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G = D8:11D10order 320 = 26·5

5th semidirect product of D8 and D10 acting via D10/C10=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D8:11D10, Q16:10D10, D20.46D4, SD16:15D10, C20.17C24, C40.43C23, Dic10.46D4, D20.12C23, Dic10.11C23, C4oD8:5D5, C4oD4:2D10, (C2xC8):14D10, C5:D4.2D4, D8:D5:6C2, D4:D5:4C22, (D5xSD16):6C2, C5:3(D4oSD16), C4.144(D4xD5), Q8:D5:3C22, D4:D10:8C2, D4:8D10:6C2, (Q8xD5):2C22, C22.9(D4xD5), (C2xC40):17C22, Q16:D5:6C2, D10.53(C2xD4), C20.350(C2xD4), (C8xD5):10C22, (C5xD8):16C22, C5:2C8.8C23, D4.D5:3C22, (D4xD5).2C22, C5:Q16:2C22, C4.17(C23xD5), C8.17(C22xD5), SD16:3D5:6C2, D4.9D10:7C2, D4:2D5:2C22, C40:C2:21C22, C8:D5:16C22, Dic5.59(C2xD4), (C5xQ16):14C22, D4.11(C22xD5), (C5xD4).11C23, (C4xD5).10C23, D4.10D10:5C2, D20.3C4:10C2, (C5xQ8).11C23, Q8.11(C22xD5), (C2xC20).534C23, C4oD20.55C22, (C5xSD16):16C22, C10.118(C22xD4), C4.Dic5:31C22, Q8:2D5.2C22, (C2xDic10):38C22, (C2xD20).187C22, C2.91(C2xD4xD5), (C5xC4oD8):7C2, (C2xC40:C2):27C2, (C2xC10).14(C2xD4), (C5xC4oD4):4C22, (C2xC4).233(C22xD5), SmallGroup(320,1442)

Series: Derived Chief Lower central Upper central

C1C20 — D8:11D10
C1C5C10C20C4xD5C4oD20D4:8D10 — D8:11D10
C5C10C20 — D8:11D10
C1C2C2xC4C4oD8

Generators and relations for D8:11D10
 G = < a,b,c,d | a8=b2=c10=d2=1, bab=a-1, ac=ca, dad=a3, cbc-1=a4b, dbd=a6b, dcd=c-1 >

Subgroups: 1046 in 258 conjugacy classes, 99 normal (53 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C8, C2xC4, C2xC4, D4, D4, Q8, Q8, C23, D5, C10, C10, C2xC8, C2xC8, M4(2), D8, D8, SD16, SD16, Q16, Q16, C2xD4, C2xQ8, C4oD4, C4oD4, Dic5, Dic5, C20, C20, D10, D10, C2xC10, C2xC10, C8oD4, C2xSD16, C4oD8, C4oD8, C8:C22, C8.C22, 2+ 1+4, 2- 1+4, C5:2C8, C40, Dic10, Dic10, Dic10, C4xD5, C4xD5, D20, D20, D20, C2xDic5, C5:D4, C5:D4, C2xC20, C2xC20, C5xD4, C5xD4, C5xQ8, C22xD5, D4oSD16, C8xD5, C8:D5, C40:C2, C4.Dic5, D4:D5, D4.D5, Q8:D5, C5:Q16, C2xC40, C5xD8, C5xSD16, C5xQ16, C2xDic10, C2xDic10, C2xD20, C2xD20, C4oD20, C4oD20, D4xD5, D4xD5, D4:2D5, D4:2D5, Q8xD5, Q8:2D5, C5xC4oD4, D20.3C4, C2xC40:C2, D8:D5, D5xSD16, SD16:3D5, Q16:D5, D4:D10, D4.9D10, C5xC4oD8, D4:8D10, D4.10D10, D8:11D10
Quotients: C1, C2, C22, D4, C23, D5, C2xD4, C24, D10, C22xD4, C22xD5, D4oSD16, D4xD5, C23xD5, C2xD4xD5, D8:11D10

Smallest permutation representation of D8:11D10
On 80 points
Generators in S80
(1 41 26 78 63 16 40 51)(2 42 27 79 64 17 31 52)(3 43 28 80 65 18 32 53)(4 44 29 71 66 19 33 54)(5 45 30 72 67 20 34 55)(6 46 21 73 68 11 35 56)(7 47 22 74 69 12 36 57)(8 48 23 75 70 13 37 58)(9 49 24 76 61 14 38 59)(10 50 25 77 62 15 39 60)
(1 51)(2 79)(3 53)(4 71)(5 55)(6 73)(7 57)(8 75)(9 59)(10 77)(11 35)(12 22)(13 37)(14 24)(15 39)(16 26)(17 31)(18 28)(19 33)(20 30)(21 46)(23 48)(25 50)(27 42)(29 44)(32 43)(34 45)(36 47)(38 49)(40 41)(52 64)(54 66)(56 68)(58 70)(60 62)(61 76)(63 78)(65 80)(67 72)(69 74)
(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 30)(2 29)(3 28)(4 27)(5 26)(6 25)(7 24)(8 23)(9 22)(10 21)(11 50)(12 49)(13 48)(14 47)(15 46)(16 45)(17 44)(18 43)(19 42)(20 41)(31 66)(32 65)(33 64)(34 63)(35 62)(36 61)(37 70)(38 69)(39 68)(40 67)(51 55)(52 54)(56 60)(57 59)(71 79)(72 78)(73 77)(74 76)

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

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

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

50 conjugacy classes

class 1 2A2B2C2D2E2F2G2H4A4B4C4D4E4F4G4H5A5B8A8B8C8D8E10A10B10C10D10E10F10G10H20A20B20C20D20E20F20G20H20I20J40A···40H
order12222222244444444558888810101010101010102020202020202020202040···40
size11244101020202244101020202222420202244888822224488884···4

50 irreducible representations

dim1111111111112222222224444
type+++++++++++++++++++++++
imageC1C2C2C2C2C2C2C2C2C2C2C2D4D4D4D5D10D10D10D10D10D4oSD16D4xD5D4xD5D8:11D10
kernelD8:11D10D20.3C4C2xC40:C2D8:D5D5xSD16SD16:3D5Q16:D5D4:D10D4.9D10C5xC4oD8D4:8D10D4.10D10Dic10D20C5:D4C4oD8C2xC8D8SD16Q16C4oD4C5C4C22C1
# reps1112222111111122224242228

Matrix representation of D8:11D10 in GL4(F41) generated by

004029
00121
2243917
1739242
,
3917112
2422940
3917224
2421739
,
14272714
14302711
28132714
28192711
,
1700
04000
2144034
03901
G:=sub<GL(4,GF(41))| [0,0,2,17,0,0,24,39,40,12,39,24,29,1,17,2],[39,24,39,24,17,2,17,2,1,29,2,17,12,40,24,39],[14,14,28,28,27,30,13,19,27,27,27,27,14,11,14,11],[1,0,2,0,7,40,14,39,0,0,40,0,0,0,34,1] >;

D8:11D10 in GAP, Magma, Sage, TeX

D_8\rtimes_{11}D_{10}
% in TeX

G:=Group("D8:11D10");
// GroupNames label

G:=SmallGroup(320,1442);
// by ID

G=gap.SmallGroup(320,1442);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,477,387,570,185,136,438,235,102,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^8=b^2=c^10=d^2=1,b*a*b=a^-1,a*c=c*a,d*a*d=a^3,c*b*c^-1=a^4*b,d*b*d=a^6*b,d*c*d=c^-1>;
// generators/relations

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