Copied to
clipboard

G = D102Q8order 160 = 25·5

2nd semidirect product of D10 and Q8 acting via Q8/C4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D102Q8, C20.11D4, C4.13D20, C4⋊C45D5, C2.7(Q8×D5), C4⋊Dic56C2, C10.8(C2×D4), C53(C22⋊Q8), (C2×C4).14D10, C2.10(C2×D20), C10.14(C2×Q8), (C2×Dic10)⋊7C2, (C2×C20).6C22, D10⋊C4.3C2, C10.27(C4○D4), (C2×C10).38C23, C2.13(D42D5), C22.52(C22×D5), (C2×Dic5).13C22, (C22×D5).25C22, (C5×C4⋊C4)⋊8C2, (C2×C4×D5).3C2, SmallGroup(160,118)

Series: Derived Chief Lower central Upper central

C1C2×C10 — D102Q8
C1C5C10C2×C10C22×D5C2×C4×D5 — D102Q8
C5C2×C10 — D102Q8
C1C22C4⋊C4

Generators and relations for D102Q8
 G = < a,b,c,d | a10=b2=c4=1, d2=c2, bab=cac-1=a-1, ad=da, cbc-1=a3b, bd=db, dcd-1=c-1 >

Subgroups: 240 in 74 conjugacy classes, 35 normal (19 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C2×C4, C2×C4, C2×C4, Q8, C23, D5, C10, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C2×Q8, Dic5, C20, C20, D10, D10, C2×C10, C22⋊Q8, Dic10, C4×D5, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×D5, C4⋊Dic5, D10⋊C4, C5×C4⋊C4, C2×Dic10, C2×C4×D5, D102Q8
Quotients: C1, C2, C22, D4, Q8, C23, D5, C2×D4, C2×Q8, C4○D4, D10, C22⋊Q8, D20, C22×D5, C2×D20, D42D5, Q8×D5, D102Q8

Smallest permutation representation of D102Q8
On 80 points
Generators in S80
(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 29)(2 28)(3 27)(4 26)(5 25)(6 24)(7 23)(8 22)(9 21)(10 30)(11 77)(12 76)(13 75)(14 74)(15 73)(16 72)(17 71)(18 80)(19 79)(20 78)(31 41)(32 50)(33 49)(34 48)(35 47)(36 46)(37 45)(38 44)(39 43)(40 42)(51 66)(52 65)(53 64)(54 63)(55 62)(56 61)(57 70)(58 69)(59 68)(60 67)
(1 33 30 45)(2 32 21 44)(3 31 22 43)(4 40 23 42)(5 39 24 41)(6 38 25 50)(7 37 26 49)(8 36 27 48)(9 35 28 47)(10 34 29 46)(11 70 73 58)(12 69 74 57)(13 68 75 56)(14 67 76 55)(15 66 77 54)(16 65 78 53)(17 64 79 52)(18 63 80 51)(19 62 71 60)(20 61 72 59)
(1 70 30 58)(2 61 21 59)(3 62 22 60)(4 63 23 51)(5 64 24 52)(6 65 25 53)(7 66 26 54)(8 67 27 55)(9 68 28 56)(10 69 29 57)(11 45 73 33)(12 46 74 34)(13 47 75 35)(14 48 76 36)(15 49 77 37)(16 50 78 38)(17 41 79 39)(18 42 80 40)(19 43 71 31)(20 44 72 32)

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

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

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

D102Q8 is a maximal subgroup of
D10.1Q16  D20.8D4  D10⋊SD16  C52C8⋊D4  D4.D20  D104Q16  Q8.D20  D10⋊Q16  C52C8.D4  D10.12SD16  C88D20  C20.(C4○D4)  C8.2D20  D10.8Q16  C83D20  D102Q16  C2.D87D5  C10.2+ 1+4  C10.102+ 1+4  C10.52- 1+4  C428D10  C42.92D10  C42.94D10  C42.98D10  D45D20  D46D20  C42.229D10  C42.115D10  Q8×D20  Q85D20  C42.232D10  C42.134D10  Dic1019D4  C4⋊C421D10  C10.392+ 1+4  D2020D4  D5×C22⋊Q8  C10.162- 1+4  D2022D4  Dic1021D4  C10.512+ 1+4  C10.1182+ 1+4  C10.242- 1+4  C10.262- 1+4  C10.612+ 1+4  C10.1222+ 1+4  C10.852- 1+4  C10.692+ 1+4  C42.148D10  D207Q8  C42.237D10  C42.150D10  C42.152D10  C42.155D10  C42.157D10  C4224D10  C42.161D10  C42.164D10  C42.165D10  C42.241D10  D209Q8  C42.177D10  C42.178D10  C60.68D4  D3010Q8  D102Dic6  D304Q8  D306Q8
D102Q8 is a maximal quotient of
C2.(C4×D20)  (C2×Dic5)⋊Q8  (C2×C20).28D4  D103(C4⋊C4)  (C2×C4).20D20  (C2×C20).33D4  Dic10.3Q8  D203Q8  D204Q8  D20.3Q8  C20.7Q16  Dic104Q8  (C2×Dic5)⋊6Q8  (C2×C20).53D4  C206(C4⋊C4)  D104(C4⋊C4)  (C2×C20).56D4  C60.68D4  D3010Q8  D102Dic6  D304Q8  D306Q8

34 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F4G4H5A5B10A···10F20A···20L
order122222444444445510···1020···20
size11111010224410102020222···24···4

34 irreducible representations

dim11111122222244
type+++++++-+++--
imageC1C2C2C2C2C2D4Q8D5C4○D4D10D20D42D5Q8×D5
kernelD102Q8C4⋊Dic5D10⋊C4C5×C4⋊C4C2×Dic10C2×C4×D5C20D10C4⋊C4C10C2×C4C4C2C2
# reps12211122226822

Matrix representation of D102Q8 in GL6(𝔽41)

100000
010000
0040000
0004000
000007
0000356
,
4000000
0400000
001000
00324000
0000357
0000366
,
900000
9320000
00323900
000900
000061
0000635
,
1390000
1400000
009000
0013200
0000400
0000040

G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,35,0,0,0,0,7,6],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,32,0,0,0,0,0,40,0,0,0,0,0,0,35,36,0,0,0,0,7,6],[9,9,0,0,0,0,0,32,0,0,0,0,0,0,32,0,0,0,0,0,39,9,0,0,0,0,0,0,6,6,0,0,0,0,1,35],[1,1,0,0,0,0,39,40,0,0,0,0,0,0,9,1,0,0,0,0,0,32,0,0,0,0,0,0,40,0,0,0,0,0,0,40] >;

D102Q8 in GAP, Magma, Sage, TeX

D_{10}\rtimes_2Q_8
% in TeX

G:=Group("D10:2Q8");
// GroupNames label

G:=SmallGroup(160,118);
// by ID

G=gap.SmallGroup(160,118);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-5,103,218,188,122,4613]);
// Polycyclic

G:=Group<a,b,c,d|a^10=b^2=c^4=1,d^2=c^2,b*a*b=c*a*c^-1=a^-1,a*d=d*a,c*b*c^-1=a^3*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

׿
×
𝔽