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G = D5×C42order 160 = 25·5

Direct product of C42 and D5

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

Aliases: D5×C42, C208(C2×C4), (C4×C20)⋊8C2, C52(C2×C42), Dic58(C2×C4), (C2×C4).95D10, (C4×Dic5)⋊17C2, D10.18(C2×C4), (C2×C10).12C23, C10.15(C22×C4), C22.9(C22×D5), (C2×C20).109C22, (C2×Dic5).59C22, (C22×D5).41C22, C2.1(C2×C4×D5), (C2×C4×D5).19C2, SmallGroup(160,92)

Series: Derived Chief Lower central Upper central

C1C5 — D5×C42
C1C5C10C2×C10C22×D5C2×C4×D5 — D5×C42
C5 — D5×C42
C1C42

Generators and relations for D5×C42
 G = < a,b,c,d | a4=b4=c5=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 264 in 108 conjugacy classes, 69 normal (8 characteristic)
C1, C2 [×3], C2 [×4], C4 [×6], C4 [×6], C22, C22 [×6], C5, C2×C4 [×3], C2×C4 [×15], C23, D5 [×4], C10 [×3], C42, C42 [×3], C22×C4 [×3], Dic5 [×6], C20 [×6], D10 [×6], C2×C10, C2×C42, C4×D5 [×12], C2×Dic5 [×3], C2×C20 [×3], C22×D5, C4×Dic5 [×3], C4×C20, C2×C4×D5 [×3], D5×C42
Quotients: C1, C2 [×7], C4 [×12], C22 [×7], C2×C4 [×18], C23, D5, C42 [×4], C22×C4 [×3], D10 [×3], C2×C42, C4×D5 [×6], C22×D5, C2×C4×D5 [×3], D5×C42

Smallest permutation representation of D5×C42
On 80 points
Generators in S80
(1 49 9 44)(2 50 10 45)(3 46 6 41)(4 47 7 42)(5 48 8 43)(11 56 16 51)(12 57 17 52)(13 58 18 53)(14 59 19 54)(15 60 20 55)(21 66 26 61)(22 67 27 62)(23 68 28 63)(24 69 29 64)(25 70 30 65)(31 76 36 71)(32 77 37 72)(33 78 38 73)(34 79 39 74)(35 80 40 75)
(1 39 14 29)(2 40 15 30)(3 36 11 26)(4 37 12 27)(5 38 13 28)(6 31 16 21)(7 32 17 22)(8 33 18 23)(9 34 19 24)(10 35 20 25)(41 76 51 66)(42 77 52 67)(43 78 53 68)(44 79 54 69)(45 80 55 70)(46 71 56 61)(47 72 57 62)(48 73 58 63)(49 74 59 64)(50 75 60 65)
(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 18)(2 17)(3 16)(4 20)(5 19)(6 11)(7 15)(8 14)(9 13)(10 12)(21 36)(22 40)(23 39)(24 38)(25 37)(26 31)(27 35)(28 34)(29 33)(30 32)(41 56)(42 60)(43 59)(44 58)(45 57)(46 51)(47 55)(48 54)(49 53)(50 52)(61 76)(62 80)(63 79)(64 78)(65 77)(66 71)(67 75)(68 74)(69 73)(70 72)

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

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

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

D5×C42 is a maximal subgroup of
C426F5  C42.282D10  C42.182D10  D10.6C42  C42.200D10  C42.202D10  C205M4(2)  C42.5F5  C42.6F5  C42.11F5  C42.12F5  C203M4(2)  C42.14F5  C42.15F5  C42.7F5  C424F5  C428F5  C429F5  C425F5  C42.188D10  C42.93D10  C42.228D10  C42.229D10  C42.232D10  C42.131D10  C42.233D10  C42.234D10  C42.236D10  C42.237D10  C42.189D10  C42.238D10  C42.240D10  C42.241D10
D5×C42 is a maximal quotient of
Dic5.15C42  Dic52C42  D102C42  D10.5C42  D10.6C42  D10.7C42

64 conjugacy classes

class 1 2A2B2C2D2E2F2G4A···4L4M···4X5A5B10A···10F20A···20X
order122222224···44···45510···1020···20
size111155551···15···5222···22···2

64 irreducible representations

dim11111222
type++++++
imageC1C2C2C2C4D5D10C4×D5
kernelD5×C42C4×Dic5C4×C20C2×C4×D5C4×D5C42C2×C4C4
# reps1313242624

Matrix representation of D5×C42 in GL3(𝔽41) generated by

900
010
001
,
100
090
009
,
100
061
0400
,
100
0040
0400
G:=sub<GL(3,GF(41))| [9,0,0,0,1,0,0,0,1],[1,0,0,0,9,0,0,0,9],[1,0,0,0,6,40,0,1,0],[1,0,0,0,0,40,0,40,0] >;

D5×C42 in GAP, Magma, Sage, TeX

D_5\times C_4^2
% in TeX

G:=Group("D5xC4^2");
// GroupNames label

G:=SmallGroup(160,92);
// by ID

G=gap.SmallGroup(160,92);
# by ID

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

G:=Group<a,b,c,d|a^4=b^4=c^5=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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