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## G = C2×D5×Dic5order 400 = 24·52

### Direct product of C2, D5 and Dic5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C52 — C2×D5×Dic5
 Chief series C1 — C5 — C52 — C5×C10 — D5×C10 — D5×Dic5 — C2×D5×Dic5
 Lower central C52 — C2×D5×Dic5
 Upper central C1 — C22

Generators and relations for C2×D5×Dic5
G = < a,b,c,d,e | a2=b5=c2=d10=1, e2=d5, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede-1=d-1 >

Subgroups: 556 in 124 conjugacy classes, 56 normal (18 characteristic)
C1, C2, C2, C2, C4, C22, C22, C5, C5, C2×C4, C23, D5, C10, C10, C10, C22×C4, Dic5, Dic5, C20, D10, C2×C10, C2×C10, C52, C4×D5, C2×Dic5, C2×Dic5, C2×C20, C22×D5, C22×C10, C5×D5, C5×C10, C5×C10, C2×C4×D5, C22×Dic5, C5×Dic5, C526C4, D5×C10, C102, D5×Dic5, C10×Dic5, C2×C526C4, D5×C2×C10, C2×D5×Dic5
Quotients: C1, C2, C4, C22, C2×C4, C23, D5, C22×C4, Dic5, D10, C4×D5, C2×Dic5, C22×D5, C2×C4×D5, C22×Dic5, D52, D5×Dic5, C2×D52, C2×D5×Dic5

Smallest permutation representation of C2×D5×Dic5
On 80 points
Generators in S80
(1 13)(2 14)(3 15)(4 16)(5 17)(6 18)(7 19)(8 20)(9 11)(10 12)(21 31)(22 32)(23 33)(24 34)(25 35)(26 36)(27 37)(28 38)(29 39)(30 40)(41 51)(42 52)(43 53)(44 54)(45 55)(46 56)(47 57)(48 58)(49 59)(50 60)(61 71)(62 72)(63 73)(64 74)(65 75)(66 76)(67 77)(68 78)(69 79)(70 80)
(1 3 5 7 9)(2 4 6 8 10)(11 13 15 17 19)(12 14 16 18 20)(21 29 27 25 23)(22 30 28 26 24)(31 39 37 35 33)(32 40 38 36 34)(41 49 47 45 43)(42 50 48 46 44)(51 59 57 55 53)(52 60 58 56 54)(61 63 65 67 69)(62 64 66 68 70)(71 73 75 77 79)(72 74 76 78 80)
(1 60)(2 51)(3 52)(4 53)(5 54)(6 55)(7 56)(8 57)(9 58)(10 59)(11 48)(12 49)(13 50)(14 41)(15 42)(16 43)(17 44)(18 45)(19 46)(20 47)(21 74)(22 75)(23 76)(24 77)(25 78)(26 79)(27 80)(28 71)(29 72)(30 73)(31 64)(32 65)(33 66)(34 67)(35 68)(36 69)(37 70)(38 61)(39 62)(40 63)
(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 32 6 37)(2 31 7 36)(3 40 8 35)(4 39 9 34)(5 38 10 33)(11 24 16 29)(12 23 17 28)(13 22 18 27)(14 21 19 26)(15 30 20 25)(41 74 46 79)(42 73 47 78)(43 72 48 77)(44 71 49 76)(45 80 50 75)(51 64 56 69)(52 63 57 68)(53 62 58 67)(54 61 59 66)(55 70 60 65)

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

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

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

64 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C 4D 4E 4F 4G 4H 5A 5B 5C 5D 5E 5F 5G 5H 10A ··· 10L 10M ··· 10X 10Y ··· 10AF 20A ··· 20H order 1 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 10 ··· 10 10 ··· 10 10 ··· 10 20 ··· 20 size 1 1 1 1 5 5 5 5 5 5 5 5 25 25 25 25 2 2 2 2 4 4 4 4 2 ··· 2 4 ··· 4 10 ··· 10 10 ··· 10

64 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + - + + + - + image C1 C2 C2 C2 C2 C4 D5 D5 D10 Dic5 D10 D10 C4×D5 D52 D5×Dic5 C2×D52 kernel C2×D5×Dic5 D5×Dic5 C10×Dic5 C2×C52⋊6C4 D5×C2×C10 D5×C10 C2×Dic5 C22×D5 Dic5 D10 D10 C2×C10 C10 C22 C2 C2 # reps 1 4 1 1 1 8 2 2 4 8 4 4 8 4 8 4

Matrix representation of C2×D5×Dic5 in GL4(𝔽41) generated by

 40 0 0 0 0 40 0 0 0 0 40 0 0 0 0 40
,
 1 0 0 0 0 1 0 0 0 0 0 40 0 0 1 6
,
 40 0 0 0 0 40 0 0 0 0 1 6 0 0 0 40
,
 0 1 0 0 40 7 0 0 0 0 1 0 0 0 0 1
,
 0 9 0 0 9 0 0 0 0 0 1 0 0 0 0 1
G:=sub<GL(4,GF(41))| [40,0,0,0,0,40,0,0,0,0,40,0,0,0,0,40],[1,0,0,0,0,1,0,0,0,0,0,1,0,0,40,6],[40,0,0,0,0,40,0,0,0,0,1,0,0,0,6,40],[0,40,0,0,1,7,0,0,0,0,1,0,0,0,0,1],[0,9,0,0,9,0,0,0,0,0,1,0,0,0,0,1] >;

C2×D5×Dic5 in GAP, Magma, Sage, TeX

C_2\times D_5\times {\rm Dic}_5
% in TeX

G:=Group("C2xD5xDic5");
// GroupNames label

G:=SmallGroup(400,172);
// by ID

G=gap.SmallGroup(400,172);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-5,-5,55,970,11525]);
// Polycyclic

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

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