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## G = Dic5×C2×C10order 400 = 24·52

### Direct product of C2×C10 and Dic5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5 — Dic5×C2×C10
 Chief series C1 — C5 — C10 — C5×C10 — C5×Dic5 — C10×Dic5 — Dic5×C2×C10
 Lower central C5 — Dic5×C2×C10
 Upper central C1 — C22×C10

Generators and relations for Dic5×C2×C10
G = < a,b,c,d | a2=b10=c10=1, d2=c5, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 260 in 140 conjugacy classes, 86 normal (14 characteristic)
C1, C2, C2, C4, C22, C5, C5, C2×C4, C23, C10, C10, C10, C22×C4, Dic5, C20, C2×C10, C2×C10, C52, C2×Dic5, C2×C20, C22×C10, C22×C10, C5×C10, C5×C10, C22×Dic5, C22×C20, C5×Dic5, C102, C10×Dic5, C2×C102, Dic5×C2×C10
Quotients: C1, C2, C4, C22, C5, C2×C4, C23, D5, C10, C22×C4, Dic5, C20, D10, C2×C10, C2×Dic5, C2×C20, C22×D5, C22×C10, C5×D5, C22×Dic5, C22×C20, C5×Dic5, D5×C10, C10×Dic5, D5×C2×C10, Dic5×C2×C10

Smallest permutation representation of Dic5×C2×C10
On 80 points
Generators in S80
(1 36)(2 37)(3 38)(4 39)(5 40)(6 31)(7 32)(8 33)(9 34)(10 35)(11 59)(12 60)(13 51)(14 52)(15 53)(16 54)(17 55)(18 56)(19 57)(20 58)(21 45)(22 46)(23 47)(24 48)(25 49)(26 50)(27 41)(28 42)(29 43)(30 44)(61 75)(62 76)(63 77)(64 78)(65 79)(66 80)(67 71)(68 72)(69 73)(70 74)
(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 22 9 30 7 28 5 26 3 24)(2 23 10 21 8 29 6 27 4 25)(11 75 13 77 15 79 17 71 19 73)(12 76 14 78 16 80 18 72 20 74)(31 41 39 49 37 47 35 45 33 43)(32 42 40 50 38 48 36 46 34 44)(51 63 53 65 55 67 57 69 59 61)(52 64 54 66 56 68 58 70 60 62)
(1 76 28 18)(2 77 29 19)(3 78 30 20)(4 79 21 11)(5 80 22 12)(6 71 23 13)(7 72 24 14)(8 73 25 15)(9 74 26 16)(10 75 27 17)(31 67 47 51)(32 68 48 52)(33 69 49 53)(34 70 50 54)(35 61 41 55)(36 62 42 56)(37 63 43 57)(38 64 44 58)(39 65 45 59)(40 66 46 60)

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

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

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

160 conjugacy classes

 class 1 2A ··· 2G 4A ··· 4H 5A 5B 5C 5D 5E ··· 5N 10A ··· 10AB 10AC ··· 10CT 20A ··· 20AF order 1 2 ··· 2 4 ··· 4 5 5 5 5 5 ··· 5 10 ··· 10 10 ··· 10 20 ··· 20 size 1 1 ··· 1 5 ··· 5 1 1 1 1 2 ··· 2 1 ··· 1 2 ··· 2 5 ··· 5

160 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 type + + + + - + image C1 C2 C2 C4 C5 C10 C10 C20 D5 Dic5 D10 C5×D5 C5×Dic5 D5×C10 kernel Dic5×C2×C10 C10×Dic5 C2×C102 C102 C22×Dic5 C2×Dic5 C22×C10 C2×C10 C22×C10 C2×C10 C2×C10 C23 C22 C22 # reps 1 6 1 8 4 24 4 32 2 8 6 8 32 24

Matrix representation of Dic5×C2×C10 in GL4(𝔽41) generated by

 1 0 0 0 0 1 0 0 0 0 40 0 0 0 0 40
,
 31 0 0 0 0 31 0 0 0 0 37 0 0 0 0 37
,
 40 0 0 0 0 1 0 0 0 0 10 0 0 0 38 37
,
 9 0 0 0 0 40 0 0 0 0 1 32 0 0 0 40
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,40,0,0,0,0,40],[31,0,0,0,0,31,0,0,0,0,37,0,0,0,0,37],[40,0,0,0,0,1,0,0,0,0,10,38,0,0,0,37],[9,0,0,0,0,40,0,0,0,0,1,0,0,0,32,40] >;

Dic5×C2×C10 in GAP, Magma, Sage, TeX

{\rm Dic}_5\times C_2\times C_{10}
% in TeX

G:=Group("Dic5xC2xC10");
// GroupNames label

G:=SmallGroup(400,189);
// by ID

G=gap.SmallGroup(400,189);
# by ID

G:=PCGroup([6,-2,-2,-2,-5,-2,-5,240,11525]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^10=c^10=1,d^2=c^5,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

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