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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, (C2×C10)⋊5D4, (C2×D20)⋊6C2, C5⋊3(C4⋊D4), C4⋊3(C5⋊D4), C4⋊Dic5⋊9C2, (C22×C4)⋊4D5, (C22×C20)⋊6C2, C2.17(C2×D20), C10.43(C2×D4), (C2×C4).85D10, D10⋊C4⋊3C2, C10.19(C4○D4), C2.19(C4○D20), (C2×C10).48C23, (C2×C20).94C22, C22.56(C22×D5), (C22×C10).40C22, (C2×Dic5).16C22, (C22×D5).10C22, (C2×C5⋊D4)⋊3C2, C2.7(C2×C5⋊D4), SmallGroup(160,151)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2×C10 — C20⋊7D4
Chief series
C1 — C5 — C10 — C2×C10 — C22×D5 — C2×D20 — C20⋊7D4
Lower central
C5 — C2×C10 — C20⋊7D4
Upper central
C1 — C22 — C22×C4

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, C2×C4, C2×C4, D4, C23, C23, D5, C10, C10, C22⋊C4, C4⋊C4, C22×C4, C2×D4, Dic5, C20, C20, D10, C2×C10, C2×C10, C2×C10, C4⋊D4, D20, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C22×D5, C22×C10, C4⋊Dic5, D10⋊C4, C2×D20, C2×C5⋊D4, C22×C20, C20⋊7D4
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C4○D4, D10, C4⋊D4, D20, C5⋊D4, C22×D5, C2×D20, C4○D20, C2×C5⋊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
 (C2×D20)⋊C4  C22.2D40  D20⋊13D4  D20⋊14D4  C23.38D20  C22.D40  C23.13D20  Dic10⋊14D4  (C2×C10).40D8  C4⋊C4.228D10  C4⋊C4.236D10  (C2×C10)⋊D8  C4⋊D4⋊D5  C5⋊2C8⋊24D4  C22⋊Q8⋊D5  C40⋊30D4  C40⋊29D4  C40⋊2D4  C40⋊3D4  (C2×C10)⋊8D8  (C5×Q8)⋊13D4  (C5×D4)⋊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  D4×D20  D20⋊23D4  Dic10⋊23D4  Dic10⋊24D4  D4⋊5D20  D4⋊6D20  C42⋊17D10  C42.116D10  C42.117D10  C42.119D10  C20⋊(C4○D4)  C10.682- 1+4  D5×C4⋊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  D4×C5⋊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  (C2×C6)⋊D20  C60⋊29D4  C20⋊S4
C20⋊7D4 is a maximal quotient of
 C20⋊7(C4⋊C4)  (C2×C4)⋊6D20  (C2×C42)⋊D5  C23.14D20  C23⋊2D20  C24.16D10  (C2×C20).53D4  (C2×C4)⋊3D20  (C2×C20).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  (C2×C6)⋊D20  C60⋊29D4

46 conjugacy classes

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

46 irreducible representations

dim111111222222222
type++++++++++++
imageC1C2C2C2C2C2D4D4D5C4○D4D10D10C5⋊D4D20C4○D20
kernelC20⋊7D4C4⋊Dic5D10⋊C4C2×D20C2×C5⋊D4C22×C20C20C2×C10C22×C4C10C2×C4C23C4C22C2
# 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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