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G = D8.Dic5order 320 = 26·5

2nd non-split extension by D8 of Dic5 acting via Dic5/C10=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C40.14D4, C20.11D8, D8.2Dic5, (C2×D8).5D5, (C5×D8).4C4, C40.66(C2×C4), (C10×D8).1C2, (C2×C8).49D10, C20.4C81C2, C40.6C42C2, C8.1(C2×Dic5), C4.14(D4⋊D5), (C2×C20).115D4, C54(M5(2)⋊C2), C8.24(C5⋊D4), (C2×C40).29C22, (C2×C10).30SD16, C4.2(C23.D5), C20.61(C22⋊C4), C2.7(D4⋊Dic5), C22.6(D4.D5), C10.42(D4⋊C4), (C2×C4).24(C5⋊D4), SmallGroup(320,121)

Series: Derived Chief Lower central Upper central

C1C40 — D8.Dic5
C1C5C10C20C2×C20C2×C40C40.6C4 — D8.Dic5
C5C10C20C40 — D8.Dic5
C1C2C2×C4C2×C8C2×D8

Generators and relations for D8.Dic5
 G = < a,b,c,d | a8=b2=1, c10=a4, d2=c5, bab=a-1, ac=ca, dad-1=a3, cbc-1=a4b, dbd-1=a5b, dcd-1=c9 >

Subgroups: 206 in 62 conjugacy classes, 27 normal (23 characteristic)
C1, C2, C2 [×3], C4 [×2], C22, C22 [×4], C5, C8 [×2], C8, C2×C4, D4 [×3], C23, C10, C10 [×3], C16, C2×C8, M4(2), D8 [×2], D8, C2×D4, C20 [×2], C2×C10, C2×C10 [×4], C8.C4, M5(2), C2×D8, C52C8, C40 [×2], C2×C20, C5×D4 [×3], C22×C10, M5(2)⋊C2, C52C16, C4.Dic5, C2×C40, C5×D8 [×2], C5×D8, D4×C10, C20.4C8, C40.6C4, C10×D8, D8.Dic5
Quotients: C1, C2 [×3], C4 [×2], C22, C2×C4, D4 [×2], D5, C22⋊C4, D8, SD16, Dic5 [×2], D10, D4⋊C4, C2×Dic5, C5⋊D4 [×2], M5(2)⋊C2, D4⋊D5, D4.D5, C23.D5, D4⋊Dic5, D8.Dic5

Smallest permutation representation of D8.Dic5
On 80 points
Generators in S80
(1 27 6 32 11 37 16 22)(2 28 7 33 12 38 17 23)(3 29 8 34 13 39 18 24)(4 30 9 35 14 40 19 25)(5 31 10 36 15 21 20 26)(41 61 56 76 51 71 46 66)(42 62 57 77 52 72 47 67)(43 63 58 78 53 73 48 68)(44 64 59 79 54 74 49 69)(45 65 60 80 55 75 50 70)
(1 22)(2 33)(3 24)(4 35)(5 26)(6 37)(7 28)(8 39)(9 30)(10 21)(11 32)(12 23)(13 34)(14 25)(15 36)(16 27)(17 38)(18 29)(19 40)(20 31)(41 56)(42 47)(43 58)(44 49)(45 60)(46 51)(48 53)(50 55)(52 57)(54 59)(62 72)(64 74)(66 76)(68 78)(70 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 46 6 51 11 56 16 41)(2 55 7 60 12 45 17 50)(3 44 8 49 13 54 18 59)(4 53 9 58 14 43 19 48)(5 42 10 47 15 52 20 57)(21 67 26 72 31 77 36 62)(22 76 27 61 32 66 37 71)(23 65 28 70 33 75 38 80)(24 74 29 79 34 64 39 69)(25 63 30 68 35 73 40 78)

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

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

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

44 conjugacy classes

class 1 2A2B2C2D4A4B5A5B8A8B8C8D8E10A···10F10G···10N16A16B16C16D20A20B20C20D40A···40H
order1222244558888810···1010···10161616162020202040···40
size11288222222440402···28···82020202044444···4

44 irreducible representations

dim111112222222224444
type+++++++++-++-
imageC1C2C2C2C4D4D4D5D8SD16D10Dic5C5⋊D4C5⋊D4M5(2)⋊C2D4⋊D5D4.D5D8.Dic5
kernelD8.Dic5C20.4C8C40.6C4C10×D8C5×D8C40C2×C20C2×D8C20C2×C10C2×C8D8C8C2×C4C5C4C22C1
# reps111141122224442228

Matrix representation of D8.Dic5 in GL4(𝔽241) generated by

2212000
2000
000120
002219
,
2212000
221900
001230
000240
,
87700
21315400
00205155
0010336
,
0010
0001
123000
4424000
G:=sub<GL(4,GF(241))| [22,2,0,0,120,0,0,0,0,0,0,2,0,0,120,219],[22,2,0,0,120,219,0,0,0,0,1,0,0,0,230,240],[87,213,0,0,7,154,0,0,0,0,205,103,0,0,155,36],[0,0,1,44,0,0,230,240,1,0,0,0,0,1,0,0] >;

D8.Dic5 in GAP, Magma, Sage, TeX

D_8.{\rm Dic}_5
% in TeX

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

G:=SmallGroup(320,121);
// by ID

G=gap.SmallGroup(320,121);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,141,387,184,675,794,80,1684,851,102,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^8=b^2=1,c^10=a^4,d^2=c^5,b*a*b=a^-1,a*c=c*a,d*a*d^-1=a^3,c*b*c^-1=a^4*b,d*b*d^-1=a^5*b,d*c*d^-1=c^9>;
// generators/relations

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