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G = D85D10order 320 = 26·5

5th semidirect product of D8 and D10 acting via D10/D5=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D85D10, SD163D10, D20.41D4, D402C22, C40.2C23, M4(2)⋊9D10, C20.21C24, Dic10.41D4, D20.14C23, Dic10.14C23, (D5×D8)⋊2C2, C54(D4○D8), C4○D43D10, C8⋊C224D5, C5⋊D4.4D4, D8⋊D53C2, (C2×D4)⋊15D10, D40⋊C22C2, C8⋊D102C2, C4.115(D4×D5), (C8×D5)⋊3C22, (C5×D8)⋊3C22, (D4×D5)⋊3C22, D48D107C2, D46D107C2, C8.2(C22×D5), D4⋊D514C22, D10.55(C2×D4), C20.242(C2×D4), C4○D208C22, C40⋊C23C22, C8⋊D53C22, Q8⋊D513C22, C4.21(C23×D5), C22.14(D4×D5), SD163D52C2, D4.8D103C2, D20.2C41C2, (C2×D20)⋊36C22, (D4×C10)⋊23C22, D42D53C22, C52C8.25C23, D4.D513C22, Dic5.61(C2×D4), Q82D53C22, (C5×SD16)⋊3C22, (C5×D4).14C23, D4.14(C22×D5), C5⋊Q1612C22, (C4×D5).13C23, Q8.14(C22×D5), (C5×Q8).14C23, (C2×C20).112C23, C10.122(C22×D4), (C5×M4(2))⋊3C22, C2.95(C2×D4×D5), (C2×D4⋊D5)⋊29C2, (C5×C8⋊C22)⋊3C2, (C2×C10).67(C2×D4), (C5×C4○D4)⋊6C22, (C2×C52C8)⋊17C22, (C2×C4).96(C22×D5), SmallGroup(320,1446)

Series: Derived Chief Lower central Upper central

C1C20 — D85D10
C1C5C10C20C4×D5C4○D20D46D10 — D85D10
C5C10C20 — D85D10
C1C2C2×C4C8⋊C22

Generators and relations for D85D10
 G = < a,b,c,d | a8=b2=c10=d2=1, bab=a-1, cac-1=a5, dad=a3, bc=cb, dbd=a6b, dcd=c-1 >

Subgroups: 1190 in 268 conjugacy classes, 99 normal (45 characteristic)
C1, C2, C2 [×9], C4 [×2], C4 [×4], C22, C22 [×14], C5, C8 [×2], C8 [×2], C2×C4, C2×C4 [×8], D4, D4 [×2], D4 [×18], Q8, Q8 [×2], C23 [×6], D5 [×5], C10, C10 [×4], C2×C8 [×3], M4(2), M4(2) [×2], D8 [×2], D8 [×7], SD16 [×2], SD16 [×4], Q16, C2×D4, C2×D4 [×11], C4○D4, C4○D4 [×8], Dic5 [×2], Dic5, C20 [×2], C20, D10 [×2], D10 [×8], C2×C10, C2×C10 [×4], C8○D4, C2×D8 [×3], C4○D8 [×3], C8⋊C22, C8⋊C22 [×5], 2+ 1+4 [×2], C52C8 [×2], C40 [×2], Dic10 [×2], C4×D5 [×2], C4×D5 [×3], D20 [×2], D20 [×2], D20 [×3], C2×Dic5 [×2], C5⋊D4 [×2], C5⋊D4 [×7], C2×C20, C2×C20, C5×D4, C5×D4 [×2], C5×D4 [×2], C5×Q8, C22×D5 [×5], C22×C10, D4○D8, C8×D5 [×2], C8⋊D5 [×2], C40⋊C2 [×2], D40 [×2], C2×C52C8, D4⋊D5, D4⋊D5 [×4], D4.D5, Q8⋊D5, C5⋊Q16, C5×M4(2), C5×D8 [×2], C5×SD16 [×2], C2×D20, C2×D20, C4○D20 [×2], C4○D20, D4×D5 [×4], D4×D5 [×3], D42D5 [×2], D42D5, Q82D5 [×2], C2×C5⋊D4 [×2], D4×C10, C5×C4○D4, D20.2C4, C8⋊D10, D5×D8 [×2], D8⋊D5 [×2], D40⋊C2 [×2], SD163D5 [×2], C2×D4⋊D5, D4.8D10, C5×C8⋊C22, D46D10, D48D10, D85D10
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D5, C2×D4 [×6], C24, D10 [×7], C22×D4, C22×D5 [×7], D4○D8, D4×D5 [×2], C23×D5, C2×D4×D5, D85D10

