Copied to
clipboard

G = D86D10order 320 = 26·5

6th semidirect product of D8 and D10 acting via D10/D5=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D86D10, SD164D10, D20.42D4, C40.3C23, C20.22C24, M4(2)⋊10D10, Dic10.42D4, Dic202C22, D20.15C23, Dic10.15C23, C4○D44D10, C8⋊C225D5, C5⋊D4.5D4, D83D52C2, D8⋊D54C2, (D5×SD16)⋊2C2, C4.116(D4×D5), C54(D4○SD16), (C8×D5)⋊4C22, (C5×D8)⋊4C22, D46D108C2, (Q8×D5)⋊3C22, C8.3(C22×D5), D4⋊D515C22, D10.56(C2×D4), C8.D102C2, C20.243(C2×D4), SD16⋊D52C2, C8⋊D54C22, C40⋊C24C22, Q8⋊D514C22, (D4×D5).3C22, C22.15(D4×D5), C4.22(C23×D5), D20.2C42C2, D4.8D104C2, (C2×D4).117D10, D42D54C22, C52C8.26C23, D4.D514C22, Dic5.62(C2×D4), (C5×SD16)⋊4C22, (C4×D5).14C23, D4.15(C22×D5), (C5×D4).15C23, C5⋊Q1613C22, D4.10D107C2, Q8.15(C22×D5), (C5×Q8).15C23, (C2×C20).113C23, C4○D20.29C22, C10.123(C22×D4), (C5×M4(2))⋊4C22, (C2×Dic10)⋊40C22, (D4×C10).168C22, C2.96(C2×D4×D5), (C5×C8⋊C22)⋊4C2, (C2×D4.D5)⋊29C2, (C2×C10).68(C2×D4), (C5×C4○D4)⋊7C22, (C2×C52C8)⋊18C22, (C2×C4).97(C22×D5), SmallGroup(320,1447)

Series: Derived Chief Lower central Upper central

C1C20 — D86D10
C1C5C10C20C4×D5C4○D20D46D10 — D86D10
C5C10C20 — D86D10
C1C2C2×C4C8⋊C22

Generators and relations for D86D10
 G = < a,b,c,d | a8=b2=c10=d2=1, bab=a-1, cac-1=dad=a3, bc=cb, dbd=a4b, dcd=c-1 >

Subgroups: 998 in 258 conjugacy classes, 99 normal (45 characteristic)
C1, C2, C2 [×7], C4 [×2], C4 [×6], C22, C22 [×9], C5, C8 [×2], C8 [×2], C2×C4, C2×C4 [×11], D4, D4 [×2], D4 [×13], Q8, Q8 [×7], C23 [×3], D5 [×3], C10, C10 [×4], C2×C8 [×3], M4(2), M4(2) [×2], D8 [×2], D8, SD16 [×2], SD16 [×8], Q16 [×3], C2×D4, C2×D4 [×5], C2×Q8 [×4], C4○D4, C4○D4 [×10], Dic5 [×2], Dic5 [×3], C20 [×2], C20, D10 [×2], D10 [×3], C2×C10, C2×C10 [×4], C8○D4, C2×SD16 [×3], C4○D8 [×3], C8⋊C22, C8⋊C22 [×2], C8.C22 [×3], 2+ 1+4, 2- 1+4, C52C8 [×2], C40 [×2], Dic10 [×2], Dic10 [×2], Dic10 [×3], C4×D5 [×2], C4×D5 [×3], D20 [×2], C2×Dic5 [×5], C5⋊D4 [×2], C5⋊D4 [×7], C2×C20, C2×C20, C5×D4, C5×D4 [×2], C5×D4 [×2], C5×Q8, C22×D5 [×2], C22×C10, D4○SD16, C8×D5 [×2], C8⋊D5 [×2], C40⋊C2 [×2], Dic20 [×2], C2×C52C8, D4⋊D5, D4.D5, D4.D5 [×4], Q8⋊D5, C5⋊Q16, C5×M4(2), C5×D8 [×2], C5×SD16 [×2], C2×Dic10, C2×Dic10, C4○D20 [×2], C4○D20, D4×D5 [×2], D4×D5, D42D5 [×4], D42D5 [×3], Q8×D5 [×2], C2×C5⋊D4 [×2], D4×C10, C5×C4○D4, D20.2C4, C8.D10, D8⋊D5 [×2], D83D5 [×2], D5×SD16 [×2], SD16⋊D5 [×2], C2×D4.D5, D4.8D10, C5×C8⋊C22, D46D10, D4.10D10, D86D10
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D5, C2×D4 [×6], C24, D10 [×7], C22×D4, C22×D5 [×7], D4○SD16, D4×D5 [×2], C23×D5, C2×D4×D5, D86D10

