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## G = D5×C42order 160 = 25·5

### Direct product of C42 and D5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5 — D5×C42
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C2×C4×D5 — D5×C42
 Lower central C5 — D5×C42
 Upper central C1 — C42

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)])

64 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A ··· 4L 4M ··· 4X 5A 5B 10A ··· 10F 20A ··· 20X order 1 2 2 2 2 2 2 2 4 ··· 4 4 ··· 4 5 5 10 ··· 10 20 ··· 20 size 1 1 1 1 5 5 5 5 1 ··· 1 5 ··· 5 2 2 2 ··· 2 2 ··· 2

64 irreducible representations

 dim 1 1 1 1 1 2 2 2 type + + + + + + image C1 C2 C2 C2 C4 D5 D10 C4×D5 kernel D5×C42 C4×Dic5 C4×C20 C2×C4×D5 C4×D5 C42 C2×C4 C4 # reps 1 3 1 3 24 2 6 24

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

 9 0 0 0 1 0 0 0 1
,
 1 0 0 0 9 0 0 0 9
,
 1 0 0 0 6 1 0 40 0
,
 1 0 0 0 0 40 0 40 0
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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