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## G = C2×C8⋊D5order 160 = 25·5

### Direct product of C2 and C8⋊D5

Series: Derived Chief Lower central Upper central

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

Generators and relations for C2×C8⋊D5
G = < a,b,c,d | a2=b8=c5=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b5, dcd=c-1 >

Subgroups: 184 in 68 conjugacy classes, 41 normal (19 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×2], C4 [×2], C22, C22 [×4], C5, C8 [×2], C8 [×2], C2×C4, C2×C4 [×5], C23, D5 [×2], C10, C10 [×2], C2×C8, C2×C8, M4(2) [×4], C22×C4, Dic5 [×2], C20 [×2], D10 [×2], D10 [×2], C2×C10, C2×M4(2), C52C8 [×2], C40 [×2], C4×D5 [×4], C2×Dic5, C2×C20, C22×D5, C8⋊D5 [×4], C2×C52C8, C2×C40, C2×C4×D5, C2×C8⋊D5
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], C23, D5, M4(2) [×2], C22×C4, D10 [×3], C2×M4(2), C4×D5 [×2], C22×D5, C8⋊D5 [×2], C2×C4×D5, C2×C8⋊D5

Smallest permutation representation of C2×C8⋊D5
On 80 points
Generators in S80
(1 59)(2 60)(3 61)(4 62)(5 63)(6 64)(7 57)(8 58)(9 27)(10 28)(11 29)(12 30)(13 31)(14 32)(15 25)(16 26)(17 40)(18 33)(19 34)(20 35)(21 36)(22 37)(23 38)(24 39)(41 66)(42 67)(43 68)(44 69)(45 70)(46 71)(47 72)(48 65)(49 77)(50 78)(51 79)(52 80)(53 73)(54 74)(55 75)(56 76)
(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 11 39 72 75)(2 12 40 65 76)(3 13 33 66 77)(4 14 34 67 78)(5 15 35 68 79)(6 16 36 69 80)(7 9 37 70 73)(8 10 38 71 74)(17 48 56 60 30)(18 41 49 61 31)(19 42 50 62 32)(20 43 51 63 25)(21 44 52 64 26)(22 45 53 57 27)(23 46 54 58 28)(24 47 55 59 29)
(1 75)(2 80)(3 77)(4 74)(5 79)(6 76)(7 73)(8 78)(9 70)(10 67)(11 72)(12 69)(13 66)(14 71)(15 68)(16 65)(17 21)(19 23)(25 43)(26 48)(27 45)(28 42)(29 47)(30 44)(31 41)(32 46)(34 38)(36 40)(49 61)(50 58)(51 63)(52 60)(53 57)(54 62)(55 59)(56 64)

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

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

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

52 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 10A ··· 10F 20A ··· 20H 40A ··· 40P order 1 2 2 2 2 2 4 4 4 4 4 4 5 5 8 8 8 8 8 8 8 8 10 ··· 10 20 ··· 20 40 ··· 40 size 1 1 1 1 10 10 1 1 1 1 10 10 2 2 2 2 2 2 10 10 10 10 2 ··· 2 2 ··· 2 2 ··· 2

52 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 type + + + + + + + + image C1 C2 C2 C2 C2 C4 C4 C4 D5 M4(2) D10 D10 C4×D5 C4×D5 C8⋊D5 kernel C2×C8⋊D5 C8⋊D5 C2×C5⋊2C8 C2×C40 C2×C4×D5 C4×D5 C2×Dic5 C22×D5 C2×C8 C10 C8 C2×C4 C4 C22 C2 # reps 1 4 1 1 1 4 2 2 2 4 4 2 4 4 16

Matrix representation of C2×C8⋊D5 in GL3(𝔽41) generated by

 40 0 0 0 40 0 0 0 40
,
 1 0 0 0 6 2 0 39 35
,
 1 0 0 0 0 40 0 1 6
,
 40 0 0 0 6 35 0 40 35
G:=sub<GL(3,GF(41))| [40,0,0,0,40,0,0,0,40],[1,0,0,0,6,39,0,2,35],[1,0,0,0,0,1,0,40,6],[40,0,0,0,6,40,0,35,35] >;

C2×C8⋊D5 in GAP, Magma, Sage, TeX

C_2\times C_8\rtimes D_5
% in TeX

G:=Group("C2xC8:D5");
// GroupNames label

G:=SmallGroup(160,121);
// by ID

G=gap.SmallGroup(160,121);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-5,362,50,69,4613]);
// Polycyclic

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

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