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G = D5×C8.C22order 320 = 26·5

Direct product of D5 and C8.C22

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

Aliases: D5×C8.C22, Q163D10, SD165D10, C40.4C23, C20.23C24, M4(2)⋊11D10, Dic203C22, D20.16C23, Dic10.16C23, (D5×Q16)⋊1C2, (C2×Q8)⋊22D10, (D5×SD16)⋊3C2, C4.191(D4×D5), Q8⋊D55C22, C8.4(C22×D5), Q16⋊D51C2, C4○D4.29D10, (C4×D5).100D4, C8.D103C2, C20.244(C2×D4), SD16⋊D53C2, C40⋊C25C22, C8⋊D55C22, (D5×M4(2))⋊3C2, D4.D56C22, (Q8×D5)⋊11C22, (C5×Q16)⋊1C22, C5⋊Q164C22, (C8×D5).1C22, C22.48(D4×D5), C4.23(C23×D5), D10.116(C2×D4), C20.C239C2, C52C8.11C23, (C2×Dic5).90D4, (Q8×C10)⋊20C22, (C5×SD16)⋊5C22, D4.16(C22×D5), (C5×D4).16C23, (D4×D5).11C22, (C4×D5).15C23, D4.9D1010C2, Q8.16(C22×D5), (C5×Q8).16C23, (C2×C20).114C23, Dic5.101(C2×D4), C4○D20.30C22, (C22×D5).139D4, C10.124(C22×D4), (C5×M4(2))⋊5C22, C4.Dic514C22, (C2×Dic10)⋊41C22, D42D5.10C22, Q82D5.10C22, (C2×Q8×D5)⋊17C2, C2.97(C2×D4×D5), C54(C2×C8.C22), (D5×C4○D4).4C2, (C2×C10).69(C2×D4), (C5×C8.C22)⋊1C2, (C2×C4×D5).171C22, (C2×C4).98(C22×D5), (C5×C4○D4).25C22, SmallGroup(320,1448)

Series: Derived Chief Lower central Upper central

C1C20 — D5×C8.C22
C1C5C10C20C4×D5C2×C4×D5C2×Q8×D5 — D5×C8.C22
C5C10C20 — D5×C8.C22
C1C2C2×C4C8.C22

Generators and relations for D5×C8.C22
 G = < a,b,c,d,e | a5=b2=c8=d2=e2=1, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd=c3, ece=c5, ede=c4d >

Subgroups: 942 in 258 conjugacy classes, 101 normal (51 characteristic)
C1, C2, C2 [×6], C4 [×2], C4 [×8], C22, C22 [×8], C5, C8 [×2], C8 [×2], C2×C4, C2×C4 [×16], D4, D4 [×6], Q8, Q8 [×2], Q8 [×10], C23 [×2], D5 [×2], D5 [×2], C10, C10 [×2], C2×C8 [×2], M4(2), M4(2) [×3], SD16 [×2], SD16 [×6], Q16 [×2], Q16 [×6], C22×C4 [×3], C2×D4 [×2], C2×Q8, C2×Q8 [×9], C4○D4, C4○D4 [×5], Dic5 [×2], Dic5 [×3], C20 [×2], C20 [×3], D10 [×2], D10 [×5], C2×C10, C2×C10, C2×M4(2), C2×SD16 [×2], C2×Q16 [×2], C8.C22, C8.C22 [×7], C22×Q8, C2×C4○D4, C52C8 [×2], C40 [×2], Dic10, Dic10 [×2], Dic10 [×6], C4×D5 [×4], C4×D5 [×7], D20, D20, C2×Dic5, C2×Dic5 [×2], C5⋊D4 [×3], C2×C20, C2×C20 [×2], C5×D4, C5×D4, C5×Q8, C5×Q8 [×2], C5×Q8, C22×D5, C22×D5, C2×C8.C22, C8×D5 [×2], C8⋊D5 [×2], C40⋊C2 [×2], Dic20 [×2], C4.Dic5, D4.D5 [×2], Q8⋊D5 [×2], C5⋊Q16 [×4], C5×M4(2), C5×SD16 [×2], C5×Q16 [×2], C2×Dic10, C2×Dic10, C2×C4×D5, C2×C4×D5 [×2], C4○D20, C4○D20, D4×D5, D4×D5, D42D5, D42D5, Q8×D5, Q8×D5 [×4], Q8×D5 [×2], Q82D5, Q8×C10, C5×C4○D4, D5×M4(2), C8.D10, D5×SD16 [×2], SD16⋊D5 [×2], D5×Q16 [×2], Q16⋊D5 [×2], C20.C23, D4.9D10, C5×C8.C22, C2×Q8×D5, D5×C4○D4, D5×C8.C22
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D5, C2×D4 [×6], C24, D10 [×7], C8.C22 [×2], C22×D4, C22×D5 [×7], C2×C8.C22, D4×D5 [×2], C23×D5, C2×D4×D5, D5×C8.C22

