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G = D5×C4○D8order 320 = 26·5

Direct product of D5 and C4○D8

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

Aliases: D5×C4○D8, D814D10, Q1612D10, SD1614D10, D4018C22, C40.45C23, C20.14C24, D20.9C23, Dic2016C22, Dic10.9C23, (D5×D8)⋊8C2, C4○D47D10, (C2×C8)⋊27D10, (D5×Q16)⋊8C2, D83D58C2, (C2×C40)⋊4C22, (D5×SD16)⋊7C2, C4.221(D4×D5), Q8.D108C2, D407C26C2, C22.4(D4×D5), D4⋊D512C22, (C4×D5).124D4, C20.380(C2×D4), C4○D205C22, (C5×D8)⋊12C22, (C8×D5)⋊16C22, Q8⋊D511C22, D4.8(C22×D5), (C5×D4).8C23, C4.14(C23×D5), C8.42(C22×D5), SD163D57C2, D10.114(C2×D4), D4.8D101C2, (C5×Q8).8C23, Q8.8(C22×D5), (Q8×D5).9C22, D42D58C22, C40⋊C220C22, C52C8.24C23, D4.D511C22, Q82D58C22, (C5×Q16)⋊10C22, (C22×D5).95D4, (C4×D5).65C23, C5⋊Q1610C22, (D4×D5).10C22, (C2×C20).531C23, (C2×Dic5).169D4, Dic5.126(C2×D4), (C5×SD16)⋊15C22, C10.115(C22×D4), (D5×C2×C8)⋊1C2, C55(C2×C4○D8), C2.88(C2×D4×D5), (D5×C4○D4)⋊1C2, (C5×C4○D8)⋊2C2, (C2×C10).11(C2×D4), (C5×C4○D4)⋊1C22, (C2×C52C8)⋊37C22, (C2×C4×D5).331C22, (C2×C4).618(C22×D5), SmallGroup(320,1439)

Series: Derived Chief Lower central Upper central

C1C20 — D5×C4○D8
C1C5C10C20C4×D5C2×C4×D5D5×C4○D4 — D5×C4○D8
C5C10C20 — D5×C4○D8
C1C4C2×C4C4○D8

Generators and relations for D5×C4○D8
 G = < a,b,c,d,e | a5=b2=c4=e2=1, d4=c2, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=c2d3 >

Subgroups: 1022 in 266 conjugacy classes, 101 normal (51 characteristic)
C1, C2, C2 [×8], C4 [×2], C4 [×6], C22, C22 [×12], C5, C8 [×2], C8 [×2], C2×C4, C2×C4 [×15], D4 [×2], D4 [×12], Q8 [×2], Q8 [×4], C23 [×3], D5 [×2], D5 [×3], C10, C10 [×3], C2×C8, C2×C8 [×5], D8, D8 [×3], SD16 [×2], SD16 [×6], Q16, Q16 [×3], C22×C4 [×3], C2×D4 [×4], C2×Q8 [×2], C4○D4 [×2], C4○D4 [×10], Dic5 [×2], Dic5 [×2], C20 [×2], C20 [×2], D10 [×2], D10 [×8], C2×C10, C2×C10 [×2], C22×C8, C2×D8, C2×SD16 [×2], C2×Q16, C4○D8, C4○D8 [×7], C2×C4○D4 [×2], C52C8 [×2], C40 [×2], Dic10 [×2], Dic10 [×2], C4×D5 [×4], C4×D5 [×6], D20 [×2], D20 [×2], C2×Dic5, C2×Dic5 [×2], C5⋊D4 [×6], C2×C20, C2×C20 [×2], C5×D4 [×2], C5×D4 [×2], C5×Q8 [×2], C22×D5, C22×D5 [×2], C2×C4○D8, C8×D5 [×4], C40⋊C2 [×2], D40, Dic20, C2×C52C8, D4⋊D5 [×2], D4.D5 [×2], Q8⋊D5 [×2], C5⋊Q16 [×2], C2×C40, C5×D8, C5×SD16 [×2], C5×Q16, C2×C4×D5, C2×C4×D5 [×2], C4○D20 [×2], C4○D20 [×2], D4×D5 [×2], D4×D5 [×2], D42D5 [×2], D42D5 [×2], Q8×D5 [×2], Q82D5 [×2], C5×C4○D4 [×2], D5×C2×C8, D407C2, D5×D8, D83D5, D5×SD16 [×2], SD163D5 [×2], D5×Q16, Q8.D10, D4.8D10 [×2], C5×C4○D8, D5×C4○D4 [×2], D5×C4○D8
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D5, C2×D4 [×6], C24, D10 [×7], C4○D8 [×2], C22×D4, C22×D5 [×7], C2×C4○D8, D4×D5 [×2], C23×D5, C2×D4×D5, D5×C4○D8

