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## G = D5×C2×C20order 400 = 24·52

### Direct product of C2×C20 and D5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5 — D5×C2×C20
 Chief series C1 — C5 — C10 — C5×C10 — D5×C10 — D5×C2×C10 — D5×C2×C20
 Lower central C5 — D5×C2×C20
 Upper central C1 — C2×C20

Generators and relations for D5×C2×C20
G = < a,b,c,d | a2=b20=c5=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 292 in 124 conjugacy classes, 70 normal (22 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C5, C5, C2×C4, C2×C4, C23, D5, C10, C10, C10, C22×C4, Dic5, C20, C20, D10, C2×C10, C2×C10, C52, C4×D5, C2×Dic5, C2×C20, C2×C20, C22×D5, C22×C10, C5×D5, C5×C10, C5×C10, C2×C4×D5, C22×C20, C5×Dic5, C5×C20, D5×C10, C102, D5×C20, C10×Dic5, C10×C20, D5×C2×C10, D5×C2×C20
Quotients: C1, C2, C4, C22, C5, C2×C4, C23, D5, C10, C22×C4, C20, D10, C2×C10, C4×D5, C2×C20, C22×D5, C22×C10, C5×D5, C2×C4×D5, C22×C20, D5×C10, D5×C20, D5×C2×C10, D5×C2×C20

Smallest permutation representation of D5×C2×C20
On 80 points
Generators in S80
(1 36)(2 37)(3 38)(4 39)(5 40)(6 21)(7 22)(8 23)(9 24)(10 25)(11 26)(12 27)(13 28)(14 29)(15 30)(16 31)(17 32)(18 33)(19 34)(20 35)(41 79)(42 80)(43 61)(44 62)(45 63)(46 64)(47 65)(48 66)(49 67)(50 68)(51 69)(52 70)(53 71)(54 72)(55 73)(56 74)(57 75)(58 76)(59 77)(60 78)
(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 9 17 5 13)(2 10 18 6 14)(3 11 19 7 15)(4 12 20 8 16)(21 29 37 25 33)(22 30 38 26 34)(23 31 39 27 35)(24 32 40 28 36)(41 53 45 57 49)(42 54 46 58 50)(43 55 47 59 51)(44 56 48 60 52)(61 73 65 77 69)(62 74 66 78 70)(63 75 67 79 71)(64 76 68 80 72)
(1 62)(2 63)(3 64)(4 65)(5 66)(6 67)(7 68)(8 69)(9 70)(10 71)(11 72)(12 73)(13 74)(14 75)(15 76)(16 77)(17 78)(18 79)(19 80)(20 61)(21 49)(22 50)(23 51)(24 52)(25 53)(26 54)(27 55)(28 56)(29 57)(30 58)(31 59)(32 60)(33 41)(34 42)(35 43)(36 44)(37 45)(38 46)(39 47)(40 48)

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

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

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

160 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C 4D 4E 4F 4G 4H 5A 5B 5C 5D 5E ··· 5N 10A ··· 10L 10M ··· 10AP 10AQ ··· 10BF 20A ··· 20P 20Q ··· 20BD 20BE ··· 20BT order 1 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 5 5 5 5 5 ··· 5 10 ··· 10 10 ··· 10 10 ··· 10 20 ··· 20 20 ··· 20 20 ··· 20 size 1 1 1 1 5 5 5 5 1 1 1 1 5 5 5 5 1 1 1 1 2 ··· 2 1 ··· 1 2 ··· 2 5 ··· 5 1 ··· 1 2 ··· 2 5 ··· 5

160 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 type + + + + + + + + image C1 C2 C2 C2 C2 C4 C5 C10 C10 C10 C10 C20 D5 D10 D10 C4×D5 C5×D5 D5×C10 D5×C10 D5×C20 kernel D5×C2×C20 D5×C20 C10×Dic5 C10×C20 D5×C2×C10 D5×C10 C2×C4×D5 C4×D5 C2×Dic5 C2×C20 C22×D5 D10 C2×C20 C20 C2×C10 C10 C2×C4 C4 C22 C2 # reps 1 4 1 1 1 8 4 16 4 4 4 32 2 4 2 8 8 16 8 32

Matrix representation of D5×C2×C20 in GL3(𝔽41) generated by

 40 0 0 0 40 0 0 0 40
,
 21 0 0 0 4 0 0 0 4
,
 1 0 0 0 18 0 0 0 16
,
 40 0 0 0 0 25 0 23 0
G:=sub<GL(3,GF(41))| [40,0,0,0,40,0,0,0,40],[21,0,0,0,4,0,0,0,4],[1,0,0,0,18,0,0,0,16],[40,0,0,0,0,23,0,25,0] >;

D5×C2×C20 in GAP, Magma, Sage, TeX

D_5\times C_2\times C_{20}
% in TeX

G:=Group("D5xC2xC20");
// GroupNames label

G:=SmallGroup(400,182);
// by ID

G=gap.SmallGroup(400,182);
# by ID

G:=PCGroup([6,-2,-2,-2,-5,-2,-5,194,11525]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^20=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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