Smallest permutation representation of D86D10
On 80 points
Generators in S80
(1 41 40 78 63 20 59 21)(2 79 60 42 64 22 31 11)(3 43 32 80 65 12 51 23)(4 71 52 44 66 24 33 13)(5 45 34 72 67 14 53 25)(6 73 54 46 68 26 35 15)(7 47 36 74 69 16 55 27)(8 75 56 48 70 28 37 17)(9 49 38 76 61 18 57 29)(10 77 58 50 62 30 39 19)
(11 79)(12 80)(13 71)(14 72)(15 73)(16 74)(17 75)(18 76)(19 77)(20 78)(21 41)(22 42)(23 43)(24 44)(25 45)(26 46)(27 47)(28 48)(29 49)(30 50)(31 60)(32 51)(33 52)(34 53)(35 54)(36 55)(37 56)(38 57)(39 58)(40 59)
(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 53)(2 52)(3 51)(4 60)(5 59)(6 58)(7 57)(8 56)(9 55)(10 54)(11 13)(14 20)(15 19)(16 18)(21 72)(22 71)(23 80)(24 79)(25 78)(26 77)(27 76)(28 75)(29 74)(30 73)(31 66)(32 65)(33 64)(34 63)(35 62)(36 61)(37 70)(38 69)(39 68)(40 67)(41 45)(42 44)(46 50)(47 49)

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

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

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

44 conjugacy classes

class 1 2A2B2C2D2E2F2G2H4A4B4C4D4E4F4G4H5A5B8A8B8C8D8E10A10B10C10D10E···10J20A20B20C20D20E20F40A40B40C40D
order1222222224444444455888881010101010···1020202020202040404040
size1124441010202241010202020224410102022448···84444888888

44 irreducible representations

dim1111111111112222222224448
type+++++++++++++++++++++++-
imageC1C2C2C2C2C2C2C2C2C2C2C2D4D4D4D5D10D10D10D10D10D4○SD16D4×D5D4×D5D86D10
kernelD86D10D20.2C4C8.D10D8⋊D5D83D5D5×SD16SD16⋊D5C2×D4.D5D4.8D10C5×C8⋊C22D46D10D4.10D10Dic10D20C5⋊D4C8⋊C22M4(2)D8SD16C2×D4C4○D4C5C4C22C1
# reps1112222111111122244222222

Matrix representation of D86D10 in GL8(𝔽41)

1013280000
011300000
034000000
3830400000
00001526260
000015261526
0000030015
0000113000
,
10000000
01000000
034000000
3830400000
00001001
000004000
000000140
000000040
,
16000000
356000000
23180350000
2007340000
00000110
00000100
000014000
0000039040
,
10000000
3540000000
23387350000
20388340000
00000110
00001010
000000400
000000391

G:=sub<GL(8,GF(41))| [1,0,0,38,0,0,0,0,0,1,3,3,0,0,0,0,13,13,40,0,0,0,0,0,28,0,0,40,0,0,0,0,0,0,0,0,15,15,0,11,0,0,0,0,26,26,30,30,0,0,0,0,26,15,0,0,0,0,0,0,0,26,15,0],[1,0,0,38,0,0,0,0,0,1,3,3,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,40,40],[1,35,23,20,0,0,0,0,6,6,18,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,35,34,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,1,40,39,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,40],[1,35,23,20,0,0,0,0,0,40,38,38,0,0,0,0,0,0,7,8,0,0,0,0,0,0,35,34,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,1,40,39,0,0,0,0,0,0,0,1] >;

D86D10 in GAP, Magma, Sage, TeX

D_8\rtimes_6D_{10}
% in TeX

G:=Group("D8:6D10");
// GroupNames label

G:=SmallGroup(320,1447);
// by ID

G=gap.SmallGroup(320,1447);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,477,570,185,136,438,235,102,12550]);
// 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=d*a*d=a^3,b*c=c*b,d*b*d=a^4*b,d*c*d=c^-1>;
// generators/relations

׿
×
𝔽