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G = SD16⋊D10order 320 = 26·5

2nd semidirect product of SD16 and D10 acting via D10/D5=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D84D10, SD162D10, C40.1C23, M4(2)⋊8D10, C20.20C24, Dic201C22, D20.13C23, Dic10.13C23, C8⋊C226D5, (C2×D4)⋊30D10, D83D51C2, D8⋊D52C2, D4⋊D56C22, (C4×D5).99D4, C4.190(D4×D5), (D4×D5)⋊9C22, (C5×D8)⋊2C22, (C8×D5)⋊2C22, C8.1(C22×D5), C4○D4.28D10, D10.88(C2×D4), SD16⋊D51C2, C20.241(C2×D4), C8.D101C2, C40⋊C22C22, C8⋊D52C22, (D5×M4(2))⋊2C2, D4.D55C22, (Q8×D5)⋊10C22, C5⋊Q163C22, C4.20(C23×D5), C22.47(D4×D5), SD163D51C2, D4.9D109C2, (D4×C10)⋊22C22, C53(D8⋊C22), C52C8.10C23, (C5×SD16)⋊2C22, (C22×D5).51D4, D4.13(C22×D5), (C5×D4).13C23, (C4×D5).66C23, D4.D1010C2, Q8.13(C22×D5), (C5×Q8).13C23, D42D510C22, (C2×C20).111C23, (C2×Dic5).251D4, Dic5.100(C2×D4), Q82D510C22, C4○D20.28C22, C10.121(C22×D4), (C5×M4(2))⋊2C22, C4.Dic513C22, (C2×Dic10)⋊39C22, C2.94(C2×D4×D5), (D5×C4○D4)⋊4C2, (C5×C8⋊C22)⋊2C2, (C2×C10).66(C2×D4), (C2×D42D5)⋊26C2, (C2×C4×D5).170C22, (C2×C4).95(C22×D5), (C5×C4○D4).24C22, SmallGroup(320,1445)

Series: Derived Chief Lower central Upper central

C1C20 — SD16⋊D10
C1C5C10C20C4×D5C2×C4×D5D5×C4○D4 — SD16⋊D10
C5C10C20 — SD16⋊D10
C1C2C2×C4C8⋊C22

Generators and relations for SD16⋊D10
 G = < a,b,c,d | a8=b2=c10=d2=1, bab=dad=a3, cac-1=a-1, cbc-1=a6b, dbd=a2b, dcd=c-1 >

Subgroups: 974 in 262 conjugacy classes, 99 normal (51 characteristic)
C1, C2, C2 [×7], C4 [×2], C4 [×6], C22, C22 [×11], C5, C8 [×2], C8 [×2], C2×C4, C2×C4 [×15], D4, D4 [×2], D4 [×11], Q8, Q8 [×5], C23 [×3], D5 [×3], C10, C10 [×4], C2×C8 [×2], M4(2), M4(2) [×3], D8 [×2], D8 [×2], SD16 [×2], SD16 [×6], Q16 [×4], C22×C4 [×3], C2×D4, C2×D4 [×3], C2×Q8 [×2], C4○D4, C4○D4 [×11], Dic5 [×2], Dic5 [×3], C20 [×2], C20, D10 [×2], D10 [×4], C2×C10, C2×C10 [×5], C2×M4(2), C4○D8 [×4], C8⋊C22, C8⋊C22 [×3], C8.C22 [×4], C2×C4○D4 [×2], C52C8 [×2], C40 [×2], Dic10, Dic10 [×2], Dic10 [×2], C4×D5 [×4], C4×D5 [×3], D20, D20, C2×Dic5, C2×Dic5 [×6], C5⋊D4 [×7], C2×C20, C2×C20, C5×D4, C5×D4 [×2], C5×D4 [×2], C5×Q8, C22×D5, C22×D5, C22×C10, D8⋊C22, C8×D5 [×2], C8⋊D5 [×2], C40⋊C2 [×2], Dic20 [×2], C4.Dic5, D4⋊D5 [×2], D4.D5 [×4], C5⋊Q16 [×2], C5×M4(2), C5×D8 [×2], C5×SD16 [×2], C2×Dic10, C2×C4×D5, C2×C4×D5, C4○D20, C4○D20, D4×D5, D4×D5, D42D5, D42D5 [×4], D42D5 [×3], Q8×D5, Q82D5, C22×Dic5, C2×C5⋊D4, D4×C10, C5×C4○D4, D5×M4(2), C8.D10, D8⋊D5 [×2], D83D5 [×2], SD16⋊D5 [×2], SD163D5 [×2], D4.D10, D4.9D10, C5×C8⋊C22, C2×D42D5, D5×C4○D4, SD16⋊D10
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D5, C2×D4 [×6], C24, D10 [×7], C22×D4, C22×D5 [×7], D8⋊C22, D4×D5 [×2], C23×D5, C2×D4×D5, SD16⋊D10

