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G = D5xC22xC10order 400 = 24·52

Direct product of C22xC10 and D5

direct product, metabelian, supersoluble, monomial, A-group

Aliases: D5xC22xC10, C52:2C24, C102:9C22, C5:(C23xC10), C10:(C22xC10), (C2xC102):4C2, (C5xC10):2C23, (C22xC10):3C10, (C2xC10):4(C2xC10), SmallGroup(400,219)

Series: Derived Chief Lower central Upper central

C1C5 — D5xC22xC10
C1C5C52C5xD5D5xC10D5xC2xC10 — D5xC22xC10
C5 — D5xC22xC10
C1C22xC10

Generators and relations for D5xC22xC10
 G = < a,b,c,d,e | a2=b2=c10=d5=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >

Subgroups: 740 in 300 conjugacy classes, 166 normal (10 characteristic)
C1, C2, C2, C22, C22, C5, C5, C23, C23, D5, C10, C10, C24, D10, C2xC10, C2xC10, C52, C22xD5, C22xC10, C22xC10, C5xD5, C5xC10, C23xD5, C23xC10, D5xC10, C102, D5xC2xC10, C2xC102, D5xC22xC10
Quotients: C1, C2, C22, C5, C23, D5, C10, C24, D10, C2xC10, C22xD5, C22xC10, C5xD5, C23xD5, C23xC10, D5xC10, D5xC2xC10, D5xC22xC10

Smallest permutation representation of D5xC22xC10
On 80 points
Generators in S80
(1 26)(2 27)(3 28)(4 29)(5 30)(6 21)(7 22)(8 23)(9 24)(10 25)(11 36)(12 37)(13 38)(14 39)(15 40)(16 31)(17 32)(18 33)(19 34)(20 35)(41 66)(42 67)(43 68)(44 69)(45 70)(46 61)(47 62)(48 63)(49 64)(50 65)(51 76)(52 77)(53 78)(54 79)(55 80)(56 71)(57 72)(58 73)(59 74)(60 75)
(1 16)(2 17)(3 18)(4 19)(5 20)(6 11)(7 12)(8 13)(9 14)(10 15)(21 36)(22 37)(23 38)(24 39)(25 40)(26 31)(27 32)(28 33)(29 34)(30 35)(41 56)(42 57)(43 58)(44 59)(45 60)(46 51)(47 52)(48 53)(49 54)(50 55)(61 76)(62 77)(63 78)(64 79)(65 80)(66 71)(67 72)(68 73)(69 74)(70 75)
(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 5 9 3 7)(2 6 10 4 8)(11 15 19 13 17)(12 16 20 14 18)(21 25 29 23 27)(22 26 30 24 28)(31 35 39 33 37)(32 36 40 34 38)(41 47 43 49 45)(42 48 44 50 46)(51 57 53 59 55)(52 58 54 60 56)(61 67 63 69 65)(62 68 64 70 66)(71 77 73 79 75)(72 78 74 80 76)
(1 50)(2 41)(3 42)(4 43)(5 44)(6 45)(7 46)(8 47)(9 48)(10 49)(11 60)(12 51)(13 52)(14 53)(15 54)(16 55)(17 56)(18 57)(19 58)(20 59)(21 70)(22 61)(23 62)(24 63)(25 64)(26 65)(27 66)(28 67)(29 68)(30 69)(31 80)(32 71)(33 72)(34 73)(35 74)(36 75)(37 76)(38 77)(39 78)(40 79)

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

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

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

160 conjugacy classes

class 1 2A···2G2H···2O5A5B5C5D5E···5N10A···10AB10AC···10CT10CU···10DZ
order12···22···255555···510···1010···1010···10
size11···15···511112···21···12···25···5

160 irreducible representations

dim1111112222
type+++++
imageC1C2C2C5C10C10D5D10C5xD5D5xC10
kernelD5xC22xC10D5xC2xC10C2xC102C23xD5C22xD5C22xC10C22xC10C2xC10C23C22
# reps11414564214856

Matrix representation of D5xC22xC10 in GL4(F11) generated by

1000
0100
00100
00010
,
10000
01000
0010
0001
,
2000
0300
0070
0007
,
1000
0100
0030
0064
,
1000
0100
0073
0064
G:=sub<GL(4,GF(11))| [1,0,0,0,0,1,0,0,0,0,10,0,0,0,0,10],[10,0,0,0,0,10,0,0,0,0,1,0,0,0,0,1],[2,0,0,0,0,3,0,0,0,0,7,0,0,0,0,7],[1,0,0,0,0,1,0,0,0,0,3,6,0,0,0,4],[1,0,0,0,0,1,0,0,0,0,7,6,0,0,3,4] >;

D5xC22xC10 in GAP, Magma, Sage, TeX

D_5\times C_2^2\times C_{10}
% in TeX

G:=Group("D5xC2^2xC10");
// GroupNames label

G:=SmallGroup(400,219);
// by ID

G=gap.SmallGroup(400,219);
# by ID

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

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

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