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G = C22×D20order 160 = 25·5

Direct product of C22 and D20

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

Aliases: C22×D20, C202C23, D101C23, C10.3C24, C23.35D10, (C2×C10)⋊6D4, C101(C2×D4), (C2×C4)⋊9D10, C51(C22×D4), C42(C22×D5), (C22×C4)⋊5D5, (C22×C20)⋊7C2, (C23×D5)⋊3C2, C2.4(C23×D5), (C2×C20)⋊12C22, (C2×C10).64C23, (C22×D5)⋊5C22, C22.30(C22×D5), (C22×C10).45C22, SmallGroup(160,215)

Series: Derived Chief Lower central Upper central

C1C10 — C22×D20
C1C5C10D10C22×D5C23×D5 — C22×D20
C5C10 — C22×D20
C1C23C22×C4

Generators and relations for C22×D20
 G = < a,b,c,d | a2=b2=c20=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 840 in 236 conjugacy classes, 105 normal (9 characteristic)
C1, C2, C2 [×6], C2 [×8], C4 [×4], C22 [×7], C22 [×32], C5, C2×C4 [×6], D4 [×16], C23, C23 [×20], D5 [×8], C10, C10 [×6], C22×C4, C2×D4 [×12], C24 [×2], C20 [×4], D10 [×8], D10 [×24], C2×C10 [×7], C22×D4, D20 [×16], C2×C20 [×6], C22×D5 [×12], C22×D5 [×8], C22×C10, C2×D20 [×12], C22×C20, C23×D5 [×2], C22×D20
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D5, C2×D4 [×6], C24, D10 [×7], C22×D4, D20 [×4], C22×D5 [×7], C2×D20 [×6], C23×D5, C22×D20

Smallest permutation representation of C22×D20
On 80 points
Generators in S80
(1 64)(2 65)(3 66)(4 67)(5 68)(6 69)(7 70)(8 71)(9 72)(10 73)(11 74)(12 75)(13 76)(14 77)(15 78)(16 79)(17 80)(18 61)(19 62)(20 63)(21 54)(22 55)(23 56)(24 57)(25 58)(26 59)(27 60)(28 41)(29 42)(30 43)(31 44)(32 45)(33 46)(34 47)(35 48)(36 49)(37 50)(38 51)(39 52)(40 53)
(1 37)(2 38)(3 39)(4 40)(5 21)(6 22)(7 23)(8 24)(9 25)(10 26)(11 27)(12 28)(13 29)(14 30)(15 31)(16 32)(17 33)(18 34)(19 35)(20 36)(41 75)(42 76)(43 77)(44 78)(45 79)(46 80)(47 61)(48 62)(49 63)(50 64)(51 65)(52 66)(53 67)(54 68)(55 69)(56 70)(57 71)(58 72)(59 73)(60 74)
(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 78)(2 77)(3 76)(4 75)(5 74)(6 73)(7 72)(8 71)(9 70)(10 69)(11 68)(12 67)(13 66)(14 65)(15 64)(16 63)(17 62)(18 61)(19 80)(20 79)(21 60)(22 59)(23 58)(24 57)(25 56)(26 55)(27 54)(28 53)(29 52)(30 51)(31 50)(32 49)(33 48)(34 47)(35 46)(36 45)(37 44)(38 43)(39 42)(40 41)

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

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

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

C22×D20 is a maximal subgroup of
(C2×C4)⋊9D20  (C2×C20)⋊5D4  (C2×Dic5)⋊3D4  D20.31D4  D2013D4  (C2×C4)⋊6D20  C232D20  (C2×D20)⋊22C4  (C2×C4)⋊3D20  C4⋊C436D10  D2016D4  D20.36D4  C23.48D20  C427D10  C429D10  D2023D4  D2019D4  D2021D4  C10.1202+ 1+4  C10.1462+ 1+4  C22×D4×D5
C22×D20 is a maximal quotient of
C42.276D10  C233D20  C10.2+ 1+4  C428D10  C429D10  C42.92D10  D45D20  D46D20  Q85D20  Q86D20  C40.9C23  D4.11D20  D4.12D20  D4.13D20

52 conjugacy classes

class 1 2A···2G2H···2O4A4B4C4D5A5B10A···10N20A···20P
order12···22···244445510···1020···20
size11···110···102222222···22···2

52 irreducible representations

dim111122222
type+++++++++
imageC1C2C2C2D4D5D10D10D20
kernelC22×D20C2×D20C22×C20C23×D5C2×C10C22×C4C2×C4C23C22
# reps112124212216

Matrix representation of C22×D20 in GL4(𝔽41) generated by

1000
04000
0010
0001
,
40000
0100
00400
00040
,
40000
04000
003230
001127
,
40000
0100
00040
00400
G:=sub<GL(4,GF(41))| [1,0,0,0,0,40,0,0,0,0,1,0,0,0,0,1],[40,0,0,0,0,1,0,0,0,0,40,0,0,0,0,40],[40,0,0,0,0,40,0,0,0,0,32,11,0,0,30,27],[40,0,0,0,0,1,0,0,0,0,0,40,0,0,40,0] >;

C22×D20 in GAP, Magma, Sage, TeX

C_2^2\times D_{20}
% in TeX

G:=Group("C2^2xD20");
// GroupNames label

G:=SmallGroup(160,215);
// by ID

G=gap.SmallGroup(160,215);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-5,579,69,4613]);
// Polycyclic

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

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