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G = C20:7D4  order 160 = 25·5

1st semidirect product of C20 and D4 acting via D4/C22=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C20:7D4, C22:1D20, C23.24D10, (C2xC10):5D4, (C2xD20):6C2, C5:3(C4:D4), C4:3(C5:D4), C4:Dic5:9C2, (C22xC4):4D5, (C22xC20):6C2, C2.17(C2xD20), C10.43(C2xD4), (C2xC4).85D10, D10:C4:3C2, C10.19(C4oD4), C2.19(C4oD20), (C2xC10).48C23, (C2xC20).94C22, C22.56(C22xD5), (C22xC10).40C22, (C2xDic5).16C22, (C22xD5).10C22, (C2xC5:D4):3C2, C2.7(C2xC5:D4), SmallGroup(160,151)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2xC10 — C20:7D4
Chief series
C1 — C5 — C10 — C2xC10 — C22xD5 — C2xD20 — C20:7D4
Lower central
C5 — C2xC10 — C20:7D4
Upper central
C1 — C22 — C22xC4

Generators and relations for C20:7D4
 G = < a,b,c | a20=b4=c2=1, bab-1=cac=a-1, cbc=b-1 >

Subgroups: 336 in 94 conjugacy classes, 37 normal (21 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C5, C2xC4, C2xC4, D4, C23, C23, D5, C10, C10, C22:C4, C4:C4, C22xC4, C2xD4, Dic5, C20, C20, D10, C2xC10, C2xC10, C2xC10, C4:D4, D20, C2xDic5, C5:D4, C2xC20, C2xC20, C22xD5, C22xC10, C4:Dic5, D10:C4, C2xD20, C2xC5:D4, C22xC20, C20:7D4
Quotients: C1, C2, C22, D4, C23, D5, C2xD4, C4oD4, D10, C4:D4, D20, C5:D4, C22xD5, C2xD20, C4oD20, C2xC5:D4, C20:7D4

Smallest permutation representation of C20:7D4
►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 48 61 26)(2 47 62 25)(3 46 63 24)(4 45 64 23)(5 44 65 22)(6 43 66 21)(7 42 67 40)(8 41 68 39)(9 60 69 38)(10 59 70 37)(11 58 71 36)(12 57 72 35)(13 56 73 34)(14 55 74 33)(15 54 75 32)(16 53 76 31)(17 52 77 30)(18 51 78 29)(19 50 79 28)(20 49 80 27)

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

C20:7D4 is a maximal subgroup of
 (C2xD20):C4  C22.2D40  D20:13D4  D20:14D4  C23.38D20  C22.D40  C23.13D20  Dic10:14D4  (C2xC10).40D8  C4:C4.228D10  C4:C4.236D10  (C2xC10):D8  C4:D4:D5  C5:2C8:24D4  C22:Q8:D5  C40:30D4  C40:29D4  C40:2D4  C40:3D4  (C2xC10):8D8  (C5xQ8):13D4  (C5xD4):14D4  C42.276D10  C42.277D10  C24.27D10  C23:3D20  C24.30D10  C10.2- 1+4  C10.2+ 1+4  C10.112+ 1+4  C10.62- 1+4  C42.95D10  C42.97D10  C42.99D10  C42.100D10  C42.104D10  D4xD20  D20:23D4  Dic10:23D4  Dic10:24D4  D4:5D20  D4:6D20  C42:17D10  C42.116D10  C42.117D10  C42.119D10  C20:(C4oD4)  C10.682- 1+4  D5xC4:D4  C10.372+ 1+4  C10.382+ 1+4  C10.472+ 1+4  C10.482+ 1+4  C22:Q8:25D5  C4:C4:26D10  C10.172- 1+4  C10.242- 1+4  C10.562+ 1+4  C10.572+ 1+4  C10.262- 1+4  C10.612+ 1+4  C10.662+ 1+4  C10.682+ 1+4  C10.692+ 1+4  C24.72D10  D4xC5:D4  C10.452- 1+4  C10.1452+ 1+4  C10.1462+ 1+4  C10.1472+ 1+4  C10.1482+ 1+4  D6:D20  C60:4D4  C60:6D4  (C2xC6):D20  C60:29D4  C20:S4
C20:7D4 is a maximal quotient of
 C20:7(C4:C4)  (C2xC4):6D20  (C2xC42):D5  C23.14D20  C23:2D20  C24.16D10  (C2xC20).53D4  (C2xC4):3D20  (C2xC20).56D4  C20:7D8  D4.1D20  D4.2D20  Q8:D20  Q8.1D20  C20:7Q16  C40:30D4  C40:29D4  C40.82D4  C40:2D4  C40:3D4  C40.4D4  D4.3D20  D4.4D20  D4.5D20  C24.64D10  C24.65D10  D6:D20  C60:4D4  C60:6D4  (C2xC6):D20  C60:29D4

46 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B4C4D4E4F5A5B10A···10N20A···20P
order122222224444445510···1020···20
size111122202022222020222···22···2

46 irreducible representations

dim111111222222222
type++++++++++++
imageC1C2C2C2C2C2D4D4D5C4oD4D10D10C5:D4D20C4oD20
kernelC20:7D4C4:Dic5D10:C4C2xD20C2xC5:D4C22xC20C20C2xC10C22xC4C10C2xC4C23C4C22C2
# reps112121222242888

Matrix representation of C20:7D4 ►in GL4(F41) generated by

1100
333400
001430
00119
,
382000
20300
00400
0071
,
343500
8700
0010
003440
G:=sub<GL(4,GF(41))| [1,33,0,0,1,34,0,0,0,0,14,11,0,0,30,9],[38,20,0,0,20,3,0,0,0,0,40,7,0,0,0,1],[34,8,0,0,35,7,0,0,0,0,1,34,0,0,0,40] >;
 

C20:7D4 in GAP, Magma, Sage, TeX 

C_{20}\rtimes_7D_4
 
% in TeX
 
G:=Group("C20:7D4");
 
// GroupNames label
 
G:=SmallGroup(160,151);
 
// by ID
 
G=gap.SmallGroup(160,151);
 
# by ID
 
G:=PCGroup([6,-2,-2,-2,-2,-2,-5,217,103,218,4613]);
 
// Polycyclic
 
G:=Group<a,b,c|a^20=b^4=c^2=1,b*a*b^-1=c*a*c=a^-1,c*b*c=b^-1>;
 
// generators/relations
 

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