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G = D48D10order 160 = 25·5

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

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D48D10, Q87D10, C522+ 1+4, D2011C22, C20.26C23, C10.12C24, D10.7C23, Dic5.7C23, Dic1012C22, C4○D43D5, (D4×D5)⋊5C2, (C2×C4)⋊4D10, C4○D208C2, (C2×D20)⋊13C2, (C2×C20)⋊5C22, Q82D55C2, (C5×D4)⋊9C22, (C4×D5)⋊2C22, C5⋊D45C22, (C5×Q8)⋊8C22, (C2×C10).4C23, C2.13(C23×D5), C4.33(C22×D5), (C22×D5)⋊4C22, C22.3(C22×D5), (C5×C4○D4)⋊4C2, SmallGroup(160,224)

Series: Derived Chief Lower central Upper central

C1C10 — D48D10
C1C5C10D10C22×D5D4×D5 — D48D10
C5C10 — D48D10
C1C2C4○D4

Generators and relations for D48D10
 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 [×9], C4, C4 [×3], C4 [×2], C22 [×3], C22 [×12], C5, C2×C4 [×3], C2×C4 [×6], D4 [×3], D4 [×15], Q8, Q8, C23 [×6], D5 [×6], C10, C10 [×3], C2×D4 [×9], C4○D4, C4○D4 [×5], Dic5 [×2], C20, C20 [×3], D10 [×6], D10 [×6], C2×C10 [×3], 2+ 1+4, Dic10, C4×D5 [×6], D20 [×9], C5⋊D4 [×6], C2×C20 [×3], C5×D4 [×3], C5×Q8, C22×D5 [×6], C2×D20 [×3], C4○D20 [×3], D4×D5 [×6], Q82D5 [×2], C5×C4○D4, D48D10
Quotients: C1, C2 [×15], C22 [×35], C23 [×15], D5, C24, D10 [×7], 2+ 1+4, C22×D5 [×7], C23×D5, D48D10

Smallest permutation representation of D48D10
On 40 points
Generators in S40
(1 21 33 15)(2 22 34 16)(3 23 35 17)(4 24 36 18)(5 25 37 19)(6 26 38 20)(7 27 39 11)(8 28 40 12)(9 29 31 13)(10 30 32 14)
(1 15)(2 22)(3 17)(4 24)(5 19)(6 26)(7 11)(8 28)(9 13)(10 30)(12 40)(14 32)(16 34)(18 36)(20 38)(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 19)(2 18)(3 17)(4 16)(5 15)(6 14)(7 13)(8 12)(9 11)(10 20)(21 37)(22 36)(23 35)(24 34)(25 33)(26 32)(27 31)(28 40)(29 39)(30 38)

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

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

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

D48D10 is a maximal subgroup of
D44D20  M4(2)⋊D10  D2018D4  D20.39D4  D4.11D20  D4.12D20  D815D10  D811D10  D85D10  C40.C23  D20.32C23  D20.34C23  C10.C25  D5×2+ 1+4  D20.39C23  Dic10.A4  D2025D6  D2029D6  D2014D6  D2017D6  D48D30
D48D10 is a maximal quotient of
C42.90D10  C427D10  C42.91D10  C429D10  C4210D10  C42.95D10  C42.97D10  C42.99D10  C42.100D10  D46Dic10  C4211D10  D4×D20  D2023D4  Dic1024D4  C4217D10  C42.116D10  C42.117D10  C42.119D10  Q8×Dic10  C42.126D10  Q86D20  D2010Q8  C42.133D10  C42.136D10  C10.372+ 1+4  C10.382+ 1+4  D2019D4  C10.462+ 1+4  C10.1152+ 1+4  C10.472+ 1+4  C10.482+ 1+4  C10.172- 1+4  D2021D4  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  C4218D10  D2010D4  C4220D10  C42.143D10  C42.144D10  C4222D10  C42.145D10  C42.148D10  D207Q8  C42.150D10  C42.153D10  C42.156D10  C42.157D10  C42.158D10  C4223D10  C4224D10  C42.161D10  C42.163D10  C42.164D10  C4225D10  C42.165D10  C10.1062- 1+4  C10.1452+ 1+4  C10.1462+ 1+4  C10.1472+ 1+4  C10.1482+ 1+4  D2025D6  D2029D6  D2014D6  D2017D6  D48D30

37 conjugacy classes

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

37 irreducible representations

dim111111222244
type++++++++++++
imageC1C2C2C2C2C2D5D10D10D102+ 1+4D48D10
kernelD48D10C2×D20C4○D20D4×D5Q82D5C5×C4○D4C4○D4C2×C4D4Q8C5C1
# reps133621266214

Matrix representation of D48D10 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] >;

D48D10 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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