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## G = C2×D5.D8order 320 = 26·5

### Direct product of C2 and D5.D8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — C2×D5.D8
 Chief series C1 — C5 — C10 — D10 — C4×D5 — C4⋊F5 — C2×C4⋊F5 — C2×D5.D8
 Lower central C5 — C10 — C20 — C2×D5.D8
 Upper central C1 — C22 — C2×C4 — C2×C8

Generators and relations for C2×D5.D8
G = < a,b,c,d,e | a2=b5=c2=d8=1, e2=b-1c, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, ebe-1=b3, cd=dc, ece-1=b2c, ede-1=d-1 >

Subgroups: 538 in 130 conjugacy classes, 60 normal (30 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×2], C4 [×6], C22, C22 [×6], C5, C8 [×2], C8 [×2], C2×C4, C2×C4 [×13], C23, D5 [×2], D5 [×2], C10, C10 [×2], C4⋊C4 [×6], C2×C8, C2×C8 [×5], C22×C4 [×3], Dic5 [×2], C20 [×2], F5 [×4], D10 [×2], D10 [×4], C2×C10, C2.D8 [×4], C2×C4⋊C4 [×2], C22×C8, C52C8 [×2], C40 [×2], C4×D5 [×4], C2×Dic5, C2×C20, C2×F5 [×8], C22×D5, C2×C2.D8, C8×D5 [×4], C2×C52C8, C2×C40, C4⋊F5 [×4], C4⋊F5 [×2], C2×C4×D5, C22×F5 [×2], D5.D8 [×4], D5×C2×C8, C2×C4⋊F5 [×2], C2×D5.D8
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], Q8 [×2], C23, C4⋊C4 [×4], D8 [×2], Q16 [×2], C22×C4, C2×D4, C2×Q8, F5, C2.D8 [×4], C2×C4⋊C4, C2×D8, C2×Q16, C2×F5 [×3], C2×C2.D8, C4⋊F5 [×2], C22×F5, D5.D8 [×2], C2×C4⋊F5, C2×D5.D8

Smallest permutation representation of C2×D5.D8
On 80 points
Generators in S80
(1 63)(2 64)(3 57)(4 58)(5 59)(6 60)(7 61)(8 62)(9 47)(10 48)(11 41)(12 42)(13 43)(14 44)(15 45)(16 46)(17 40)(18 33)(19 34)(20 35)(21 36)(22 37)(23 38)(24 39)(25 75)(26 76)(27 77)(28 78)(29 79)(30 80)(31 73)(32 74)(49 72)(50 65)(51 66)(52 67)(53 68)(54 69)(55 70)(56 71)
(1 75 39 12 66)(2 76 40 13 67)(3 77 33 14 68)(4 78 34 15 69)(5 79 35 16 70)(6 80 36 9 71)(7 73 37 10 72)(8 74 38 11 65)(17 43 52 64 26)(18 44 53 57 27)(19 45 54 58 28)(20 46 55 59 29)(21 47 56 60 30)(22 48 49 61 31)(23 41 50 62 32)(24 42 51 63 25)
(1 66)(2 67)(3 68)(4 69)(5 70)(6 71)(7 72)(8 65)(9 80)(10 73)(11 74)(12 75)(13 76)(14 77)(15 78)(16 79)(25 42)(26 43)(27 44)(28 45)(29 46)(30 47)(31 48)(32 41)(49 61)(50 62)(51 63)(52 64)(53 57)(54 58)(55 59)(56 60)
(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)
(2 8)(3 7)(4 6)(9 78 36 69)(10 77 37 68)(11 76 38 67)(12 75 39 66)(13 74 40 65)(14 73 33 72)(15 80 34 71)(16 79 35 70)(17 50 43 32)(18 49 44 31)(19 56 45 30)(20 55 46 29)(21 54 47 28)(22 53 48 27)(23 52 41 26)(24 51 42 25)(57 61)(58 60)(62 64)

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

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

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

44 conjugacy classes

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

44 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 4 4 4 4 4 4 type + + + + + - - + + - + + + image C1 C2 C2 C2 C4 C4 C4 D4 Q8 Q8 D4 D8 Q16 F5 C2×F5 C2×F5 C4⋊F5 C4⋊F5 D5.D8 kernel C2×D5.D8 D5.D8 D5×C2×C8 C2×C4⋊F5 C8×D5 C2×C5⋊2C8 C2×C40 C4×D5 C4×D5 C2×Dic5 C22×D5 D10 D10 C2×C8 C8 C2×C4 C4 C22 C2 # reps 1 4 1 2 4 2 2 1 1 1 1 4 4 1 2 1 2 2 8

Matrix representation of C2×D5.D8 in GL6(𝔽41)

 40 0 0 0 0 0 0 40 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 40 40 40 40 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0
,
 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40 40 40 40 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0
,
 35 31 0 0 0 0 16 6 0 0 0 0 0 0 27 0 37 37 0 0 4 31 4 0 0 0 0 4 31 4 0 0 37 37 0 27
,
 9 0 0 0 0 0 22 32 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 40 40 40 40

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,1,0,0,0,0,40,0,1,0,0,0,40,0,0,1,0,0,40,0,0,0],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,40,0,0,1,0,0,40,0,1,0,0,0,40,1,0,0],[35,16,0,0,0,0,31,6,0,0,0,0,0,0,27,4,0,37,0,0,0,31,4,37,0,0,37,4,31,0,0,0,37,0,4,27],[9,22,0,0,0,0,0,32,0,0,0,0,0,0,1,0,0,40,0,0,0,0,1,40,0,0,0,0,0,40,0,0,0,1,0,40] >;

C2×D5.D8 in GAP, Magma, Sage, TeX

C_2\times D_5.D_8
% in TeX

G:=Group("C2xD5.D8");
// GroupNames label

G:=SmallGroup(320,1058);
// by ID

G=gap.SmallGroup(320,1058);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,422,268,1684,102,6278,1595]);
// Polycyclic

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

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