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## G = D4.Dic5order 160 = 25·5

### The non-split extension by D4 of Dic5 acting through Inn(D4)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — D4.Dic5
 Chief series C1 — C5 — C10 — C20 — C5⋊2C8 — C2×C5⋊2C8 — D4.Dic5
 Lower central C5 — C10 — D4.Dic5
 Upper central C1 — C4 — C4○D4

Generators and relations for D4.Dic5
G = < a,b,c,d | a4=1, b2=c10=a2, d2=c5, bab-1=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c9 >

Subgroups: 112 in 62 conjugacy classes, 45 normal (12 characteristic)
C1, C2, C2, C4, C4, C22, C5, C8, C2×C4, D4, Q8, C10, C10, C2×C8, M4(2), C4○D4, C20, C20, C2×C10, C8○D4, C52C8, C52C8, C2×C20, C5×D4, C5×Q8, C2×C52C8, C4.Dic5, C5×C4○D4, D4.Dic5
Quotients: C1, C2, C4, C22, C2×C4, C23, D5, C22×C4, Dic5, D10, C8○D4, C2×Dic5, C22×D5, C22×Dic5, D4.Dic5

Smallest permutation representation of D4.Dic5
On 80 points
Generators in S80
(1 38 11 28)(2 39 12 29)(3 40 13 30)(4 21 14 31)(5 22 15 32)(6 23 16 33)(7 24 17 34)(8 25 18 35)(9 26 19 36)(10 27 20 37)(41 76 51 66)(42 77 52 67)(43 78 53 68)(44 79 54 69)(45 80 55 70)(46 61 56 71)(47 62 57 72)(48 63 58 73)(49 64 59 74)(50 65 60 75)
(1 6 11 16)(2 7 12 17)(3 8 13 18)(4 9 14 19)(5 10 15 20)(21 36 31 26)(22 37 32 27)(23 38 33 28)(24 39 34 29)(25 40 35 30)(41 46 51 56)(42 47 52 57)(43 48 53 58)(44 49 54 59)(45 50 55 60)(61 76 71 66)(62 77 72 67)(63 78 73 68)(64 79 74 69)(65 80 75 70)
(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 63 26 68 31 73 36 78)(22 72 27 77 32 62 37 67)(23 61 28 66 33 71 38 76)(24 70 29 75 34 80 39 65)(25 79 30 64 35 69 40 74)

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

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

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

40 conjugacy classes

 class 1 2A 2B 2C 2D 4A 4B 4C 4D 4E 5A 5B 8A 8B 8C 8D 8E ··· 8J 10A 10B 10C ··· 10H 20A 20B 20C 20D 20E ··· 20J order 1 2 2 2 2 4 4 4 4 4 5 5 8 8 8 8 8 ··· 8 10 10 10 ··· 10 20 20 20 20 20 ··· 20 size 1 1 2 2 2 1 1 2 2 2 2 2 5 5 5 5 10 ··· 10 2 2 4 ··· 4 2 2 2 2 4 ··· 4

40 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 4 type + + + + + + - - image C1 C2 C2 C2 C4 C4 D5 D10 Dic5 Dic5 C8○D4 D4.Dic5 kernel D4.Dic5 C2×C5⋊2C8 C4.Dic5 C5×C4○D4 C5×D4 C5×Q8 C4○D4 C2×C4 D4 Q8 C5 C1 # reps 1 3 3 1 6 2 2 6 6 2 4 4

Matrix representation of D4.Dic5 in GL4(𝔽41) generated by

 1 0 0 0 0 1 0 0 0 0 9 23 0 0 0 32
,
 1 0 0 0 0 1 0 0 0 0 9 0 0 0 9 32
,
 40 1 0 0 33 7 0 0 0 0 9 0 0 0 0 9
,
 0 35 0 0 34 0 0 0 0 0 38 0 0 0 0 38
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,9,0,0,0,23,32],[1,0,0,0,0,1,0,0,0,0,9,9,0,0,0,32],[40,33,0,0,1,7,0,0,0,0,9,0,0,0,0,9],[0,34,0,0,35,0,0,0,0,0,38,0,0,0,0,38] >;

D4.Dic5 in GAP, Magma, Sage, TeX

D_4.{\rm Dic}_5
% in TeX

G:=Group("D4.Dic5");
// GroupNames label

G:=SmallGroup(160,169);
// by ID

G=gap.SmallGroup(160,169);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-5,48,188,69,4613]);
// Polycyclic

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

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