Smallest permutation representation of SD16⋊D10
On 80 points
Generators in S80
(1 62 50 76 16 71 45 67)(2 68 46 72 17 77 41 63)(3 64 42 78 18 73 47 69)(4 70 48 74 19 79 43 65)(5 66 44 80 20 75 49 61)(6 59 39 24 15 29 34 54)(7 55 35 30 11 25 40 60)(8 51 31 26 12 21 36 56)(9 57 37 22 13 27 32 52)(10 53 33 28 14 23 38 58)
(1 28)(2 24)(3 30)(4 26)(5 22)(6 63)(7 69)(8 65)(9 61)(10 67)(11 78)(12 74)(13 80)(14 76)(15 72)(16 58)(17 54)(18 60)(19 56)(20 52)(21 43)(23 45)(25 47)(27 49)(29 41)(31 79)(32 66)(33 71)(34 68)(35 73)(36 70)(37 75)(38 62)(39 77)(40 64)(42 55)(44 57)(46 59)(48 51)(50 53)
(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 5)(3 4)(6 9)(7 8)(11 12)(13 15)(17 20)(18 19)(21 60)(22 59)(23 58)(24 57)(25 56)(26 55)(27 54)(28 53)(29 52)(30 51)(31 40)(32 39)(33 38)(34 37)(35 36)(41 44)(42 43)(45 50)(46 49)(47 48)(61 77)(62 76)(63 75)(64 74)(65 73)(66 72)(67 71)(68 80)(69 79)(70 78)

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

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

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

44 conjugacy classes

class 1 2A2B2C2D2E2F2G2H4A4B4C4D4E4F4G4H4I5A5B8A8B8C8D10A10B10C10D10E···10J20A20B20C20D20E20F40A40B40C40D
order1222222224444444445588881010101010···1020202020202040404040
size11244410102022455102020202244202022448···84444888888

44 irreducible representations

dim1111111111112222222224448
type+++++++++++++++++++++++-
imageC1C2C2C2C2C2C2C2C2C2C2C2D4D4D4D5D10D10D10D10D10D8⋊C22D4×D5D4×D5SD16⋊D10
kernelSD16⋊D10D5×M4(2)C8.D10D8⋊D5D83D5SD16⋊D5SD163D5D4.D10D4.9D10C5×C8⋊C22C2×D42D5D5×C4○D4C4×D5C2×Dic5C22×D5C8⋊C22M4(2)D8SD16C2×D4C4○D4C5C4C22C1
# reps1112222111112112244222222

Matrix representation of SD16⋊D10 in GL8(𝔽41)

001400000
71270000
3184000000
4184000000
00000001
00000010
00001000
000004000
,
001400000
71270000
004000000
4004000000
00000009
000000320
00000900
000032000
,
07000000
356000000
8634340000
1613710000
00001000
000004000
00000001
00000010
,
357000000
366000000
8634340000
1613170000
00001000
000004000
000000040
000000400

G:=sub<GL(8,GF(41))| [0,7,3,4,0,0,0,0,0,1,18,18,0,0,0,0,1,2,40,40,0,0,0,0,40,7,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0],[0,7,0,40,0,0,0,0,0,1,0,0,0,0,0,0,1,2,40,40,0,0,0,0,40,7,0,0,0,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,9,0,0,0,0,0,0,32,0,0,0,0,0,0,9,0,0,0],[0,35,8,16,0,0,0,0,7,6,6,13,0,0,0,0,0,0,34,7,0,0,0,0,0,0,34,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0],[35,36,8,16,0,0,0,0,7,6,6,13,0,0,0,0,0,0,34,1,0,0,0,0,0,0,34,7,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0] >;

SD16⋊D10 in GAP, Magma, Sage, TeX

{\rm SD}_{16}\rtimes D_{10}
% in TeX

G:=Group("SD16:D10");
// GroupNames label

G:=SmallGroup(320,1445);
// by ID

G=gap.SmallGroup(320,1445);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,387,1123,185,438,235,102,12550]);
// Polycyclic

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

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