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G = D4×C2×C10order 160 = 25·5

Direct product of C2×C10 and D4

direct product, metabelian, nilpotent (class 2), monomial, 2-elementary

Aliases: D4×C2×C10, C204C23, C243C10, C10.16C24, C4⋊(C22×C10), (C2×C10)⋊2C23, C233(C2×C10), (C23×C10)⋊2C2, (C22×C4)⋊5C10, C22⋊(C22×C10), (C2×C20)⋊15C22, (C22×C20)⋊12C2, C2.1(C23×C10), (C22×C10)⋊6C22, (C2×C4)⋊4(C2×C10), SmallGroup(160,229)

Series: Derived Chief Lower central Upper central

C1C2 — D4×C2×C10
C1C2C10C2×C10C5×D4D4×C10 — D4×C2×C10
C1C2 — D4×C2×C10
C1C22×C10 — D4×C2×C10

Generators and relations for D4×C2×C10
 G = < a,b,c,d | a2=b10=c4=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 316 in 236 conjugacy classes, 156 normal (10 characteristic)
C1, C2, C2 [×6], C2 [×8], C4 [×4], C22 [×15], C22 [×24], C5, C2×C4 [×6], D4 [×16], C23, C23 [×12], C23 [×8], C10, C10 [×6], C10 [×8], C22×C4, C2×D4 [×12], C24 [×2], C20 [×4], C2×C10 [×15], C2×C10 [×24], C22×D4, C2×C20 [×6], C5×D4 [×16], C22×C10, C22×C10 [×12], C22×C10 [×8], C22×C20, D4×C10 [×12], C23×C10 [×2], D4×C2×C10
Quotients: C1, C2 [×15], C22 [×35], C5, D4 [×4], C23 [×15], C10 [×15], C2×D4 [×6], C24, C2×C10 [×35], C22×D4, C5×D4 [×4], C22×C10 [×15], D4×C10 [×6], C23×C10, D4×C2×C10

Smallest permutation representation of D4×C2×C10
On 80 points
Generators in S80
(1 42)(2 43)(3 44)(4 45)(5 46)(6 47)(7 48)(8 49)(9 50)(10 41)(11 52)(12 53)(13 54)(14 55)(15 56)(16 57)(17 58)(18 59)(19 60)(20 51)(21 33)(22 34)(23 35)(24 36)(25 37)(26 38)(27 39)(28 40)(29 31)(30 32)(61 73)(62 74)(63 75)(64 76)(65 77)(66 78)(67 79)(68 80)(69 71)(70 72)
(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 27 74)(2 19 28 75)(3 20 29 76)(4 11 30 77)(5 12 21 78)(6 13 22 79)(7 14 23 80)(8 15 24 71)(9 16 25 72)(10 17 26 73)(31 64 44 51)(32 65 45 52)(33 66 46 53)(34 67 47 54)(35 68 48 55)(36 69 49 56)(37 70 50 57)(38 61 41 58)(39 62 42 59)(40 63 43 60)
(1 74)(2 75)(3 76)(4 77)(5 78)(6 79)(7 80)(8 71)(9 72)(10 73)(11 30)(12 21)(13 22)(14 23)(15 24)(16 25)(17 26)(18 27)(19 28)(20 29)(31 51)(32 52)(33 53)(34 54)(35 55)(36 56)(37 57)(38 58)(39 59)(40 60)(41 61)(42 62)(43 63)(44 64)(45 65)(46 66)(47 67)(48 68)(49 69)(50 70)

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

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

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

D4×C2×C10 is a maximal subgroup of
(D4×C10)⋊18C4  (C2×C10)⋊8D8  (C5×D4).31D4  C24.18D10  C24.19D10  C24.20D10  C24.21D10  C24.38D10  C248D10  C24.41D10  C24.42D10

100 conjugacy classes

class 1 2A···2G2H···2O4A4B4C4D5A5B5C5D10A···10AB10AC···10BH20A···20P
order12···22···24444555510···1010···1020···20
size11···12···2222211111···12···22···2

100 irreducible representations

dim1111111122
type+++++
imageC1C2C2C2C5C10C10C10D4C5×D4
kernelD4×C2×C10C22×C20D4×C10C23×C10C22×D4C22×C4C2×D4C24C2×C10C22
# reps1112244488416

Matrix representation of D4×C2×C10 in GL4(𝔽41) generated by

1000
04000
00400
00040
,
40000
0100
00250
00025
,
40000
04000
00040
0010
,
1000
0100
00040
00400
G:=sub<GL(4,GF(41))| [1,0,0,0,0,40,0,0,0,0,40,0,0,0,0,40],[40,0,0,0,0,1,0,0,0,0,25,0,0,0,0,25],[40,0,0,0,0,40,0,0,0,0,0,1,0,0,40,0],[1,0,0,0,0,1,0,0,0,0,0,40,0,0,40,0] >;

D4×C2×C10 in GAP, Magma, Sage, TeX

D_4\times C_2\times C_{10}
% in TeX

G:=Group("D4xC2xC10");
// GroupNames label

G:=SmallGroup(160,229);
// by ID

G=gap.SmallGroup(160,229);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-5,-2,985]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^10=c^4=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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