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G = C2×C8⋊D5order 160 = 25·5

Direct product of C2 and C8⋊D5

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C2×C8⋊D5, C89D10, C4011C22, C103M4(2), C20.36C23, (C2×C8)⋊6D5, (C2×C40)⋊9C2, (C4×D5).3C4, C4.24(C4×D5), C54(C2×M4(2)), C20.48(C2×C4), (C2×C4).98D10, C52C810C22, D10.10(C2×C4), (C2×Dic5).6C4, (C22×D5).4C4, C22.14(C4×D5), C4.36(C22×D5), C10.26(C22×C4), Dic5.12(C2×C4), (C4×D5).22C22, (C2×C20).111C22, C2.14(C2×C4×D5), (C2×C4×D5).12C2, (C2×C52C8)⋊11C2, (C2×C10).35(C2×C4), SmallGroup(160,121)

Series: Derived Chief Lower central Upper central

C1C10 — C2×C8⋊D5
C1C5C10C20C4×D5C2×C4×D5 — C2×C8⋊D5
C5C10 — C2×C8⋊D5
C1C2×C4C2×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)])

C2×C8⋊D5 is a maximal subgroup of
C8.25D20  C20.10M4(2)  (C2×C8)⋊F5  C20.24C42  C20.25C42  C86D20  D10.5C42  C89D20  D10.6C42  D10.7C42  D107M4(2)  C22⋊C8⋊D5  Dic52M4(2)  C52C826D4  (D4×D5)⋊C4  D4⋊(C4×D5)  C52C8⋊D4  C5⋊(C82D4)  (Q8×D5)⋊C4  Q8⋊(C4×D5)  C5⋊(C8⋊D4)  C52C8.D4  D105M4(2)  C205M4(2)  C206M4(2)  C42.30D10  C8⋊(C4×D5)  C82D20  C8.2D20  C4020(C2×C4)  C83D20  M4(2).25D10  C4032D4  C4018D4  C4012D4  C408D4  C40.36D4  C2×D5×M4(2)  C20.72C24  Q16⋊D10
C2×C8⋊D5 is a maximal quotient of
C4011Q8  C42.282D10  C86D20  Dic5.9M4(2)  D107M4(2)  C52C826D4  C42.198D10  C42.202D10  C205M4(2)  C206M4(2)  C4032D4

52 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F5A5B8A8B8C8D8E8F8G8H10A···10F20A···20H40A···40P
order122222444444558888888810···1020···2040···40
size1111101011111010222222101010102···22···22···2

52 irreducible representations

dim111111112222222
type++++++++
imageC1C2C2C2C2C4C4C4D5M4(2)D10D10C4×D5C4×D5C8⋊D5
kernelC2×C8⋊D5C8⋊D5C2×C52C8C2×C40C2×C4×D5C4×D5C2×Dic5C22×D5C2×C8C10C8C2×C4C4C22C2
# reps1411142224424416

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

4000
0400
0040
,
100
062
03935
,
100
0040
016
,
4000
0635
04035
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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