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G = D4.9D10order 160 = 25·5

4th non-split extension by D4 of D10 acting via D10/C10=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D4.9D10, C20.50D4, Q8.9D10, C20.19C23, Dic10.12C22, D4.D56C2, C4○D4.2D5, C5⋊Q166C2, (C2×C10).10D4, (C2×C4).23D10, C10.61(C2×D4), C55(C8.C22), C4.25(C5⋊D4), C4.Dic510C2, C52C8.4C22, (C5×D4).9C22, C4.19(C22×D5), (C2×Dic10)⋊11C2, (C5×Q8).9C22, (C2×C20).44C22, C22.6(C5⋊D4), (C5×C4○D4).3C2, C2.25(C2×C5⋊D4), SmallGroup(160,172)

Series: Derived Chief Lower central Upper central

C1C20 — D4.9D10
C1C5C10C20Dic10C2×Dic10 — D4.9D10
C5C10C20 — D4.9D10
C1C2C2×C4C4○D4

Generators and relations for D4.9D10
 G = < a,b,c,d | a4=b2=c10=1, d2=a2, bab=dad-1=a-1, ac=ca, cbc-1=a2b, dbd-1=a-1b, dcd-1=c-1 >

Subgroups: 160 in 60 conjugacy classes, 29 normal (21 characteristic)
C1, C2, C2 [×2], C4 [×2], C4 [×3], C22, C22, C5, C8 [×2], C2×C4, C2×C4 [×2], D4, D4, Q8, Q8 [×3], C10, C10 [×2], M4(2), SD16 [×2], Q16 [×2], C2×Q8, C4○D4, Dic5 [×2], C20 [×2], C20, C2×C10, C2×C10, C8.C22, C52C8 [×2], Dic10 [×2], Dic10, C2×Dic5, C2×C20, C2×C20, C5×D4, C5×D4, C5×Q8, C4.Dic5, D4.D5 [×2], C5⋊Q16 [×2], C2×Dic10, C5×C4○D4, D4.9D10
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, D5, C2×D4, D10 [×3], C8.C22, C5⋊D4 [×2], C22×D5, C2×C5⋊D4, D4.9D10

Smallest permutation representation of D4.9D10
On 80 points
Generators in S80
(1 41 33 15)(2 42 34 16)(3 43 35 17)(4 44 36 18)(5 45 37 19)(6 46 38 20)(7 47 39 11)(8 48 40 12)(9 49 31 13)(10 50 32 14)(21 62 53 78)(22 63 54 79)(23 64 55 80)(24 65 56 71)(25 66 57 72)(26 67 58 73)(27 68 59 74)(28 69 60 75)(29 70 51 76)(30 61 52 77)
(1 15)(2 42)(3 17)(4 44)(5 19)(6 46)(7 11)(8 48)(9 13)(10 50)(12 40)(14 32)(16 34)(18 36)(20 38)(21 53)(23 55)(25 57)(27 59)(29 51)(31 49)(33 41)(35 43)(37 45)(39 47)(61 77)(63 79)(65 71)(67 73)(69 75)
(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 76 33 70)(2 75 34 69)(3 74 35 68)(4 73 36 67)(5 72 37 66)(6 71 38 65)(7 80 39 64)(8 79 40 63)(9 78 31 62)(10 77 32 61)(11 23 47 55)(12 22 48 54)(13 21 49 53)(14 30 50 52)(15 29 41 51)(16 28 42 60)(17 27 43 59)(18 26 44 58)(19 25 45 57)(20 24 46 56)

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

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

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

D4.9D10 is a maximal subgroup of
M4(2)⋊D10  D4.9D20  D4.3D20  D4.5D20  M4(2).13D10  M4(2).16D10  2+ 1+4.D5  2- 1+4.2D5  D811D10  D20.47D4  SD16⋊D10  D5×C8.C22  C20.C24  D20.33C23  D20.35C23  D12.37D10  D12.33D10  C60.10C23  Dic10.26D6  D4.9D30  C20.2S4
D4.9D10 is a maximal quotient of
C20.64(C4⋊C4)  C4⋊C4.233D10  C4.(C2×D20)  C20.50D8  C42.51D10  D4.2D20  C20.48SD16  C42.59D10  C207Q16  (C2×C10).D8  Dic1017D4  C4.(D4×D5)  C22⋊Q8.D5  Dic10.37D4  C5⋊(C8.D4)  C42.61D10  C42.62D10  C42.65D10  Dic10.4Q8  C42.68D10  C42.71D10  C4○D4⋊Dic5  (C5×D4).32D4  D12.37D10  D12.33D10  C60.10C23  Dic10.26D6  D4.9D30

31 conjugacy classes

class 1 2A2B2C4A4B4C4D4E5A5B8A8B10A10B10C···10H20A20B20C20D20E···20J
order1222444445588101010···102020202020···20
size11242242020222020224···422224···4

31 irreducible representations

dim1111112222222244
type++++++++++++--
imageC1C2C2C2C2C2D4D4D5D10D10D10C5⋊D4C5⋊D4C8.C22D4.9D10
kernelD4.9D10C4.Dic5D4.D5C5⋊Q16C2×Dic10C5×C4○D4C20C2×C10C4○D4C2×C4D4Q8C4C22C5C1
# reps1122111122224414

Matrix representation of D4.9D10 in GL4(𝔽41) generated by

10028
011313
33400
380040
,
10028
011313
00400
00040
,
25141821
2913538
39401427
7401430
,
25362910
27162231
3737300
24283611
G:=sub<GL(4,GF(41))| [1,0,3,38,0,1,3,0,0,13,40,0,28,13,0,40],[1,0,0,0,0,1,0,0,0,13,40,0,28,13,0,40],[25,29,39,7,14,13,40,40,18,5,14,14,21,38,27,30],[25,27,37,24,36,16,37,28,29,22,30,36,10,31,0,11] >;

D4.9D10 in GAP, Magma, Sage, TeX

D_4._9D_{10}
% in TeX

G:=Group("D4.9D10");
// GroupNames label

G:=SmallGroup(160,172);
// by ID

G=gap.SmallGroup(160,172);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-5,96,218,188,579,159,69,4613]);
// Polycyclic

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

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