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G = D4.Dic5order 160 = 25·5

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

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D4.Dic5, Q8.Dic5, C20.42C23, C55(C8○D4), C4○D4.3D5, (C5×D4).2C4, (C5×Q8).2C4, C20.36(C2×C4), (C2×C4).58D10, C4.Dic58C2, C4.5(C2×Dic5), C4.42(C22×D5), (C2×C20).41C22, C10.40(C22×C4), C52C8.13C22, C2.8(C22×Dic5), C22.1(C2×Dic5), (C2×C52C8)⋊7C2, (C5×C4○D4).2C2, (C2×C10).27(C2×C4), SmallGroup(160,169)

Series: Derived Chief Lower central Upper central

C1C10 — D4.Dic5
C1C5C10C20C52C8C2×C52C8 — D4.Dic5
C5C10 — D4.Dic5
C1C4C4○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)]])

D4.Dic5 is a maximal subgroup of
D4.(C5⋊C8)  M4(2).22D10  C42.196D10  D85Dic5  D84Dic5  M4(2).D10  M4(2).13D10  M4(2).15D10  M4(2).16D10  C5⋊C16.C22  D5×C8○D4  C20.72C24  C20.76C24  D20.32C23  D20.33C23  D20.34C23  D20.35C23  SL2(𝔽3).Dic5  D12.2Dic5  D12.Dic5  D4.Dic15
D4.Dic5 is a maximal quotient of
C20.35C42  C42.43D10  C42.187D10  D4×C52C8  C42.47D10  C207M4(2)  Q8×C52C8  C42.210D10  (D4×C10).24C4  D12.2Dic5  D12.Dic5  D4.Dic15

40 conjugacy classes

class 1 2A2B2C2D4A4B4C4D4E5A5B8A8B8C8D8E···8J10A10B10C···10H20A20B20C20D20E···20J
order12222444445588888···8101010···102020202020···20
size112221122222555510···10224···422224···4

40 irreducible representations

dim111111222224
type++++++--
imageC1C2C2C2C4C4D5D10Dic5Dic5C8○D4D4.Dic5
kernelD4.Dic5C2×C52C8C4.Dic5C5×C4○D4C5×D4C5×Q8C4○D4C2×C4D4Q8C5C1
# reps133162266244

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

1000
0100
00923
00032
,
1000
0100
0090
00932
,
40100
33700
0090
0009
,
03500
34000
00380
00038
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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