Smallest permutation representation of D85D10
On 80 points
Generators in S80
(1 24 57 46 63 73 40 11)(2 74 58 12 64 25 31 47)(3 26 59 48 65 75 32 13)(4 76 60 14 66 27 33 49)(5 28 51 50 67 77 34 15)(6 78 52 16 68 29 35 41)(7 30 53 42 69 79 36 17)(8 80 54 18 70 21 37 43)(9 22 55 44 61 71 38 19)(10 72 56 20 62 23 39 45)
(11 24)(12 25)(13 26)(14 27)(15 28)(16 29)(17 30)(18 21)(19 22)(20 23)(31 58)(32 59)(33 60)(34 51)(35 52)(36 53)(37 54)(38 55)(39 56)(40 57)(41 78)(42 79)(43 80)(44 71)(45 72)(46 73)(47 74)(48 75)(49 76)(50 77)
(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 23)(2 22)(3 21)(4 30)(5 29)(6 28)(7 27)(8 26)(9 25)(10 24)(11 56)(12 55)(13 54)(14 53)(15 52)(16 51)(17 60)(18 59)(19 58)(20 57)(31 44)(32 43)(33 42)(34 41)(35 50)(36 49)(37 48)(38 47)(39 46)(40 45)(61 74)(62 73)(63 72)(64 71)(65 80)(66 79)(67 78)(68 77)(69 76)(70 75)

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

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

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

44 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I2J4A4B4C4D4E4F5A5B8A8B8C8D8E10A10B10C10D10E···10J20A20B20C20D20E20F40A40B40C40D
order1222222222244444455888881010101010···1020202020202040404040
size1124441010202020224101020224410102022448···84444888888

44 irreducible representations

dim1111111111112222222224448
type+++++++++++++++++++++++++
imageC1C2C2C2C2C2C2C2C2C2C2C2D4D4D4D5D10D10D10D10D10D4○D8D4×D5D4×D5D85D10
kernelD85D10D20.2C4C8⋊D10D5×D8D8⋊D5D40⋊C2SD163D5C2×D4⋊D5D4.8D10C5×C8⋊C22D46D10D48D10Dic10D20C5⋊D4C8⋊C22M4(2)D8SD16C2×D4C4○D4C5C4C22C1
# reps1112222111111122244222222

Matrix representation of D85D10 in GL8(𝔽41)

004000000
000400000
10000000
01000000
0000292900
0000122900
00001821034
0000335624
,
10000000
01000000
004000000
000400000
00000100
00001000
000025254037
00000001
,
66000000
351000000
00660000
003510000
000073790
00001627325
00002063025
000011018
,
00660000
001350000
66000000
135000000
00001111022
00002633019
00002937275
00003034260

G:=sub<GL(8,GF(41))| [0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,29,12,18,3,0,0,0,0,29,29,21,35,0,0,0,0,0,0,0,6,0,0,0,0,0,0,34,24],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,1,25,0,0,0,0,0,1,0,25,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,37,1],[6,35,0,0,0,0,0,0,6,1,0,0,0,0,0,0,0,0,6,35,0,0,0,0,0,0,6,1,0,0,0,0,0,0,0,0,7,16,20,1,0,0,0,0,37,27,6,1,0,0,0,0,9,32,30,0,0,0,0,0,0,5,25,18],[0,0,6,1,0,0,0,0,0,0,6,35,0,0,0,0,6,1,0,0,0,0,0,0,6,35,0,0,0,0,0,0,0,0,0,0,11,26,29,30,0,0,0,0,11,3,37,34,0,0,0,0,0,30,27,26,0,0,0,0,22,19,5,0] >;

D85D10 in GAP, Magma, Sage, TeX

D_8\rtimes_5D_{10}
% in TeX

G:=Group("D8:5D10");
// GroupNames label

G:=SmallGroup(320,1446);
// by ID

G=gap.SmallGroup(320,1446);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,477,570,185,438,235,102,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^8=b^2=c^10=d^2=1,b*a*b=a^-1,c*a*c^-1=a^5,d*a*d=a^3,b*c=c*b,d*b*d=a^6*b,d*c*d=c^-1>;
// generators/relations

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