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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 [×4], C4 [×2], C4 [×2], C22, C22 [×4], C5, C8 [×2], C8 [×6], C2×C4, C2×C4 [×5], C23, D5 [×2], D5, C10, C10, C2×C8, C2×C8 [×7], M4(2) [×2], M4(2) [×4], C22×C4, Dic5 [×2], C20 [×2], D10 [×2], D10 [×2], C2×C10, C8.C4, C8.C4 [×3], C22×C8, C2×M4(2) [×2], C52C8 [×2], C52C8 [×2], C40 [×2], C40 [×2], C4×D5 [×4], C2×Dic5, C2×C20, C22×D5, C2×C8.C4, C8×D5 [×4], C8×D5 [×2], C8⋊D5 [×2], C2×C52C8, C4.Dic5 [×2], C2×C40, C5×M4(2) [×2], C2×C4×D5, C40.6C4, C20.53D4 [×2], C5×C8.C4, D5×C2×C8, D5×M4(2) [×2], D5×C8.C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], Q8 [×2], C23, D5, C4⋊C4 [×4], C22×C4, C2×D4, C2×Q8, D10 [×3], C8.C4 [×2], C2×C4⋊C4, C4×D5 [×2], 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 77 17 37 57)(2 78 18 38 58)(3 79 19 39 59)(4 80 20 40 60)(5 73 21 33 61)(6 74 22 34 62)(7 75 23 35 63)(8 76 24 36 64)(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 61)(2 62)(3 63)(4 64)(5 57)(6 58)(7 59)(8 60)(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 77)(34 78)(35 79)(36 80)(37 73)(38 74)(39 75)(40 76)(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 63 11 61 13 59 15 57)(10 62 12 60 14 58 16 64)(17 41 23 43 21 45 19 47)(18 48 24 42 22 44 20 46)(25 38 27 36 29 34 31 40)(26 37 28 35 30 33 32 39)(49 79 51 77 53 75 55 73)(50 78 52 76 54 74 56 80)

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

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

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

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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