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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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — D4×C2×C10
 Chief series C1 — C2 — C10 — C2×C10 — C5×D4 — D4×C10 — D4×C2×C10
 Lower central C1 — C2 — D4×C2×C10
 Upper central C1 — C22×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 ··· 2G 2H ··· 2O 4A 4B 4C 4D 5A 5B 5C 5D 10A ··· 10AB 10AC ··· 10BH 20A ··· 20P order 1 2 ··· 2 2 ··· 2 4 4 4 4 5 5 5 5 10 ··· 10 10 ··· 10 20 ··· 20 size 1 1 ··· 1 2 ··· 2 2 2 2 2 1 1 1 1 1 ··· 1 2 ··· 2 2 ··· 2

100 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 type + + + + + image C1 C2 C2 C2 C5 C10 C10 C10 D4 C5×D4 kernel D4×C2×C10 C22×C20 D4×C10 C23×C10 C22×D4 C22×C4 C2×D4 C24 C2×C10 C22 # reps 1 1 12 2 4 4 48 8 4 16

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

 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 40 0 0 1 0
,
 1 0 0 0 0 1 0 0 0 0 0 40 0 0 40 0
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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