Smallest permutation representation of D5×C8.C22
On 80 points
Generators in S80
(1 75 62 35 11)(2 76 63 36 12)(3 77 64 37 13)(4 78 57 38 14)(5 79 58 39 15)(6 80 59 40 16)(7 73 60 33 9)(8 74 61 34 10)(17 26 69 52 44)(18 27 70 53 45)(19 28 71 54 46)(20 29 72 55 47)(21 30 65 56 48)(22 31 66 49 41)(23 32 67 50 42)(24 25 68 51 43)
(1 11)(2 12)(3 13)(4 14)(5 15)(6 16)(7 9)(8 10)(17 52)(18 53)(19 54)(20 55)(21 56)(22 49)(23 50)(24 51)(25 68)(26 69)(27 70)(28 71)(29 72)(30 65)(31 66)(32 67)(33 73)(34 74)(35 75)(36 76)(37 77)(38 78)(39 79)(40 80)
(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 4)(3 7)(6 8)(9 13)(10 16)(12 14)(17 23)(19 21)(20 24)(25 29)(26 32)(28 30)(33 37)(34 40)(36 38)(42 44)(43 47)(46 48)(50 52)(51 55)(54 56)(57 63)(59 61)(60 64)(65 71)(67 69)(68 72)(73 77)(74 80)(76 78)
(1 68)(2 65)(3 70)(4 67)(5 72)(6 69)(7 66)(8 71)(9 31)(10 28)(11 25)(12 30)(13 27)(14 32)(15 29)(16 26)(17 40)(18 37)(19 34)(20 39)(21 36)(22 33)(23 38)(24 35)(41 60)(42 57)(43 62)(44 59)(45 64)(46 61)(47 58)(48 63)(49 73)(50 78)(51 75)(52 80)(53 77)(54 74)(55 79)(56 76)

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

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

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

44 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B4C4D4E4F4G4H4I4J5A5B8A8B8C8D10A10B10C10D10E10F20A20B20C20D20E···20J40A40B40C40D
order1222222244444444445588881010101010102020202020···2040404040
size11245510202244410102020202244202022448844448···88888

44 irreducible representations

dim1111111111112222222224448
type+++++++++++++++++++++-++-
imageC1C2C2C2C2C2C2C2C2C2C2C2D4D4D4D5D10D10D10D10D10C8.C22D4×D5D4×D5D5×C8.C22
kernelD5×C8.C22D5×M4(2)C8.D10D5×SD16SD16⋊D5D5×Q16Q16⋊D5C20.C23D4.9D10C5×C8.C22C2×Q8×D5D5×C4○D4C4×D5C2×Dic5C22×D5C8.C22M4(2)SD16Q16C2×Q8C4○D4D5C4C22C1
# reps1112222111112112244222222

Matrix representation of D5×C8.C22 in GL8(𝔽41)

01000000
4034000000
00010000
0040340000
00001000
00000100
00000010
00000001
,
01000000
10000000
00010000
00100000
00001000
00000100
00000010
00000001
,
004000000
000400000
10000000
01000000
00002031394
0000926010
0000230021
00003631036
,
10000000
01000000
004000000
000400000
00001000
000054000
0000038402
0000283801
,
400000000
040000000
004000000
000400000
000039202120
000031163218
00002031394
00003617529

G:=sub<GL(8,GF(41))| [0,40,0,0,0,0,0,0,1,34,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,1,34,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,20,9,2,36,0,0,0,0,31,26,30,3,0,0,0,0,39,0,0,10,0,0,0,0,4,10,21,36],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,5,0,28,0,0,0,0,0,40,38,38,0,0,0,0,0,0,40,0,0,0,0,0,0,0,2,1],[40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,39,31,20,36,0,0,0,0,20,16,31,17,0,0,0,0,21,32,39,5,0,0,0,0,20,18,4,29] >;

D5×C8.C22 in GAP, Magma, Sage, TeX

D_5\times C_8.C_2^2
% in TeX

G:=Group("D5xC8.C2^2");
// GroupNames label

G:=SmallGroup(320,1448);
// by ID

G=gap.SmallGroup(320,1448);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,184,570,185,438,235,102,12550]);
// Polycyclic

G:=Group<a,b,c,d,e|a^5=b^2=c^8=d^2=e^2=1,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,d*c*d=c^3,e*c*e=c^5,e*d*e=c^4*d>;
// generators/relations

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