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G = D2013D4order 320 = 26·5

1st semidirect product of D20 and D4 acting via D4/C22=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D2013D4, C222D40, C23.37D20, (C2×C10)⋊1D8, (C2×C8)⋊1D10, (C2×D40)⋊2C2, C22⋊C83D5, C2.7(C2×D40), C10.5(C2×D8), C207D41C2, C51(C22⋊D8), (C2×C40)⋊1C22, (C2×C4).32D20, (C2×C20).43D4, C4.120(D4×D5), D205C44C2, C10.8C22≀C2, C20.332(C2×D4), (C2×D20)⋊1C22, (C22×D20)⋊2C2, C4⋊Dic52C22, C10.9(C8⋊C22), (C22×C4).82D10, (C22×C10).52D4, C2.12(C8⋊D10), (C2×C20).742C23, C22.105(C2×D20), C2.11(C22⋊D20), (C22×C20).51C22, (C5×C22⋊C8)⋊5C2, (C2×C10).125(C2×D4), (C2×C4).687(C22×D5), SmallGroup(320,359)

Series: Derived Chief Lower central Upper central

C1C2×C20 — D2013D4
C1C5C10C20C2×C20C2×D20C22×D20 — D2013D4
C5C10C2×C20 — D2013D4
C1C22C22×C4C22⋊C8

Generators and relations for D2013D4
 G = < a,b,c,d | a20=b2=c4=d2=1, bab=cac-1=a-1, ad=da, cbc-1=a3b, bd=db, dcd=c-1 >

Subgroups: 1214 in 198 conjugacy classes, 47 normal (25 characteristic)
C1, C2 [×3], C2 [×7], C4 [×2], C4 [×2], C22, C22 [×2], C22 [×21], C5, C8 [×2], C2×C4 [×2], C2×C4 [×3], D4 [×14], C23, C23 [×11], D5 [×5], C10 [×3], C10 [×2], C22⋊C4, C4⋊C4, C2×C8 [×2], D8 [×4], C22×C4, C2×D4 [×9], C24, Dic5, C20 [×2], C20, D10 [×19], C2×C10, C2×C10 [×2], C2×C10 [×2], C22⋊C8, D4⋊C4 [×2], C4⋊D4, C2×D8 [×2], C22×D4, C40 [×2], D20 [×4], D20 [×8], C2×Dic5, C5⋊D4 [×2], C2×C20 [×2], C2×C20 [×2], C22×D5 [×11], C22×C10, C22⋊D8, D40 [×4], C4⋊Dic5, D10⋊C4, C2×C40 [×2], C2×D20, C2×D20 [×2], C2×D20 [×5], C2×C5⋊D4, C22×C20, C23×D5, D205C4 [×2], C5×C22⋊C8, C2×D40 [×2], C207D4, C22×D20, D2013D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, D5, D8 [×2], C2×D4 [×3], D10 [×3], C22≀C2, C2×D8, C8⋊C22, D20 [×2], C22×D5, C22⋊D8, D40 [×2], C2×D20, D4×D5 [×2], C22⋊D20, C2×D40, C8⋊D10, D2013D4

Smallest permutation representation of D2013D4
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 20)(2 19)(3 18)(4 17)(5 16)(6 15)(7 14)(8 13)(9 12)(10 11)(22 40)(23 39)(24 38)(25 37)(26 36)(27 35)(28 34)(29 33)(30 32)(41 43)(44 60)(45 59)(46 58)(47 57)(48 56)(49 55)(50 54)(51 53)(61 72)(62 71)(63 70)(64 69)(65 68)(66 67)(73 80)(74 79)(75 78)(76 77)
(1 29 77 50)(2 28 78 49)(3 27 79 48)(4 26 80 47)(5 25 61 46)(6 24 62 45)(7 23 63 44)(8 22 64 43)(9 21 65 42)(10 40 66 41)(11 39 67 60)(12 38 68 59)(13 37 69 58)(14 36 70 57)(15 35 71 56)(16 34 72 55)(17 33 73 54)(18 32 74 53)(19 31 75 52)(20 30 76 51)
(1 67)(2 68)(3 69)(4 70)(5 71)(6 72)(7 73)(8 74)(9 75)(10 76)(11 77)(12 78)(13 79)(14 80)(15 61)(16 62)(17 63)(18 64)(19 65)(20 66)(21 31)(22 32)(23 33)(24 34)(25 35)(26 36)(27 37)(28 38)(29 39)(30 40)(41 51)(42 52)(43 53)(44 54)(45 55)(46 56)(47 57)(48 58)(49 59)(50 60)

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

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

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

59 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I2J4A4B4C4D5A5B8A8B8C8D10A···10F10G10H10I10J20A···20H20I20J20K20L40A···40P
order12222222222444455888810···101010101020···202020202040···40
size1111222020202040224402244442···244442···244444···4

59 irreducible representations

dim1111112222222222444
type+++++++++++++++++++
imageC1C2C2C2C2C2D4D4D4D5D8D10D10D20D20D40C8⋊C22D4×D5C8⋊D10
kernelD2013D4D205C4C5×C22⋊C8C2×D40C207D4C22×D20D20C2×C20C22×C10C22⋊C8C2×C10C2×C8C22×C4C2×C4C23C22C10C4C2
# reps12121141124424416144

Matrix representation of D2013D4 in GL4(𝔽41) generated by

163000
27200
00400
00040
,
393000
4200
00400
00141
,
183900
182300
00284
001913
,
40000
04000
00400
00141
G:=sub<GL(4,GF(41))| [16,27,0,0,30,2,0,0,0,0,40,0,0,0,0,40],[39,4,0,0,30,2,0,0,0,0,40,14,0,0,0,1],[18,18,0,0,39,23,0,0,0,0,28,19,0,0,4,13],[40,0,0,0,0,40,0,0,0,0,40,14,0,0,0,1] >;

D2013D4 in GAP, Magma, Sage, TeX

D_{20}\rtimes_{13}D_4
% in TeX

G:=Group("D20:13D4");
// GroupNames label

G:=SmallGroup(320,359);
// by ID

G=gap.SmallGroup(320,359);
# by ID

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

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

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