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G = D4:5D20order 320 = 26·5

1st semidirect product of D4 and D20 acting through Inn(D4)

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D4:5D20, C42:15D10, C10.172+ 1+4, C4:C4:48D10, (C5xD4):10D4, (C4xD4):16D5, (C4xD20):30C2, (D4xC20):18C2, C20:7D4:9C2, C5:3(D4:5D4), C20.54(C2xD4), C4.22(C2xD20), D10:7(C4oD4), C22:D20:6C2, C4:D20:15C2, (C4xC20):20C22, C22:C4:47D10, (C2xD20):6C22, (C22xC4):13D10, C4:Dic5:8C22, C22.1(C2xD20), D10:2Q8:14C2, (C2xD4).248D10, C4.D20:18C2, (C2xC10).98C24, C10.16(C22xD4), C2.18(C22xD20), (C2xC20).159C23, (C22xC20):10C22, C22.D20:5C2, C2.18(D4:6D10), D10:C4:52C22, (C2xDic10):17C22, (D4xC10).259C22, (C2xDic5).42C23, (C22xDic5):9C22, (C22xD5).33C23, (C23xD5).40C22, C22.123(C23xD5), C23.172(C22xD5), (C22xC10).168C23, (C2xD4xD5):4C2, (C2xC4xD5):3C22, C2.22(D5xC4oD4), (C2xC10).1(C2xD4), (C2xD4:2D5):3C2, (C5xC4:C4):60C22, (C2xC5:D4):4C22, C10.139(C2xC4oD4), (C2xD10:C4):21C2, (C5xC22:C4):50C22, (C2xC4).160(C22xD5), SmallGroup(320,1226)

Series: Derived Chief Lower central Upper central

C1C2xC10 — D4:5D20
C1C5C10C2xC10C22xD5C23xD5C2xD4xD5 — D4:5D20
C5C2xC10 — D4:5D20
C1C22C4xD4

Generators and relations for D4:5D20
 G = < a,b,c,d | a4=b2=c20=d2=1, bab=cac-1=a-1, ad=da, cbc-1=dbd=a2b, dcd=c-1 >

Subgroups: 1462 in 334 conjugacy classes, 113 normal (43 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C5, C2xC4, C2xC4, C2xC4, D4, D4, Q8, C23, C23, D5, C10, C10, C42, C22:C4, C22:C4, C4:C4, C4:C4, C22xC4, C22xC4, C2xD4, C2xD4, C2xQ8, C4oD4, C24, Dic5, C20, C20, D10, D10, C2xC10, C2xC10, C2xC10, C2xC22:C4, C4xD4, C4xD4, C22wrC2, C4:D4, C22:Q8, C22.D4, C4.4D4, C22xD4, C2xC4oD4, Dic10, C4xD5, D20, C2xDic5, C2xDic5, C2xDic5, C5:D4, C2xC20, C2xC20, C2xC20, C5xD4, C22xD5, C22xD5, C22xD5, C22xC10, D4:5D4, C4:Dic5, C4:Dic5, D10:C4, D10:C4, C4xC20, C5xC22:C4, C5xC4:C4, C2xDic10, C2xC4xD5, C2xD20, C2xD20, D4xD5, D4:2D5, C22xDic5, C2xC5:D4, C22xC20, D4xC10, C23xD5, C4xD20, C4.D20, C22:D20, C22.D20, C4:D20, D10:2Q8, C2xD10:C4, C20:7D4, D4xC20, C2xD4xD5, C2xD4:2D5, D4:5D20
Quotients: C1, C2, C22, D4, C23, D5, C2xD4, C4oD4, C24, D10, C22xD4, C2xC4oD4, 2+ 1+4, D20, C22xD5, D4:5D4, C2xD20, C23xD5, C22xD20, D4:6D10, D5xC4oD4, D4:5D20

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

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

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

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

65 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I2J2K2L4A4B4C4D4E4F4G4H4I4J4K4L5A5B10A···10F10G···10N20A···20H20I···20X
order12222222222224444444444445510···1010···1020···2020···20
size11112222101020202022224441010202020222···24···42···24···4

65 irreducible representations

dim111111111111222222222444
type+++++++++++++++++++++
imageC1C2C2C2C2C2C2C2C2C2C2C2D4D5C4oD4D10D10D10D10D10D202+ 1+4D4:6D10D5xC4oD4
kernelD4:5D20C4xD20C4.D20C22:D20C22.D20C4:D20D10:2Q8C2xD10:C4C20:7D4D4xC20C2xD4xD5C2xD4:2D5C5xD4C4xD4D10C42C22:C4C4:C4C22xC4C2xD4D4C10C2C2
# reps1112211221114242424216144

Matrix representation of D4:5D20 in GL6(F41)

100000
010000
001000
000100
00004039
000011
,
100000
010000
001000
000100
00004039
000001
,
1400000
8340000
0013200
00234000
0000320
000099
,
100000
8400000
0013200
0004000
00003223
000099

G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,1,0,0,0,0,39,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,39,1],[1,8,0,0,0,0,40,34,0,0,0,0,0,0,1,23,0,0,0,0,32,40,0,0,0,0,0,0,32,9,0,0,0,0,0,9],[1,8,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,32,40,0,0,0,0,0,0,32,9,0,0,0,0,23,9] >;

D4:5D20 in GAP, Magma, Sage, TeX

D_4\rtimes_5D_{20}
% in TeX

G:=Group("D4:5D20");
// GroupNames label

G:=SmallGroup(320,1226);
// by ID

G=gap.SmallGroup(320,1226);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,387,675,192,12550]);
// Polycyclic

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

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