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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 [×3], C2 [×6], C4 [×2], C4 [×3], C22, C22 [×20], C5, C8 [×2], C2×C4, C2×C4 [×5], D4 [×2], D4 [×8], Q8 [×2], C23 [×11], D5 [×4], C10 [×3], C10 [×2], C22⋊C4, C4⋊C4 [×2], C2×C8, C2×C8, SD16 [×4], C22×C4, C2×D4, C2×D4 [×6], C2×Q8, C24, Dic5 [×2], C20 [×2], C20, D10 [×2], D10 [×14], C2×C10, C2×C10 [×4], C22⋊C8, D4⋊C4 [×2], C22⋊Q8, C2×SD16, C2×SD16, C22×D4, C52C8, C40, C4×D5 [×2], D20 [×2], D20, C2×Dic5, C2×Dic5, C5⋊D4 [×4], C2×C20, C2×C20, C5×D4 [×2], C5×D4, C5×Q8 [×2], C22×D5, C22×D5 [×9], C22×C10, C22⋊SD16, C2×C52C8, C10.D4, C4⋊Dic5, D10⋊C4, Q8⋊D5 [×2], C2×C40, C5×SD16 [×2], C2×C4×D5, C2×D20, D4×D5 [×4], 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 [×7], C22 [×7], D4 [×6], C23, D5, SD16 [×2], C2×D4 [×3], D10 [×3], C22≀C2, C2×SD16, C8⋊C22, C5⋊D4 [×2], C22×D5, C22⋊SD16, D4×D5 [×2], 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 14)(12 13)(15 20)(16 19)(17 18)(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)(52 60)(53 59)(54 58)(55 57)(61 67)(62 66)(63 65)(68 70)(71 79)(72 78)(73 77)(74 76)
(1 26 18 73 33 62 50 54)(2 27 19 74 34 63 41 55)(3 28 20 75 35 64 42 56)(4 29 11 76 36 65 43 57)(5 30 12 77 37 66 44 58)(6 21 13 78 38 67 45 59)(7 22 14 79 39 68 46 60)(8 23 15 80 40 69 47 51)(9 24 16 71 31 70 48 52)(10 25 17 72 32 61 49 53)
(11 43)(12 44)(13 45)(14 46)(15 47)(16 48)(17 49)(18 50)(19 41)(20 42)(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)(11,14)(12,13)(15,20)(16,19)(17,18)(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)(52,60)(53,59)(54,58)(55,57)(61,67)(62,66)(63,65)(68,70)(71,79)(72,78)(73,77)(74,76), (1,26,18,73,33,62,50,54)(2,27,19,74,34,63,41,55)(3,28,20,75,35,64,42,56)(4,29,11,76,36,65,43,57)(5,30,12,77,37,66,44,58)(6,21,13,78,38,67,45,59)(7,22,14,79,39,68,46,60)(8,23,15,80,40,69,47,51)(9,24,16,71,31,70,48,52)(10,25,17,72,32,61,49,53), (11,43)(12,44)(13,45)(14,46)(15,47)(16,48)(17,49)(18,50)(19,41)(20,42)(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)(11,14)(12,13)(15,20)(16,19)(17,18)(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)(52,60)(53,59)(54,58)(55,57)(61,67)(62,66)(63,65)(68,70)(71,79)(72,78)(73,77)(74,76), (1,26,18,73,33,62,50,54)(2,27,19,74,34,63,41,55)(3,28,20,75,35,64,42,56)(4,29,11,76,36,65,43,57)(5,30,12,77,37,66,44,58)(6,21,13,78,38,67,45,59)(7,22,14,79,39,68,46,60)(8,23,15,80,40,69,47,51)(9,24,16,71,31,70,48,52)(10,25,17,72,32,61,49,53), (11,43)(12,44)(13,45)(14,46)(15,47)(16,48)(17,49)(18,50)(19,41)(20,42)(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),(11,14),(12,13),(15,20),(16,19),(17,18),(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),(52,60),(53,59),(54,58),(55,57),(61,67),(62,66),(63,65),(68,70),(71,79),(72,78),(73,77),(74,76)], [(1,26,18,73,33,62,50,54),(2,27,19,74,34,63,41,55),(3,28,20,75,35,64,42,56),(4,29,11,76,36,65,43,57),(5,30,12,77,37,66,44,58),(6,21,13,78,38,67,45,59),(7,22,14,79,39,68,46,60),(8,23,15,80,40,69,47,51),(9,24,16,71,31,70,48,52),(10,25,17,72,32,61,49,53)], [(11,43),(12,44),(13,45),(14,46),(15,47),(16,48),(17,49),(18,50),(19,41),(20,42),(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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