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G = D85Dic5order 320 = 26·5

The semidirect product of D8 and Dic5 acting through Inn(D8)

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D85Dic5, Q165Dic5, SD163Dic5, (C5×D8)⋊9C4, C58(C8○D8), (C5×Q16)⋊9C4, C4○D8.5D5, C40.61(C2×C4), (C8×Dic5)⋊2C2, (C5×SD16)⋊8C4, C4.217(D4×D5), C52C8.35D4, C40.6C48C2, C4○D4.22D10, C10.129(C4×D4), (C2×C8).254D10, C20.376(C2×D4), D4.Dic53C2, Q8.3(C2×Dic5), D4.3(C2×Dic5), C8.11(C2×Dic5), C2.16(D4×Dic5), D42Dic54C2, (C2×C40).44C22, C4.7(C22×Dic5), C20.136(C22×C4), (C2×C20).467C23, C22.3(D42D5), C4.Dic5.22C22, (C4×Dic5).277C22, (C5×C4○D8).2C2, (C5×D4).24(C2×C4), (C5×Q8).25(C2×C4), (C5×C4○D4).9C22, (C2×C10).11(C4○D4), (C2×C4).554(C22×D5), (C2×C52C8).288C22, SmallGroup(320,823)

Series: Derived Chief Lower central Upper central

C1C20 — D85Dic5
C1C5C10C20C2×C20C2×C52C8D4.Dic5 — D85Dic5
C5C10C20 — D85Dic5
C1C4C2×C4C4○D8

Generators and relations for D85Dic5
 G = < a,b,c,d | a8=b2=c10=1, d2=c5, bab=a-1, ac=ca, ad=da, cbc-1=a4b, dbd-1=a2b, dcd-1=c-1 >

Subgroups: 278 in 106 conjugacy classes, 53 normal (31 characteristic)
C1, C2, C2 [×3], C4 [×2], C4 [×4], C22, C22 [×2], C5, C8 [×2], C8 [×4], C2×C4, C2×C4 [×3], D4 [×2], D4 [×2], Q8 [×2], C10, C10 [×3], C42, C2×C8, C2×C8 [×3], M4(2) [×4], D8, SD16 [×2], Q16, C4○D4 [×2], Dic5 [×2], C20 [×2], C20 [×2], C2×C10, C2×C10 [×2], C4×C8, C4≀C2 [×2], C8.C4, C8○D4 [×2], C4○D8, C52C8 [×2], C52C8 [×2], C40 [×2], C2×Dic5, C2×C20, C2×C20 [×2], C5×D4 [×2], C5×D4 [×2], C5×Q8 [×2], C8○D8, C2×C52C8, C2×C52C8 [×2], C4.Dic5 [×2], C4.Dic5 [×2], C4×Dic5, C2×C40, C5×D8, C5×SD16 [×2], C5×Q16, C5×C4○D4 [×2], C8×Dic5, C40.6C4, D42Dic5 [×2], D4.Dic5 [×2], C5×C4○D8, D85Dic5
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], C23, D5, C22×C4, C2×D4, C4○D4, Dic5 [×4], D10 [×3], C4×D4, C2×Dic5 [×6], C22×D5, C8○D8, D4×D5, D42D5, C22×Dic5, D4×Dic5, D85Dic5

Smallest permutation representation of D85Dic5
On 80 points
Generators in S80
(1 24 14 28 31 38 19 9)(2 25 15 29 32 39 20 10)(3 21 11 30 33 40 16 6)(4 22 12 26 34 36 17 7)(5 23 13 27 35 37 18 8)(41 69 56 74 46 64 51 79)(42 70 57 75 47 65 52 80)(43 61 58 76 48 66 53 71)(44 62 59 77 49 67 54 72)(45 63 60 78 50 68 55 73)
(1 41)(2 47)(3 43)(4 49)(5 45)(6 61)(7 67)(8 63)(9 69)(10 65)(11 53)(12 59)(13 55)(14 51)(15 57)(16 58)(17 54)(18 60)(19 56)(20 52)(21 71)(22 77)(23 73)(24 79)(25 75)(26 62)(27 68)(28 64)(29 70)(30 66)(31 46)(32 42)(33 48)(34 44)(35 50)(36 72)(37 78)(38 74)(39 80)(40 76)
(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 31)(2 35)(3 34)(4 33)(5 32)(6 26)(7 30)(8 29)(9 28)(10 27)(11 17)(12 16)(13 20)(14 19)(15 18)(21 36)(22 40)(23 39)(24 38)(25 37)(41 51 46 56)(42 60 47 55)(43 59 48 54)(44 58 49 53)(45 57 50 52)(61 77 66 72)(62 76 67 71)(63 75 68 80)(64 74 69 79)(65 73 70 78)

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

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

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

56 conjugacy classes

class 1 2A2B2C2D4A4B4C4D4E4F4G4H4I5A5B8A8B8C8D8E8F8G8H8I8J8K8L8M8N10A10B10C10D10E10F10G10H20A20B20C20D20E20F20G20H20I20J40A···40H
order12222444444444558888888888888810101010101010102020202020202020202040···40
size11244112441010101022222255551010202020202244888822224488884···4

56 irreducible representations

dim111111111222222222444
type+++++++++---++-
imageC1C2C2C2C2C2C4C4C4D4D5C4○D4D10Dic5Dic5Dic5D10C8○D8D4×D5D42D5D85Dic5
kernelD85Dic5C8×Dic5C40.6C4D42Dic5D4.Dic5C5×C4○D8C5×D8C5×SD16C5×Q16C52C8C4○D8C2×C10C2×C8D8SD16Q16C4○D4C5C4C22C1
# reps111221242222224248228

Matrix representation of D85Dic5 in GL4(𝔽41) generated by

142600
0300
0010
0001
,
31500
353800
00400
00040
,
404000
0100
0001
00406
,
9500
04000
0010
00640
G:=sub<GL(4,GF(41))| [14,0,0,0,26,3,0,0,0,0,1,0,0,0,0,1],[3,35,0,0,15,38,0,0,0,0,40,0,0,0,0,40],[40,0,0,0,40,1,0,0,0,0,0,40,0,0,1,6],[9,0,0,0,5,40,0,0,0,0,1,6,0,0,0,40] >;

D85Dic5 in GAP, Magma, Sage, TeX

D_8\rtimes_5{\rm Dic}_5
% in TeX

G:=Group("D8:5Dic5");
// GroupNames label

G:=SmallGroup(320,823);
// by ID

G=gap.SmallGroup(320,823);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,219,136,851,438,102,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^8=b^2=c^10=1,d^2=c^5,b*a*b=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=a^4*b,d*b*d^-1=a^2*b,d*c*d^-1=c^-1>;
// generators/relations

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