51)(31,52)(32,53)(33,57)(34,58)(35,59)(36,60)(37,61)(38,62)(39,63)(40,64)(41,45)(42,46)(43,47)(44,48), (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)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64)(65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80), (1,66,7,68,5,70,3,72)(2,65,8,67,6,69,4,71)(9,73,11,79,13,77,15,75)(10,80,12,78,14,76,16,74)(17,42,23,44,21,46,19,48)(18,41,24,43,22,45,20,47)(25,38,27,36,29,34,31,40)(26,37,28,35,30,33,32,39)(49,59,51,57,53,63,55,61)(50,58,52,64,54,62,56,60) );

G=PermutationGroup([[(1,57,18,37,75),(2,58,19,38,76),(3,59,20,39,77),(4,60,21,40,78),(5,61,22,33,79),(6,62,23,34,80),(7,63,24,35,73),(8,64,17,36,74),(9,66,53,41,28),(10,67,54,42,29),(11,68,55,43,30),(12,69,56,44,31),(13,70,49,45,32),(14,71,50,46,25),(15,72,51,47,26),(16,65,52,48,27)], [(1,79),(2,80),(3,73),(4,74),(5,75),(6,76),(7,77),(8,78),(9,70),(10,71),(11,72),(12,65),(13,66),(14,67),(15,68),(16,69),(17,21),(18,22),(19,23),(20,24),(25,54),(26,55),(27,56),(28,49),(29,50),(30,51),(31,52),(32,53),(33,57),(34,58),(35,59),(36,60),(37,61),(38,62),(39,63),(40,64),(41,45),(42,46),(43,47),(44,48)], [(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),(41,42,43,44,45,46,47,48),(49,50,51,52,53,54,55,56),(57,58,59,60,61,62,63,64),(65,66,67,68,69,70,71,72),(73,74,75,76,77,78,79,80)], [(1,66,7,68,5,70,3,72),(2,65,8,67,6,69,4,71),(9,73,11,79,13,77,15,75),(10,80,12,78,14,76,16,74),(17,42,23,44,21,46,19,48),(18,41,24,43,22,45,20,47),(25,38,27,36,29,34,31,40),(26,37,28,35,30,33,32,39),(49,59,51,57,53,63,55,61),(50,58,52,64,54,62,56,60)]])

56 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 8I 8J 8K 8L 8M 8N 8O 8P 10A 10B 10C 10D 20A 20B 20C 20D 20E 20F 40A ··· 40H 40I ··· 40P order 1 2 2 2 2 2 4 4 4 4 4 4 5 5 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 10 10 10 10 20 20 20 20 20 20 40 ··· 40 40 ··· 40 size 1 1 2 5 5 10 1 1 2 5 5 10 2 2 2 2 2 2 4 4 4 4 10 10 10 10 20 20 20 20 2 2 4 4 2 2 2 2 4 4 4 ··· 4 8 ··· 8

56 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + - - + + + + - image C1 C2 C2 C2 C2 C2 C4 D4 Q8 Q8 D5 D10 D10 C8.C4 C4×D5 D4×D5 Q8×D5 D5×C8.C4 kernel D5×C8.C4 C40.6C4 C20.53D4 C5×C8.C4 D5×C2×C8 D5×M4(2) C8×D5 C4×D5 C2×Dic5 C22×D5 C8.C4 C2×C8 M4(2) D5 C8 C4 C22 C1 # reps 1 1 2 1 1 2 8 2 1 1 2 2 4 8 8 2 2 8

Matrix representation of D5×C8.C4 in GL4(𝔽41) generated by

 6 1 0 0 40 0 0 0 0 0 1 0 0 0 0 1
,
 1 6 0 0 0 40 0 0 0 0 40 0 0 0 0 40
,
 1 0 0 0 0 1 0 0 0 0 3 0 0 0 0 14
,
 1 0 0 0 0 1 0 0 0 0 0 1 0 0 32 0
G:=sub<GL(4,GF(41))| [6,40,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,6,40,0,0,0,0,40,0,0,0,0,40],[1,0,0,0,0,1,0,0,0,0,3,0,0,0,0,14],[1,0,0,0,0,1,0,0,0,0,0,32,0,0,1,0] >;

D5×C8.C4 in GAP, Magma, Sage, TeX

D_5\times C_8.C_4
% in TeX

G:=Group("D5xC8.C4");
// GroupNames label

G:=SmallGroup(320,519);
// by ID

G=gap.SmallGroup(320,519);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,120,219,58,136,438,102,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^5=b^2=c^8=1,d^4=c^4,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

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