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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, C4, C4, C22, C22, C5, C8, C8, C2×C4, C2×C4, D4, D4, D4, Q8, Q8, C23, D5, C10, C10, C2×C8, M4(2), M4(2), D8, D8, SD16, SD16, Q16, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C4○D4, Dic5, Dic5, C20, C20, D10, D10, C2×C10, C2×C10, C2×M4(2), C4○D8, C8⋊C22, C8⋊C22, C8.C22, C2×C4○D4, C52C8, C40, Dic10, Dic10, Dic10, C4×D5, C4×D5, D20, D20, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C5×D4, C5×D4, C5×D4, C5×Q8, C22×D5, C22×D5, C22×C10, D8⋊C22, C8×D5, C8⋊D5, C40⋊C2, Dic20, C4.Dic5, D4⋊D5, D4.D5, C5⋊Q16, C5×M4(2), C5×D8, C5×SD16, C2×Dic10, C2×C4×D5, C2×C4×D5, C4○D20, C4○D20, D4×D5, D4×D5, D42D5, D42D5, D42D5, Q8×D5, Q82D5, C22×Dic5, C2×C5⋊D4, D4×C10, C5×C4○D4, D5×M4(2), C8.D10, D8⋊D5, D83D5, SD16⋊D5, SD163D5, D4.D10, D4.9D10, C5×C8⋊C22, C2×D42D5, D5×C4○D4, SD16⋊D10
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C24, D10, C22×D4, C22×D5, D8⋊C22, D4×D5, C23×D5, C2×D4×D5, SD16⋊D10

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

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

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

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

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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