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G = SD16⋊D10order 320 = 26·5

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — SD16⋊D10
 Chief series C1 — C5 — C10 — C20 — C4×D5 — C2×C4×D5 — D5×C4○D4 — SD16⋊D10
 Lower central C5 — C10 — C20 — SD16⋊D10
 Upper central C1 — C2 — C2×C4 — C8⋊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 2A 2B 2C 2D 2E 2F 2G 2H 4A 4B 4C 4D 4E 4F 4G 4H 4I 5A 5B 8A 8B 8C 8D 10A 10B 10C 10D 10E ··· 10J 20A 20B 20C 20D 20E 20F 40A 40B 40C 40D order 1 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 4 5 5 8 8 8 8 10 10 10 10 10 ··· 10 20 20 20 20 20 20 40 40 40 40 size 1 1 2 4 4 4 10 10 20 2 2 4 5 5 10 20 20 20 2 2 4 4 20 20 2 2 4 4 8 ··· 8 4 4 4 4 8 8 8 8 8 8

44 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 8 type + + + + + + + + + + + + + + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 D4 D4 D4 D5 D10 D10 D10 D10 D10 D8⋊C22 D4×D5 D4×D5 SD16⋊D10 kernel SD16⋊D10 D5×M4(2) C8.D10 D8⋊D5 D8⋊3D5 SD16⋊D5 SD16⋊3D5 D4.D10 D4.9D10 C5×C8⋊C22 C2×D4⋊2D5 D5×C4○D4 C4×D5 C2×Dic5 C22×D5 C8⋊C22 M4(2) D8 SD16 C2×D4 C4○D4 C5 C4 C22 C1 # reps 1 1 1 2 2 2 2 1 1 1 1 1 2 1 1 2 2 4 4 2 2 2 2 2 2

Matrix representation of SD16⋊D10 in GL8(𝔽41)

 0 0 1 40 0 0 0 0 7 1 2 7 0 0 0 0 3 18 40 0 0 0 0 0 4 18 40 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 40 0 0
,
 0 0 1 40 0 0 0 0 7 1 2 7 0 0 0 0 0 0 40 0 0 0 0 0 40 0 40 0 0 0 0 0 0 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 0 0 32 0 0 0
,
 0 7 0 0 0 0 0 0 35 6 0 0 0 0 0 0 8 6 34 34 0 0 0 0 16 13 7 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 7 0 0 0 0 0 0 36 6 0 0 0 0 0 0 8 6 34 34 0 0 0 0 16 13 1 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

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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