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G = D5×C2×C8order 160 = 25·5

Direct product of C2×C8 and D5

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

Aliases: D5×C2×C8, C4010C22, C20.35C23, (C2×C40)⋊8C2, C103(C2×C8), C53(C22×C8), C4.23(C4×D5), C20.47(C2×C4), (C4×D5).10C4, (C2×C4).97D10, C52C813C22, D10.20(C2×C4), C22.13(C4×D5), C4.35(C22×D5), C10.25(C22×C4), Dic5.22(C2×C4), (C2×Dic5).16C4, (C4×D5).37C22, (C22×D5).10C4, (C2×C20).110C22, C2.2(C2×C4×D5), (C2×C4×D5).20C2, (C2×C52C8)⋊13C2, (C2×C10).34(C2×C4), SmallGroup(160,120)

Series: Derived Chief Lower central Upper central

C1C5 — D5×C2×C8
C1C5C10C20C4×D5C2×C4×D5 — D5×C2×C8
C5 — D5×C2×C8
C1C2×C8

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

Subgroups: 184 in 76 conjugacy classes, 49 normal (19 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×2], C4 [×2], C22, C22 [×6], C5, C8 [×2], C8 [×2], C2×C4, C2×C4 [×5], C23, D5 [×4], C10, C10 [×2], C2×C8, C2×C8 [×5], C22×C4, Dic5 [×2], C20 [×2], D10 [×6], C2×C10, C22×C8, C52C8 [×2], C40 [×2], C4×D5 [×4], C2×Dic5, C2×C20, C22×D5, C8×D5 [×4], C2×C52C8, C2×C40, C2×C4×D5, D5×C2×C8
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C8 [×4], C2×C4 [×6], C23, D5, C2×C8 [×6], C22×C4, D10 [×3], C22×C8, C4×D5 [×2], C22×D5, C8×D5 [×2], C2×C4×D5, D5×C2×C8

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

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

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

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

D5×C2×C8 is a maximal subgroup of
D101C16  D10⋊C16  D10.3M4(2)  D10.10D8  C20.10C42  D10.5C42  C89D20  D10.7C42  C55(C8×D4)  C22⋊C8⋊D5  D104M4(2)  D42D5⋊C4  D10⋊D8  D10⋊SD16  Q82D5⋊C4  D102SD16  D10⋊Q16  D205C8  D105M4(2)  C42.30D10  (C8×D5)⋊C4  C88D20  C8.27(C4×D5)  C87D20  D102Q16  C40⋊D4  C406D4  C4014D4  D103Q16  D5⋊M5(2)  C20.12C42  (C2×C8)⋊6F5  (C8×D5).C4
D5×C2×C8 is a maximal quotient of
C42.282D10  Dic5.14M4(2)  C55(C8×D4)  Dic105C8  C42.200D10  D205C8  D20.6C8  D20.5C8

64 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B4C4D4E4F4G4H5A5B8A···8H8I···8P10A···10F20A···20H40A···40P
order1222222244444444558···88···810···1020···2040···40
size1111555511115555221···15···52···22···22···2

64 irreducible representations

dim111111111222222
type++++++++
imageC1C2C2C2C2C4C4C4C8D5D10D10C4×D5C4×D5C8×D5
kernelD5×C2×C8C8×D5C2×C52C8C2×C40C2×C4×D5C4×D5C2×Dic5C22×D5D10C2×C8C8C2×C4C4C22C2
# reps14111422162424416

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

4000
010
001
,
100
030
003
,
100
001
0406
,
4000
0040
0400
G:=sub<GL(3,GF(41))| [40,0,0,0,1,0,0,0,1],[1,0,0,0,3,0,0,0,3],[1,0,0,0,0,40,0,1,6],[40,0,0,0,0,40,0,40,0] >;

D5×C2×C8 in GAP, Magma, Sage, TeX

D_5\times C_2\times C_8
% in TeX

G:=Group("D5xC2xC8");
// GroupNames label

G:=SmallGroup(160,120);
// by ID

G=gap.SmallGroup(160,120);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-5,50,69,4613]);
// 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,b*d=d*b,d*c*d=c^-1>;
// generators/relations

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