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G = D4:8D10  order 160 = 25·5

4th semidirect product of D4 and D10 acting through Inn(D4)

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D4:8D10, Q8:7D10, C5:22+ 1+4, D20:11C22, C20.26C23, C10.12C24, D10.7C23, Dic5.7C23, Dic10:12C22, C4oD4:3D5, (D4xD5):5C2, (C2xC4):4D10, C4oD20:8C2, (C2xD20):13C2, (C2xC20):5C22, Q8:2D5:5C2, (C5xD4):9C22, (C4xD5):2C22, C5:D4:5C22, (C5xQ8):8C22, (C2xC10).4C23, C2.13(C23xD5), C4.33(C22xD5), (C22xD5):4C22, C22.3(C22xD5), (C5xC4oD4):4C2, SmallGroup(160,224)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C10 — D4:8D10
Chief series
C1 — C5 — C10 — D10 — C22xD5 — D4xD5 — D4:8D10
Lower central
C5 — C10 — D4:8D10
Upper central
C1 — C2 — C4oD4

Generators and relations for D4:8D10
 G = < a,b,c,d | a4=b2=c10=d2=1, bab=dad=a-1, ac=ca, cbc-1=a2b, bd=db, dcd=c-1 >

Subgroups: 568 in 166 conjugacy classes, 85 normal (12 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C5, C2xC4, C2xC4, D4, D4, Q8, Q8, C23, D5, C10, C10, C2xD4, C4oD4, C4oD4, Dic5, C20, C20, D10, D10, C2xC10, 2+ 1+4, Dic10, C4xD5, D20, C5:D4, C2xC20, C5xD4, C5xQ8, C22xD5, C2xD20, C4oD20, D4xD5, Q8:2D5, C5xC4oD4, D4:8D10
Quotients: C1, C2, C22, C23, D5, C24, D10, 2+ 1+4, C22xD5, C23xD5, D4:8D10

Smallest permutation representation of D4:8D10
►On 40 points
Generators in S40
(1 21 33 19)(2 22 34 20)(3 23 35 11)(4 24 36 12)(5 25 37 13)(6 26 38 14)(7 27 39 15)(8 28 40 16)(9 29 31 17)(10 30 32 18)

(1 19)(2 22)(3 11)(4 24)(5 13)(6 26)(7 15)(8 28)(9 17)(10 30)(12 36)(14 38)(16 40)(18 32)(20 34)(21 33)(23 35)(25 37)(27 39)(29 31)

(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)

(1 13)(2 12)(3 11)(4 20)(5 19)(6 18)(7 17)(8 16)(9 15)(10 14)(21 37)(22 36)(23 35)(24 34)(25 33)(26 32)(27 31)(28 40)(29 39)(30 38)
 
G:=sub<Sym(40)| (1,21,33,19)(2,22,34,20)(3,23,35,11)(4,24,36,12)(5,25,37,13)(6,26,38,14)(7,27,39,15)(8,28,40,16)(9,29,31,17)(10,30,32,18), (1,19)(2,22)(3,11)(4,24)(5,13)(6,26)(7,15)(8,28)(9,17)(10,30)(12,36)(14,38)(16,40)(18,32)(20,34)(21,33)(23,35)(25,37)(27,39)(29,31), (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), (1,13)(2,12)(3,11)(4,20)(5,19)(6,18)(7,17)(8,16)(9,15)(10,14)(21,37)(22,36)(23,35)(24,34)(25,33)(26,32)(27,31)(28,40)(29,39)(30,38)>;
 
G:=Group( (1,21,33,19)(2,22,34,20)(3,23,35,11)(4,24,36,12)(5,25,37,13)(6,26,38,14)(7,27,39,15)(8,28,40,16)(9,29,31,17)(10,30,32,18), (1,19)(2,22)(3,11)(4,24)(5,13)(6,26)(7,15)(8,28)(9,17)(10,30)(12,36)(14,38)(16,40)(18,32)(20,34)(21,33)(23,35)(25,37)(27,39)(29,31), (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), (1,13)(2,12)(3,11)(4,20)(5,19)(6,18)(7,17)(8,16)(9,15)(10,14)(21,37)(22,36)(23,35)(24,34)(25,33)(26,32)(27,31)(28,40)(29,39)(30,38) );
 