Smallest permutation representation of D5×C4○D8
On 80 points
Generators in S80
(1 55 59 43 66)(2 56 60 44 67)(3 49 61 45 68)(4 50 62 46 69)(5 51 63 47 70)(6 52 64 48 71)(7 53 57 41 72)(8 54 58 42 65)(9 38 18 30 76)(10 39 19 31 77)(11 40 20 32 78)(12 33 21 25 79)(13 34 22 26 80)(14 35 23 27 73)(15 36 24 28 74)(16 37 17 29 75)
(1 66)(2 67)(3 68)(4 69)(5 70)(6 71)(7 72)(8 65)(9 38)(10 39)(11 40)(12 33)(13 34)(14 35)(15 36)(16 37)(17 75)(18 76)(19 77)(20 78)(21 79)(22 80)(23 73)(24 74)(41 53)(42 54)(43 55)(44 56)(45 49)(46 50)(47 51)(48 52)
(1 40 5 36)(2 33 6 37)(3 34 7 38)(4 35 8 39)(9 68 13 72)(10 69 14 65)(11 70 15 66)(12 71 16 67)(17 56 21 52)(18 49 22 53)(19 50 23 54)(20 51 24 55)(25 64 29 60)(26 57 30 61)(27 58 31 62)(28 59 32 63)(41 76 45 80)(42 77 46 73)(43 78 47 74)(44 79 48 75)
(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 13)(10 12)(14 16)(17 23)(18 22)(19 21)(25 31)(26 30)(27 29)(33 39)(34 38)(35 37)(41 45)(42 44)(46 48)(49 53)(50 52)(54 56)(57 61)(58 60)(62 64)(65 67)(68 72)(69 71)(73 75)(76 80)(77 79)

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

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

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

56 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I4A4B4C4D4E4F4G4H4I4J5A5B8A8B8C8D8E8F8G8H10A10B10C10D10E10F10G10H20A20B20C20D20E20F20G20H20I20J40A···40H
order12222222224444444444558888888810101010101010102020202020202020202040···40
size11244551020201124455102020222222101010102244888822224488884···4

56 irreducible representations

dim1111111111112222222222444
type+++++++++++++++++++++++
imageC1C2C2C2C2C2C2C2C2C2C2C2D4D4D4D5D10D10D10D10D10C4○D8D4×D5D4×D5D5×C4○D8
kernelD5×C4○D8D5×C2×C8D407C2D5×D8D83D5D5×SD16SD163D5D5×Q16Q8.D10D4.8D10C5×C4○D8D5×C4○D4C4×D5C2×Dic5C22×D5C4○D8C2×C8D8SD16Q16C4○D4D5C4C22C1
# reps1111122112122112224248228

Matrix representation of D5×C4○D8 in GL4(𝔽41) generated by

7100
334000
0010
0001
,
404000
0100
0010
0001
,
40000
04000
00320
00032
,
1000
0100
001229
001212
,
1000
0100
0010
00040
G:=sub<GL(4,GF(41))| [7,33,0,0,1,40,0,0,0,0,1,0,0,0,0,1],[40,0,0,0,40,1,0,0,0,0,1,0,0,0,0,1],[40,0,0,0,0,40,0,0,0,0,32,0,0,0,0,32],[1,0,0,0,0,1,0,0,0,0,12,12,0,0,29,12],[1,0,0,0,0,1,0,0,0,0,1,0,0,0,0,40] >;

D5×C4○D8 in GAP, Magma, Sage, TeX

D_5\times C_4\circ D_8
% in TeX

G:=Group("D5xC4oD8");
// GroupNames label

G:=SmallGroup(320,1439);
// by ID

G=gap.SmallGroup(320,1439);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,387,570,185,438,235,102,12550]);
// Polycyclic

G:=Group<a,b,c,d,e|a^5=b^2=c^4=e^2=1,d^4=c^2,b*a*b=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=c^2*d^3>;
// generators/relations

׿
×
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