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G = D20.15D4order 320 = 26·5

15th non-split extension by D20 of D4 acting via D4/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D20.15D4, C42.86D10, Dic10.15D4, C4⋊Q88D5, C4.58(D4×D5), (C2×C20).12D4, C20.41(C2×D4), (C2×Q8).49D10, D204C414C2, C10.54C22≀C2, C20.C233C2, C20.10D46C2, C53(D4.10D4), (C4×C20).142C22, (C2×C20).413C23, C4○D20.22C22, (Q8×C10).67C22, C2.22(C23⋊D10), Q8.10D10.2C2, C4.Dic5.15C22, (C5×C4⋊Q8)⋊8C2, (C2×C10).544(C2×D4), (C2×C4).11(C5⋊D4), C22.34(C2×C5⋊D4), (C2×C4).119(C22×D5), SmallGroup(320,722)

Series: Derived Chief Lower central Upper central

C1C2×C20 — D20.15D4
C1C5C10C20C2×C20C4○D20Q8.10D10 — D20.15D4
C5C10C2×C20 — D20.15D4
C1C2C2×C4C4⋊Q8

Generators and relations for D20.15D4
 G = < a,b,c,d | a20=b2=d2=1, c4=a10, bab=cac-1=a-1, dad=a9, cbc-1=a3b, dbd=a18b, dcd=c3 >

Subgroups: 558 in 142 conjugacy classes, 39 normal (19 characteristic)
C1, C2, C2 [×3], C4 [×2], C4 [×7], C22, C22 [×2], C5, C8 [×2], C2×C4, C2×C4 [×2], C2×C4 [×8], D4 [×6], Q8 [×8], D5 [×2], C10, C10, C42, C4⋊C4 [×2], M4(2) [×2], SD16 [×2], Q16 [×2], C2×Q8 [×2], C2×Q8 [×2], C4○D4 [×6], Dic5 [×2], C20 [×2], C20 [×5], D10 [×2], C2×C10, C4.10D4, C4≀C2 [×2], C4⋊Q8, C8.C22 [×2], 2- 1+4, C52C8 [×2], Dic10 [×2], Dic10 [×2], C4×D5 [×6], D20 [×2], D20 [×2], C5⋊D4 [×2], C2×C20, C2×C20 [×2], C2×C20 [×2], C5×Q8 [×4], D4.10D4, C4.Dic5 [×2], Q8⋊D5 [×2], C5⋊Q16 [×2], C4×C20, C5×C4⋊C4 [×2], C4○D20 [×2], C4○D20 [×2], Q8×D5 [×2], Q82D5 [×2], Q8×C10 [×2], D204C4 [×2], C20.10D4, C20.C23 [×2], C5×C4⋊Q8, Q8.10D10, D20.15D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, D5, C2×D4 [×3], D10 [×3], C22≀C2, C5⋊D4 [×2], C22×D5, D4.10D4, D4×D5 [×2], C2×C5⋊D4, C23⋊D10, D20.15D4

Smallest permutation representation of D20.15D4
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 65)(2 64)(3 63)(4 62)(5 61)(6 80)(7 79)(8 78)(9 77)(10 76)(11 75)(12 74)(13 73)(14 72)(15 71)(16 70)(17 69)(18 68)(19 67)(20 66)(21 55)(22 54)(23 53)(24 52)(25 51)(26 50)(27 49)(28 48)(29 47)(30 46)(31 45)(32 44)(33 43)(34 42)(35 41)(36 60)(37 59)(38 58)(39 57)(40 56)
(1 76 16 61 11 66 6 71)(2 75 17 80 12 65 7 70)(3 74 18 79 13 64 8 69)(4 73 19 78 14 63 9 68)(5 72 20 77 15 62 10 67)(21 46 36 51 31 56 26 41)(22 45 37 50 32 55 27 60)(23 44 38 49 33 54 28 59)(24 43 39 48 34 53 29 58)(25 42 40 47 35 52 30 57)
(1 46)(2 55)(3 44)(4 53)(5 42)(6 51)(7 60)(8 49)(9 58)(10 47)(11 56)(12 45)(13 54)(14 43)(15 52)(16 41)(17 50)(18 59)(19 48)(20 57)(21 66)(22 75)(23 64)(24 73)(25 62)(26 71)(27 80)(28 69)(29 78)(30 67)(31 76)(32 65)(33 74)(34 63)(35 72)(36 61)(37 70)(38 79)(39 68)(40 77)

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

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

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

44 conjugacy classes

class 1 2A2B2C2D4A4B4C4D4E4F4G4H4I5A5B8A8B10A···10F20A···20L20M···20T
order12222444444444558810···1020···2020···20
size1122020224444820202240402···24···48···8

44 irreducible representations

dim1111112222222444
type++++++++++++-+
imageC1C2C2C2C2C2D4D4D4D5D10D10C5⋊D4D4.10D4D4×D5D20.15D4
kernelD20.15D4D204C4C20.10D4C20.C23C5×C4⋊Q8Q8.10D10Dic10D20C2×C20C4⋊Q8C42C2×Q8C2×C4C5C4C1
# reps1212112222248248

Matrix representation of D20.15D4 in GL4(𝔽41) generated by

03700
4000
00010
00310
,
002924
002412
282600
261300
,
003414
00147
273400
341400
,
0001
00400
04000
1000
G:=sub<GL(4,GF(41))| [0,4,0,0,37,0,0,0,0,0,0,31,0,0,10,0],[0,0,28,26,0,0,26,13,29,24,0,0,24,12,0,0],[0,0,27,34,0,0,34,14,34,14,0,0,14,7,0,0],[0,0,0,1,0,0,40,0,0,40,0,0,1,0,0,0] >;

D20.15D4 in GAP, Magma, Sage, TeX

D_{20}._{15}D_4
% in TeX

G:=Group("D20.15D4");
// GroupNames label

G:=SmallGroup(320,722);
// by ID

G=gap.SmallGroup(320,722);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,254,219,184,1123,570,297,136,1684,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^20=b^2=d^2=1,c^4=a^10,b*a*b=c*a*c^-1=a^-1,d*a*d=a^9,c*b*c^-1=a^3*b,d*b*d=a^18*b,d*c*d=c^3>;
// generators/relations

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