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G = C4×D20order 160 = 25·5

Direct product of C4 and D20

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C4×D20, C205D4, C424D5, C52(C4×D4), C42(C4×D5), C207(C2×C4), (C4×C20)⋊7C2, D102(C2×C4), C2.1(C2×D20), C10.2(C2×D4), C4⋊Dic516C2, (C2×C4).75D10, (C2×D20).11C2, C10.4(C4○D4), C2.3(C4○D20), D10⋊C417C2, (C2×C20).86C22, C10.17(C22×C4), (C2×C10).14C23, C22.11(C22×D5), (C2×Dic5).28C22, (C22×D5).18C22, (C2×C4×D5)⋊7C2, C2.6(C2×C4×D5), SmallGroup(160,94)

Series: Derived Chief Lower central Upper central

C1C10 — C4×D20
C1C5C10C2×C10C22×D5C2×D20 — C4×D20
C5C10 — C4×D20
C1C2×C4C42

Generators and relations for C4×D20
 G = < a,b,c | a4=b20=c2=1, ab=ba, ac=ca, cbc=b-1 >

Subgroups: 312 in 94 conjugacy classes, 45 normal (21 characteristic)
C1, C2 [×3], C2 [×4], C4 [×4], C4 [×3], C22, C22 [×8], C5, C2×C4 [×3], C2×C4 [×6], D4 [×4], C23 [×2], D5 [×4], C10 [×3], C42, C22⋊C4 [×2], C4⋊C4, C22×C4 [×2], C2×D4, Dic5 [×2], C20 [×4], C20, D10 [×4], D10 [×4], C2×C10, C4×D4, C4×D5 [×4], D20 [×4], C2×Dic5 [×2], C2×C20 [×3], C22×D5 [×2], C4⋊Dic5, D10⋊C4 [×2], C4×C20, C2×C4×D5 [×2], C2×D20, C4×D20
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], C23, D5, C22×C4, C2×D4, C4○D4, D10 [×3], C4×D4, C4×D5 [×2], D20 [×2], C22×D5, C2×C4×D5, C2×D20, C4○D20, C4×D20

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

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

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

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

C4×D20 is a maximal subgroup of
D203C8  D204C8  C86D20  C89D20  C42.16D10  D409C4  D205C8  D105M4(2)  C206M4(2)  C20⋊SD16  D203Q8  C4⋊D40  D20.19D4  D204Q8  D20.3Q8  C42.48D10  C42.56D10  D20.23D4  D20.4Q8  C202D8  C205SD16  D205Q8  D206Q8  C42.276D10  C42.277D10  C427D10  C42.91D10  C428D10  C4210D10  C42.93D10  C42.95D10  C42.99D10  C42.100D10  C4×D4×D5  C4211D10  C4212D10  C42.228D10  D2023D4  D2024D4  D45D20  D46D20  C42.113D10  C42.116D10  C42.117D10  C42.119D10  C42.126D10  Q85D20  Q86D20  D2010Q8  C42.131D10  C42.132D10  C42.133D10  C42.135D10  C42.136D10  D2010D4  Dic1010D4  C4220D10  C42.143D10  D207Q8  C42.150D10  C42.152D10  C42.153D10  C4223D10  C4224D10  C42.161D10  C42.163D10  D2011D4  Dic1011D4  D2012D4  D208Q8  D209Q8  C42.177D10  C42.179D10  Dic34D20  D6014C4
C4×D20 is a maximal quotient of
C2.(C4×D20)  (C2×C4)⋊9D20  D103(C4⋊C4)  C10.55(C4×D4)  C86D20  D4017C4  C89D20  C42.16D10  D409C4  Dic209C4  D4010C4  C207(C4⋊C4)  (C2×C4)⋊6D20  (C2×C42)⋊D5  Dic34D20  D6014C4

52 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B4C4D4E4F4G4H4I4J4K4L5A5B10A···10F20A···20X
order122222224444444444445510···1020···20
size1111101010101111222210101010222···22···2

52 irreducible representations

dim11111112222222
type++++++++++
imageC1C2C2C2C2C2C4D4D5C4○D4D10C4×D5D20C4○D20
kernelC4×D20C4⋊Dic5D10⋊C4C4×C20C2×C4×D5C2×D20D20C20C42C10C2×C4C4C4C2
# reps11212182226888

Matrix representation of C4×D20 in GL3(𝔽41) generated by

3200
090
009
,
100
03230
01127
,
4000
0400
071
G:=sub<GL(3,GF(41))| [32,0,0,0,9,0,0,0,9],[1,0,0,0,32,11,0,30,27],[40,0,0,0,40,7,0,0,1] >;

C4×D20 in GAP, Magma, Sage, TeX

C_4\times D_{20}
% in TeX

G:=Group("C4xD20");
// GroupNames label

G:=SmallGroup(160,94);
// by ID

G=gap.SmallGroup(160,94);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-5,217,103,50,4613]);
// Polycyclic

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

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