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G = C4⋊D20order 160 = 25·5

The semidirect product of C4 and D20 acting via D20/D10=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C201D4, C42D20, D102D4, C4⋊C43D5, (C2×D20)⋊4C2, C52(C4⋊D4), C2.13(D4×D5), C2.9(C2×D20), C10.7(C2×D4), (C2×C4).12D10, D10⋊C48C2, (C2×C20).5C22, C10.34(C4○D4), (C2×C10).36C23, C2.6(Q82D5), (C22×D5).7C22, C22.50(C22×D5), (C2×Dic5).34C22, (C2×C4×D5)⋊1C2, (C5×C4⋊C4)⋊6C2, SmallGroup(160,116)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C4⋊D20
C1C5C10C2×C10C22×D5C2×C4×D5 — C4⋊D20
C5C2×C10 — C4⋊D20
C1C22C4⋊C4

Generators and relations for C4⋊D20
 G = < a,b,c | a4=b20=c2=1, bab-1=cac=a-1, cbc=b-1 >

Subgroups: 384 in 94 conjugacy classes, 35 normal (19 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×3], C22, C22 [×10], C5, C2×C4, C2×C4 [×2], C2×C4 [×3], D4 [×6], C23 [×3], D5 [×4], C10 [×3], C22⋊C4 [×2], C4⋊C4, C22×C4, C2×D4 [×3], Dic5, C20 [×2], C20 [×2], D10 [×2], D10 [×8], C2×C10, C4⋊D4, C4×D5 [×2], D20 [×6], C2×Dic5, C2×C20, C2×C20 [×2], C22×D5, C22×D5 [×2], D10⋊C4 [×2], C5×C4⋊C4, C2×C4×D5, C2×D20, C2×D20 [×2], C4⋊D20
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], C23, D5, C2×D4 [×2], C4○D4, D10 [×3], C4⋊D4, D20 [×2], C22×D5, C2×D20, D4×D5, Q82D5, C4⋊D20

Smallest permutation representation of C4⋊D20
On 80 points
Generators in S80
(1 53 63 28)(2 29 64 54)(3 55 65 30)(4 31 66 56)(5 57 67 32)(6 33 68 58)(7 59 69 34)(8 35 70 60)(9 41 71 36)(10 37 72 42)(11 43 73 38)(12 39 74 44)(13 45 75 40)(14 21 76 46)(15 47 77 22)(16 23 78 48)(17 49 79 24)(18 25 80 50)(19 51 61 26)(20 27 62 52)
(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 15)(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)(16 20)(17 19)(21 54)(22 53)(23 52)(24 51)(25 50)(26 49)(27 48)(28 47)(29 46)(30 45)(31 44)(32 43)(33 42)(34 41)(35 60)(36 59)(37 58)(38 57)(39 56)(40 55)(61 79)(62 78)(63 77)(64 76)(65 75)(66 74)(67 73)(68 72)(69 71)

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

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

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

C4⋊D20 is a maximal subgroup of
D10.1D8  D4⋊D20  D10⋊D8  D43D20  C5⋊(C82D4)  Q82D20  D102SD16  D204D4  C5⋊(C8⋊D4)  D10.17SD16  C88D20  C82D20  C4.Q8⋊D5  D10.13D8  C87D20  C2.D8⋊D5  C83D20  C10.2- 1+4  C10.2+ 1+4  C10.112+ 1+4  C428D10  C429D10  C42.95D10  C42.97D10  C42.228D10  D4×D20  D45D20  C42.116D10  Q85D20  Q86D20  C42.131D10  C42.133D10  Dic1020D4  D5×C4⋊D4  C10.382+ 1+4  D2019D4  C4⋊C426D10  C10.172- 1+4  D2021D4  Dic1022D4  C10.562+ 1+4  C10.262- 1+4  C10.1202+ 1+4  C10.1212+ 1+4  C10.662+ 1+4  C10.682+ 1+4  C42.237D10  C42.150D10  C42.153D10  C42.156D10  C42.158D10  C4223D10  C42.163D10  C4225D10  C42.240D10  D2012D4  C42.178D10  C42.179D10  Dic3⋊D20  C127D20  D302D4  C122D20  C4⋊D60
C4⋊D20 is a maximal quotient of
C10.(C4⋊Q8)  (C2×C4)⋊9D20  C10.55(C4×D4)  (C2×C20)⋊5D4  (C2×Dic5)⋊3D4  (C2×C4).21D20  C20⋊SD16  C4⋊D40  D20.19D4  C42.36D10  Dic108D4  C4⋊Dic20  C88D20  C82D20  C8.2D20  C87D20  C83D20  D102Q16  C8.20D20  C8.21D20  C8.24D20  C206(C4⋊C4)  D104(C4⋊C4)  (C2×D20)⋊22C4  (C2×C4)⋊3D20  (C2×C20).56D4  Dic3⋊D20  C127D20  D302D4  C122D20  C4⋊D60

34 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B4C4D4E4F5A5B10A···10F20A···20L
order122222224444445510···1020···20
size11111010202022441010222···24···4

34 irreducible representations

dim1111122222244
type++++++++++++
imageC1C2C2C2C2D4D4D5C4○D4D10D20D4×D5Q82D5
kernelC4⋊D20D10⋊C4C5×C4⋊C4C2×C4×D5C2×D20C20D10C4⋊C4C10C2×C4C4C2C2
# reps1211322226822

Matrix representation of C4⋊D20 in GL4(𝔽41) generated by

1000
0100
00139
00140
,
143900
163000
0010
00140
,
1100
04000
0010
00140
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,1,1,0,0,39,40],[14,16,0,0,39,30,0,0,0,0,1,1,0,0,0,40],[1,0,0,0,1,40,0,0,0,0,1,1,0,0,0,40] >;

C4⋊D20 in GAP, Magma, Sage, TeX

C_4\rtimes D_{20}
% in TeX

G:=Group("C4:D20");
// GroupNames label

G:=SmallGroup(160,116);
// by ID

G=gap.SmallGroup(160,116);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-5,103,218,188,50,4613]);
// Polycyclic

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

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