G=PermutationGroup([[(1,21,33,19),(2,22,34,20),(3,23,35,11),(4,24,36,12),(5,25,37,13),(6,26,38,14),(7,27,39,15),(8,28,40,16),(9,29,31,17),(10,30,32,18)], [(1,19),(2,22),(3,11),(4,24),(5,13),(6,26),(7,15),(8,28),(9,17),(10,30),(12,36),(14,38),(16,40),(18,32),(20,34),(21,33),(23,35),(25,37),(27,39),(29,31)], [(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)], [(1,13),(2,12),(3,11),(4,20),(5,19),(6,18),(7,17),(8,16),(9,15),(10,14),(21,37),(22,36),(23,35),(24,34),(25,33),(26,32),(27,31),(28,40),(29,39),(30,38)]])
 

D4:8D10 is a maximal subgroup of
 D4:4D20  M4(2):D10  D20:18D4  D20.39D4  D4.11D20  D4.12D20  D8:15D10  D8:11D10  D8:5D10  C40.C23  D20.32C23  D20.34C23  C10.C25  D5x2+ 1+4  D20.39C23  Dic10.A4  D20:25D6  D20:29D6  D20:14D6  D20:17D6  D4:8D30
D4:8D10 is a maximal quotient of
 C42.90D10  C42:7D10  C42.91D10  C42:9D10  C42:10D10  C42.95D10  C42.97D10  C42.99D10  C42.100D10  D4:6Dic10  C42:11D10  D4xD20  D20:23D4  Dic10:24D4  C42:17D10  C42.116D10  C42.117D10  C42.119D10  Q8xDic10  C42.126D10  Q8:6D20  D20:10Q8  C42.133D10  C42.136D10  C10.372+ 1+4  C10.382+ 1+4  D20:19D4  C10.462+ 1+4  C10.1152+ 1+4  C10.472+ 1+4  C10.482+ 1+4  C10.172- 1+4  D20:21D4  C10.512+ 1+4  C10.1182+ 1+4  C10.242- 1+4  C10.562+ 1+4  C10.262- 1+4  C10.1202+ 1+4  C10.1212+ 1+4  C10.612+ 1+4  C10.1222+ 1+4  C10.662+ 1+4  C10.852- 1+4  C10.682+ 1+4  C10.692+ 1+4  C42:18D10  D20:10D4  C42:20D10  C42.143D10  C42.144D10  C42:22D10  C42.145D10  C42.148D10  D20:7Q8  C42.150D10  C42.153D10  C42.156D10  C42.157D10  C42.158D10  C42:23D10  C42:24D10  C42.161D10  C42.163D10  C42.164D10  C42:25D10  C42.165D10  C10.1062- 1+4  C10.1452+ 1+4  C10.1462+ 1+4  C10.1472+ 1+4  C10.1482+ 1+4  D20:25D6  D20:29D6  D20:14D6  D20:17D6  D4:8D30

37 conjugacy classes

class 1 2A2B2C2D2E···2J4A4B4C4D4E4F5A5B10A10B10C···10H20A20B20C20D20E···20J
order122222···244444455101010···102020202020···20
size1122210···102222101022224···422224···4

37 irreducible representations

dim111111222244
type++++++++++++
imageC1C2C2C2C2C2D5D10D10D102+ 1+4D4:8D10
kernelD4:8D10C2xD20C4oD20D4xD5Q8:2D5C5xC4oD4C4oD4C2xC4D4Q8C5C1
# reps133621266214

Matrix representation of D4:8D10 ►in GL6(F41)

4000000
0400000
0013005
00704028
0001040
0070028
,
100000
010000
0013005
00704028
00340040
00320028
,
770000
34400000
001050
0000281
0000400
0001280
,
770000
40340000
0013005
00320128
0001040
00320028

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,13,7,0,7,0,0,0,0,1,0,0,0,0,40,0,0,0,0,5,28,40,28],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,13,7,3,32,0,0,0,0,40,0,0,0,0,40,0,0,0,0,5,28,40,28],[7,34,0,0,0,0,7,40,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,5,28,40,28,0,0,0,1,0,0],[7,40,0,0,0,0,7,34,0,0,0,0,0,0,13,32,0,32,0,0,0,0,1,0,0,0,0,1,0,0,0,0,5,28,40,28] >;
 

D4:8D10 in GAP, Magma, Sage, TeX 

D_4\rtimes_8D_{10}
 
% in TeX
 
G:=Group("D4:8D10");
 
// GroupNames label
 
G:=SmallGroup(160,224);
 
// by ID
 
G=gap.SmallGroup(160,224);
 
# by ID
 
G:=PCGroup([6,-2,-2,-2,-2,-2,-5,188,579,69,4613]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^4=b^2=c^10=d^2=1,b*a*b=d*a*d=a^-1,a*c=c*a,c*b*c^-1=a^2*b,b*d=d*b,d*c*d=c^-1>;
 
// generators/relations
 

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