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## G = D10⋊6SD16order 320 = 26·5

### 2nd semidirect product of D10 and SD16 acting via SD16/D4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C20 — D10⋊6SD16
 Chief series C1 — C5 — C10 — C2×C10 — C2×C20 — C2×C4×D5 — C2×D4×D5 — D10⋊6SD16
 Lower central C5 — C10 — C2×C20 — D10⋊6SD16
 Upper central C1 — C22 — C2×C4 — C2×SD16

Generators and relations for D106SD16
G = < a,b,c,d | a10=b2=c8=d2=1, bab=a-1, ac=ca, ad=da, cbc-1=a5b, bd=db, dcd=c3 >

Subgroups: 990 in 188 conjugacy classes, 45 normal (37 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, D4, Q8, C23, D5, C10, C10, C22⋊C4, C4⋊C4, C2×C8, C2×C8, SD16, C22×C4, C2×D4, C2×D4, C2×Q8, C24, Dic5, C20, C20, D10, D10, C2×C10, C2×C10, C22⋊C8, D4⋊C4, C22⋊Q8, C2×SD16, C2×SD16, C22×D4, C52C8, C40, C4×D5, D20, D20, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C5×D4, C5×D4, C5×Q8, C22×D5, C22×D5, C22×C10, C22⋊SD16, C2×C52C8, C10.D4, C4⋊Dic5, D10⋊C4, Q8⋊D5, C2×C40, C5×SD16, C2×C4×D5, C2×D20, D4×D5, C2×C5⋊D4, D4×C10, Q8×C10, C23×D5, D101C8, D205C4, D4⋊Dic5, C2×Q8⋊D5, D103Q8, C10×SD16, C2×D4×D5, D106SD16
Quotients: C1, C2, C22, D4, C23, D5, SD16, C2×D4, D10, C22≀C2, C2×SD16, C8⋊C22, C5⋊D4, C22×D5, C22⋊SD16, D4×D5, C2×C5⋊D4, D5×SD16, D40⋊C2, C23⋊D10, D106SD16

Smallest permutation representation of D106SD16
On 80 points
Generators in S80
(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 10)(2 9)(3 8)(4 7)(5 6)(11 12)(13 20)(14 19)(15 18)(16 17)(21 25)(22 24)(26 30)(27 29)(31 34)(32 33)(35 40)(36 39)(37 38)(41 48)(42 47)(43 46)(44 45)(49 50)(51 57)(52 56)(53 55)(58 60)(61 63)(64 70)(65 69)(66 68)(71 79)(72 78)(73 77)(74 76)
(1 26 12 73 33 70 50 52)(2 27 13 74 34 61 41 53)(3 28 14 75 35 62 42 54)(4 29 15 76 36 63 43 55)(5 30 16 77 37 64 44 56)(6 21 17 78 38 65 45 57)(7 22 18 79 39 66 46 58)(8 23 19 80 40 67 47 59)(9 24 20 71 31 68 48 60)(10 25 11 72 32 69 49 51)
(11 49)(12 50)(13 41)(14 42)(15 43)(16 44)(17 45)(18 46)(19 47)(20 48)(21 78)(22 79)(23 80)(24 71)(25 72)(26 73)(27 74)(28 75)(29 76)(30 77)(51 69)(52 70)(53 61)(54 62)(55 63)(56 64)(57 65)(58 66)(59 67)(60 68)

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

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

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

47 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 4A 4B 4C 4D 4E 5A 5B 8A 8B 8C 8D 10A ··· 10F 10G 10H 10I 10J 20A 20B 20C 20D 20E 20F 20G 20H 40A ··· 40H order 1 2 2 2 2 2 2 2 2 2 4 4 4 4 4 5 5 8 8 8 8 10 ··· 10 10 10 10 10 20 20 20 20 20 20 20 20 40 ··· 40 size 1 1 1 1 4 4 10 10 20 20 2 2 8 20 40 2 2 4 4 20 20 2 ··· 2 8 8 8 8 4 4 4 4 8 8 8 8 4 ··· 4

47 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 type + + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 D4 D4 D4 D4 D5 SD16 D10 D10 D10 C5⋊D4 C8⋊C22 D4×D5 D4×D5 D5×SD16 D40⋊C2 kernel D10⋊6SD16 D10⋊1C8 D20⋊5C4 D4⋊Dic5 C2×Q8⋊D5 D10⋊3Q8 C10×SD16 C2×D4×D5 D20 C2×Dic5 C5×D4 C22×D5 C2×SD16 D10 C2×C8 C2×D4 C2×Q8 D4 C10 C4 C22 C2 C2 # reps 1 1 1 1 1 1 1 1 2 1 2 1 2 4 2 2 2 8 1 2 2 4 4

Matrix representation of D106SD16 in GL6(𝔽41)

 40 7 0 0 0 0 34 7 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 40 0 0 0 0 0 34 1 0 0 0 0 0 0 40 0 0 0 0 0 24 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 40 0 0 0 0 0 0 40 0 0 0 0 0 0 12 1 0 0 0 0 19 29 0 0 0 0 0 0 15 15 0 0 0 0 26 15
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 17 40 0 0 0 0 0 0 1 0 0 0 0 0 0 40

G:=sub<GL(6,GF(41))| [40,34,0,0,0,0,7,7,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[40,34,0,0,0,0,0,1,0,0,0,0,0,0,40,24,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,12,19,0,0,0,0,1,29,0,0,0,0,0,0,15,26,0,0,0,0,15,15],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,17,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,40] >;

D106SD16 in GAP, Magma, Sage, TeX

D_{10}\rtimes_6{\rm SD}_{16}
% in TeX

G:=Group("D10:6SD16");
// GroupNames label

G:=SmallGroup(320,796);
// by ID

G=gap.SmallGroup(320,796);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,254,219,184,851,438,102,12550]);
// Polycyclic

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

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