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G = D20:11D4order 320 = 26·5

4th semidirect product of D20 and D4 acting via D4/C4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D20:11D4, C42:27D10, C10.772+ 1+4, C5:4D42, C4:2(D4xD5), C20:3(C2xD4), D10:8(C2xD4), C4:1D4:6D5, (C4xD20):49C2, (C2xD4):26D10, C20:2D4:36C2, (C4xC20):27C22, C23:D10:27C2, (D4xC10):33C22, C4:Dic5:74C22, C10.95(C22xD4), (C2xC10).261C24, (C2xC20).509C23, (C23xD5):13C22, C2.81(D4:6D10), C23.D5:37C22, D10:C4:70C22, C23.67(C22xD5), (C2xD20).277C22, (C22xC10).75C23, C22.282(C23xD5), (C2xDic5).136C23, (C22xD5).239C23, (C2xD4xD5):20C2, C2.68(C2xD4xD5), (C5xC4:1D4):8C2, (C2xC4xD5):29C22, (C2xC5:D4):27C22, (C2xC4).214(C22xD5), SmallGroup(320,1389)

Series: Derived Chief Lower central Upper central

C1C2xC10 — D20:11D4
C1C5C10C2xC10C22xD5C23xD5C2xD4xD5 — D20:11D4
C5C2xC10 — D20:11D4
C1C22C4:1D4

Generators and relations for D20:11D4
 G = < a,b,c,d | a20=b2=c4=d2=1, bab=a-1, ac=ca, dad=a11, cbc-1=a10b, bd=db, dcd=c-1 >

Subgroups: 1974 in 428 conjugacy classes, 115 normal (13 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C5, C2xC4, C2xC4, C2xC4, D4, C23, C23, D5, C10, C10, C10, C42, C22:C4, C4:C4, C22xC4, C2xD4, C2xD4, C24, Dic5, C20, C20, D10, D10, C2xC10, C2xC10, C4xD4, C22wrC2, C4:D4, C4:1D4, C22xD4, C4xD5, D20, C2xDic5, C5:D4, C2xC20, C2xC20, C5xD4, C22xD5, C22xD5, C22xC10, D42, C4:Dic5, D10:C4, C23.D5, C4xC20, C2xC4xD5, C2xD20, D4xD5, C2xC5:D4, D4xC10, C23xD5, C4xD20, C23:D10, C20:2D4, C5xC4:1D4, C2xD4xD5, D20:11D4
Quotients: C1, C2, C22, D4, C23, D5, C2xD4, C24, D10, C22xD4, 2+ 1+4, C22xD5, D42, D4xD5, C23xD5, C2xD4xD5, D4:6D10, D20:11D4

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

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

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

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

53 conjugacy classes

class 1 2A2B2C2D2E2F2G2H···2O4A4B4C4D4E4F4G4H4I5A5B10A···10F10G···10N20A···20L
order122222222···24444444445510···1010···1020···20
size1111444410···102222420202020222···28···84···4

53 irreducible representations

dim1111112222444
type++++++++++++
imageC1C2C2C2C2C2D4D5D10D102+ 1+4D4xD5D4:6D10
kernelD20:11D4C4xD20C23:D10C20:2D4C5xC4:1D4C2xD4xD5D20C4:1D4C42C2xD4C10C4C2
# reps12441482212184

Matrix representation of D20:11D4 in GL6(F41)

100000
010000
000100
0040600
00004039
000011
,
4000000
0400000
000100
001000
000012
0000040
,
0400000
100000
0040000
0004000
00004039
000011
,
100000
0400000
001000
000100
00004039
000001

G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,40,0,0,0,0,1,6,0,0,0,0,0,0,40,1,0,0,0,0,39,1],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,2,40],[0,1,0,0,0,0,40,0,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,1,0,0,0,0,39,1],[1,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,39,1] >;

D20:11D4 in GAP, Magma, Sage, TeX

D_{20}\rtimes_{11}D_4
% in TeX

G:=Group("D20:11D4");
// GroupNames label

G:=SmallGroup(320,1389);
// by ID

G=gap.SmallGroup(320,1389);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,219,1571,570,297,136,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^20=b^2=c^4=d^2=1,b*a*b=a^-1,a*c=c*a,d*a*d=a^11,c*b*c^-1=a^10*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations

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