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## G = D8.Dic5order 320 = 26·5

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C40 — D8.Dic5
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C2×C40 — C40.6C4 — D8.Dic5
 Lower central C5 — C10 — C20 — C40 — D8.Dic5
 Upper central C1 — C2 — C2×C4 — C2×C8 — C2×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, C4, C22, C22, C5, C8, C8, C2×C4, D4, C23, C10, C10, C16, C2×C8, M4(2), D8, D8, C2×D4, C20, C2×C10, C2×C10, C8.C4, M5(2), C2×D8, C52C8, C40, C2×C20, C5×D4, C22×C10, M5(2)⋊C2, C52C16, C4.Dic5, C2×C40, C5×D8, C5×D8, D4×C10, C20.4C8, C40.6C4, C10×D8, D8.Dic5
Quotients: C1, C2, C4, C22, C2×C4, D4, D5, C22⋊C4, D8, SD16, Dic5, D10, D4⋊C4, C2×Dic5, C5⋊D4, 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 21 6 26 11 31 16 36)(2 22 7 27 12 32 17 37)(3 23 8 28 13 33 18 38)(4 24 9 29 14 34 19 39)(5 25 10 30 15 35 20 40)(41 76 56 71 51 66 46 61)(42 77 57 72 52 67 47 62)(43 78 58 73 53 68 48 63)(44 79 59 74 54 69 49 64)(45 80 60 75 55 70 50 65)
(1 36)(2 27)(3 38)(4 29)(5 40)(6 31)(7 22)(8 33)(9 24)(10 35)(11 26)(12 37)(13 28)(14 39)(15 30)(16 21)(17 32)(18 23)(19 34)(20 25)(41 46)(42 57)(43 48)(44 59)(45 50)(47 52)(49 54)(51 56)(53 58)(55 60)(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 41 6 46 11 51 16 56)(2 50 7 55 12 60 17 45)(3 59 8 44 13 49 18 54)(4 48 9 53 14 58 19 43)(5 57 10 42 15 47 20 52)(21 71 26 76 31 61 36 66)(22 80 27 65 32 70 37 75)(23 69 28 74 33 79 38 64)(24 78 29 63 34 68 39 73)(25 67 30 72 35 77 40 62)

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

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

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

44 conjugacy classes

 class 1 2A 2B 2C 2D 4A 4B 5A 5B 8A 8B 8C 8D 8E 10A ··· 10F 10G ··· 10N 16A 16B 16C 16D 20A 20B 20C 20D 40A ··· 40H order 1 2 2 2 2 4 4 5 5 8 8 8 8 8 10 ··· 10 10 ··· 10 16 16 16 16 20 20 20 20 40 ··· 40 size 1 1 2 8 8 2 2 2 2 2 2 4 40 40 2 ··· 2 8 ··· 8 20 20 20 20 4 4 4 4 4 ··· 4

44 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + - + + - image C1 C2 C2 C2 C4 D4 D4 D5 D8 SD16 D10 Dic5 C5⋊D4 C5⋊D4 M5(2)⋊C2 D4⋊D5 D4.D5 D8.Dic5 kernel D8.Dic5 C20.4C8 C40.6C4 C10×D8 C5×D8 C40 C2×C20 C2×D8 C20 C2×C10 C2×C8 D8 C8 C2×C4 C5 C4 C22 C1 # reps 1 1 1 1 4 1 1 2 2 2 2 4 4 4 2 2 2 8

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

 22 120 0 0 2 0 0 0 0 0 0 120 0 0 2 219
,
 22 120 0 0 2 219 0 0 0 0 1 230 0 0 0 240
,
 87 7 0 0 213 154 0 0 0 0 205 155 0 0 103 36
,
 0 0 1 0 0 0 0 1 1 230 0 0 44 240 0 0
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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