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G = D4.9D20order 320 = 26·5

4th non-split extension by D4 of D20 acting via D20/D10=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D4.9D20, Q8.9D20, D20.34D4, C42.25D10, Dic10.34D4, M4(2).7D10, C4≀C23D5, (C5×D4).4D4, C20.5(C2×D4), (C5×Q8).4D4, C4○D4.3D10, C4.127(D4×D5), C4.11(C2×D20), D204C49C2, C202Q810C2, C8.D109C2, (C2×Dic5).2D4, C22.31(D4×D5), C10.29C22≀C2, D4.9D102C2, C4.12D201C2, C52(D4.10D4), (C4×C20).52C22, (C2×C20).266C23, C4○D20.15C22, C2.32(C22⋊D20), D4.10D10.1C2, (C5×M4(2)).4C22, C4.Dic5.10C22, (C2×Dic10).80C22, (C5×C4≀C2)⋊3C2, (C2×C10).28(C2×D4), (C5×C4○D4).7C22, (C2×C4).111(C22×D5), SmallGroup(320,453)

Series: Derived Chief Lower central Upper central

C1C2×C20 — D4.9D20
C1C5C10C20C2×C20C4○D20D4.10D10 — D4.9D20
C5C10C2×C20 — D4.9D20
C1C2C2×C4C4≀C2

Generators and relations for D4.9D20
 G = < a,b,c,d | a4=b2=1, c20=d2=a2, bab=cac-1=dad-1=a-1, cbc-1=a-1b, dbd-1=ab, dcd-1=c19 >

Subgroups: 590 in 142 conjugacy classes, 39 normal (37 characteristic)
C1, C2, C2 [×3], C4 [×2], C4 [×7], C22, C22 [×2], C5, C8 [×2], C2×C4, C2×C4 [×10], D4, D4 [×5], Q8, Q8 [×7], D5, C10, C10 [×2], C42, C4⋊C4 [×2], M4(2), M4(2), SD16 [×2], Q16 [×2], C2×Q8 [×4], C4○D4, C4○D4 [×5], Dic5 [×4], C20 [×2], C20 [×3], D10, C2×C10, C2×C10, C4.10D4, C4≀C2, C4≀C2, C4⋊Q8, C8.C22 [×2], 2- 1+4, C52C8, C40, Dic10, Dic10 [×6], C4×D5 [×3], D20, C2×Dic5 [×2], C2×Dic5 [×3], C5⋊D4 [×3], C2×C20, C2×C20 [×2], C5×D4, C5×D4, C5×Q8, D4.10D4, C40⋊C2, Dic20, C4.Dic5, C4⋊Dic5 [×2], D4.D5, C5⋊Q16, C4×C20, C5×M4(2), C2×Dic10 [×2], C2×Dic10, C4○D20, C4○D20, D42D5 [×3], Q8×D5, C5×C4○D4, D204C4, C4.12D20, C5×C4≀C2, C202Q8, C8.D10, D4.9D10, D4.10D10, D4.9D20
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, D5, C2×D4 [×3], D10 [×3], C22≀C2, D20 [×2], C22×D5, D4.10D4, C2×D20, D4×D5 [×2], C22⋊D20, D4.9D20

Smallest permutation representation of D4.9D20
On 80 points
Generators in S80
(1 31 21 11)(2 12 22 32)(3 33 23 13)(4 14 24 34)(5 35 25 15)(6 16 26 36)(7 37 27 17)(8 18 28 38)(9 39 29 19)(10 20 30 40)(41 71 61 51)(42 52 62 72)(43 73 63 53)(44 54 64 74)(45 75 65 55)(46 56 66 76)(47 77 67 57)(48 58 68 78)(49 79 69 59)(50 60 70 80)
(1 36)(2 7)(3 38)(4 9)(5 40)(6 11)(8 13)(10 15)(12 17)(14 19)(16 21)(18 23)(20 25)(22 27)(24 29)(26 31)(28 33)(30 35)(32 37)(34 39)(41 76)(42 47)(43 78)(44 49)(45 80)(46 51)(48 53)(50 55)(52 57)(54 59)(56 61)(58 63)(60 65)(62 67)(64 69)(66 71)(68 73)(70 75)(72 77)(74 79)
(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 55 21 75)(2 74 22 54)(3 53 23 73)(4 72 24 52)(5 51 25 71)(6 70 26 50)(7 49 27 69)(8 68 28 48)(9 47 29 67)(10 66 30 46)(11 45 31 65)(12 64 32 44)(13 43 33 63)(14 62 34 42)(15 41 35 61)(16 60 36 80)(17 79 37 59)(18 58 38 78)(19 77 39 57)(20 56 40 76)

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

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

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

44 conjugacy classes

class 1 2A2B2C2D4A4B4C4D4E4F4G4H4I5A5B8A8B10A10B10C10D10E10F20A20B20C20D20E···20N20O20P40A40B40C40D
order1222244444444455881010101010102020202020···20202040404040
size11242022444202020402284022448822224···4888888

44 irreducible representations

dim11111111222222222224444
type+++++++++++++++++++-++-
imageC1C2C2C2C2C2C2C2D4D4D4D4D4D5D10D10D10D20D20D4.10D4D4×D5D4×D5D4.9D20
kernelD4.9D20D204C4C4.12D20C5×C4≀C2C202Q8C8.D10D4.9D10D4.10D10Dic10D20C2×Dic5C5×D4C5×Q8C4≀C2C42M4(2)C4○D4D4Q8C5C4C22C1
# reps11111111112112222442228

Matrix representation of D4.9D20 in GL4(𝔽41) generated by

113200
93000
00309
003211
,
00400
00040
40000
04000
,
00341
00400
273000
113200
,
192200
322200
0033
002438
G:=sub<GL(4,GF(41))| [11,9,0,0,32,30,0,0,0,0,30,32,0,0,9,11],[0,0,40,0,0,0,0,40,40,0,0,0,0,40,0,0],[0,0,27,11,0,0,30,32,34,40,0,0,1,0,0,0],[19,32,0,0,22,22,0,0,0,0,3,24,0,0,3,38] >;

D4.9D20 in GAP, Magma, Sage, TeX

D_4._9D_{20}
% in TeX

G:=Group("D4.9D20");
// GroupNames label

G:=SmallGroup(320,453);
// by ID

G=gap.SmallGroup(320,453);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,254,219,58,1123,136,851,438,102,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^2=1,c^20=d^2=a^2,b*a*b=c*a*c^-1=d*a*d^-1=a^-1,c*b*c^-1=a^-1*b,d*b*d^-1=a*b,d*c*d^-1=c^19>;
// generators/relations

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