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G = D4.10D20order 320 = 26·5

5th non-split extension by D4 of D20 acting via D20/D10=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D4.10D20, D20.35D4, Q8.10D20, C42.26D10, Dic10.35D4, M4(2).8D10, C4≀C24D5, (C5×D4).5D4, C20.6(C2×D4), (C5×Q8).5D4, C8⋊D109C2, C4○D4.4D10, C4.128(D4×D5), C4.12(C2×D20), D4⋊D102C2, C52(D4.8D4), (C2×Dic5).3D4, D204C410C2, C22.32(D4×D5), C10.30C22≀C2, C4.12D202C2, C4.D2011C2, (C4×C20).53C22, D4.10D101C2, (C2×C20).267C23, C4○D20.16C22, (C2×D20).75C22, C2.33(C22⋊D20), (C5×M4(2)).5C22, C4.Dic5.11C22, (C2×Dic10).81C22, (C5×C4≀C2)⋊4C2, (C2×C10).29(C2×D4), (C5×C4○D4).8C22, (C2×C4).112(C22×D5), SmallGroup(320,454)

Series: Derived Chief Lower central Upper central

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

Generators and relations for D4.10D20
 G = < a,b,c,d | a20=b2=c4=1, d2=a10, bab=a-1, ac=ca, ad=da, cbc-1=a15b, bd=db, dcd-1=a5c-1 >

Subgroups: 686 in 146 conjugacy classes, 39 normal (37 characteristic)
C1, C2, C2 [×4], C4 [×2], C4 [×5], C22, C22 [×5], C5, C8 [×2], C2×C4, C2×C4 [×9], D4, D4 [×7], Q8, Q8 [×5], C23, D5 [×2], C10, C10 [×2], C42, C22⋊C4 [×2], M4(2), M4(2), D8 [×2], SD16 [×2], C2×D4, C2×Q8 [×3], C4○D4, C4○D4 [×5], Dic5 [×3], C20 [×2], C20 [×2], D10 [×4], C2×C10, C2×C10, C4.10D4, C4≀C2, C4≀C2, C4.4D4, C8⋊C22 [×2], 2- 1+4, C52C8, C40, Dic10, Dic10 [×4], C4×D5 [×3], D20, D20 [×2], C2×Dic5 [×2], C2×Dic5 [×2], C5⋊D4 [×3], C2×C20, C2×C20 [×2], C5×D4, C5×D4, C5×Q8, C22×D5, D4.8D4, C40⋊C2, D40, C4.Dic5, D10⋊C4 [×2], D4⋊D5, Q8⋊D5, C4×C20, C5×M4(2), C2×Dic10, C2×Dic10, C2×D20, C4○D20, C4○D20, D42D5 [×3], Q8×D5, C5×C4○D4, D204C4, C4.12D20, C5×C4≀C2, C4.D20, C8⋊D10, D4⋊D10, D4.10D10, D4.10D20
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, D5, C2×D4 [×3], D10 [×3], C22≀C2, D20 [×2], C22×D5, D4.8D4, C2×D20, D4×D5 [×2], C22⋊D20, D4.10D20

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

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

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

44 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F4G4H5A5B8A8B10A10B10C10D10E10F20A20B20C20D20E···20N20O20P40A40B40C40D
order1222224444444455881010101010102020202020···20202040404040
size11242040224442020202284022448822224···4888888

44 irreducible representations

dim11111111222222222224444
type+++++++++++++++++++++
imageC1C2C2C2C2C2C2C2D4D4D4D4D4D5D10D10D10D20D20D4.8D4D4×D5D4×D5D4.10D20
kernelD4.10D20D204C4C4.12D20C5×C4≀C2C4.D20C8⋊D10D4⋊D10D4.10D10Dic10D20C2×Dic5C5×D4C5×Q8C4≀C2C42M4(2)C4○D4D4Q8C5C4C22C1
# reps11111111112112222442228

Matrix representation of D4.10D20 in GL4(𝔽41) generated by

21400
111600
00214
001116
,
00394
00302
39400
30200
,
18500
12300
00320
00032
,
002336
004018
18500
12300
G:=sub<GL(4,GF(41))| [2,11,0,0,14,16,0,0,0,0,2,11,0,0,14,16],[0,0,39,30,0,0,4,2,39,30,0,0,4,2,0,0],[18,1,0,0,5,23,0,0,0,0,32,0,0,0,0,32],[0,0,18,1,0,0,5,23,23,40,0,0,36,18,0,0] >;

D4.10D20 in GAP, Magma, Sage, TeX

D_4._{10}D_{20}
% in TeX

G:=Group("D4.10D20");
// GroupNames label

G:=SmallGroup(320,454);
// by ID

G=gap.SmallGroup(320,454);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,254,219,226,1123,136,851,438,102,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^20=b^2=c^4=1,d^2=a^10,b*a*b=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=a^15*b,b*d=d*b,d*c*d^-1=a^5*c^-1>;
// generators/relations

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