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G = D84Dic5order 320 = 26·5

4th semidirect product of D8 and Dic5 acting via Dic5/C10=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D84Dic5, Q164Dic5, SD162Dic5, (C5×D8)⋊12C4, C4○D8.4D5, C408C42C2, C58(C8.26D4), C40.72(C2×C4), (C5×Q16)⋊12C4, (C5×SD16)⋊6C4, C4.218(D4×D5), C52C8.52D4, C40.6C49C2, C8.6(C2×Dic5), C4○D4.23D10, C20.377(C2×D4), (C2×C8).100D10, C10.130(C4×D4), D4.Dic54C2, Q8.4(C2×Dic5), D4.4(C2×Dic5), C2.17(D4×Dic5), D42Dic55C2, (C2×C40).45C22, C4.8(C22×Dic5), (C2×C20).468C23, C20.137(C22×C4), C22.4(D42D5), (C4×Dic5).63C22, C4.Dic5.23C22, (C5×C4○D8).3C2, (C5×D4).25(C2×C4), (C5×Q8).26(C2×C4), (C2×C10).12(C4○D4), (C5×C4○D4).10C22, (C2×C4).555(C22×D5), (C2×C52C8).169C22, SmallGroup(320,824)

Series: Derived Chief Lower central Upper central

C1C20 — D84Dic5
C1C5C10C20C2×C20C2×C52C8D4.Dic5 — D84Dic5
C5C10C20 — D84Dic5
C1C4C2×C4C4○D8

Generators and relations for D84Dic5
 G = < a,b,c,d | a8=b2=c10=1, d2=c5, bab=a-1, ac=ca, dad-1=a5, cbc-1=a4b, dbd-1=a2b, dcd-1=c-1 >

Subgroups: 278 in 104 conjugacy classes, 53 normal (29 characteristic)
C1, C2, C2 [×3], C4 [×2], C4 [×3], 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, C20 [×2], C20 [×2], C2×C10, C2×C10 [×2], C8⋊C4, 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], C8.26D4, 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], C408C4, C40.6C4, D42Dic5 [×2], D4.Dic5 [×2], C5×C4○D8, D84Dic5
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, C8.26D4, D4×D5, D42D5, C22×Dic5, D4×Dic5, D84Dic5

Smallest permutation representation of D84Dic5
On 80 points
Generators in S80
(1 29 14 23 36 33 19 6)(2 30 15 24 37 34 20 7)(3 26 11 25 38 35 16 8)(4 27 12 21 39 31 17 9)(5 28 13 22 40 32 18 10)(41 75 70 60 46 80 65 55)(42 76 61 51 47 71 66 56)(43 77 62 52 48 72 67 57)(44 78 63 53 49 73 68 58)(45 79 64 54 50 74 69 59)
(1 45)(2 41)(3 47)(4 43)(5 49)(6 79)(7 75)(8 71)(9 77)(10 73)(11 61)(12 67)(13 63)(14 69)(15 65)(16 66)(17 62)(18 68)(19 64)(20 70)(21 72)(22 78)(23 74)(24 80)(25 76)(26 51)(27 57)(28 53)(29 59)(30 55)(31 52)(32 58)(33 54)(34 60)(35 56)(36 50)(37 46)(38 42)(39 48)(40 44)
(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 23)(7 22)(8 21)(9 25)(10 24)(11 12)(13 15)(16 17)(18 20)(26 31)(27 35)(28 34)(29 33)(30 32)(37 40)(38 39)(41 68 46 63)(42 67 47 62)(43 66 48 61)(44 65 49 70)(45 64 50 69)(51 72 56 77)(52 71 57 76)(53 80 58 75)(54 79 59 74)(55 78 60 73)

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

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

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

50 conjugacy classes

class 1 2A2B2C2D4A4B4C4D4E4F4G5A5B8A8B8C8D8E8F8G8H8I8J10A10B10C10D10E10F10G10H20A20B20C20D20E20F20G20H20I20J40A···40H
order12222444444455888888888810101010101010102020202020202020202040···40
size11244112442020224410101010202020202244888822224488884···4

50 irreducible representations

dim111111111222222224444
type+++++++++---++-
imageC1C2C2C2C2C2C4C4C4D4D5C4○D4D10Dic5Dic5Dic5D10C8.26D4D4×D5D42D5D84Dic5
kernelD84Dic5C408C4C40.6C4D42Dic5D4.Dic5C5×C4○D8C5×D8C5×SD16C5×Q16C52C8C4○D8C2×C10C2×C8D8SD16Q16C4○D4C5C4C22C1
# reps111221242222224242228

Matrix representation of D84Dic5 in GL4(𝔽41) generated by

103800
33100
003327
00148
,
001038
00331
332700
14800
,
04000
1700
0001
004034
,
193200
222200
0071
003434
G:=sub<GL(4,GF(41))| [10,3,0,0,38,31,0,0,0,0,33,14,0,0,27,8],[0,0,33,14,0,0,27,8,10,3,0,0,38,31,0,0],[0,1,0,0,40,7,0,0,0,0,0,40,0,0,1,34],[19,22,0,0,32,22,0,0,0,0,7,34,0,0,1,34] >;

D84Dic5 in GAP, Magma, Sage, TeX

D_8\rtimes_4{\rm Dic}_5
% in TeX

G:=Group("D8:4Dic5");
// GroupNames label

G:=SmallGroup(320,824);
// by ID

G=gap.SmallGroup(320,824);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,758,219,136,851,438,102,12550]);
// 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,d*a*d^-1=a^5,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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