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## G = D5×C22×C10order 400 = 24·52

### Direct product of C22×C10 and D5

Aliases: D5×C22×C10, C522C24, C1029C22, C5⋊(C23×C10), C10⋊(C22×C10), (C2×C102)⋊4C2, (C5×C10)⋊2C23, (C22×C10)⋊3C10, (C2×C10)⋊4(C2×C10), SmallGroup(400,219)

Series: Derived Chief Lower central Upper central

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

Generators and relations for D5×C22×C10
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, C2×C10, C2×C10, C52, C22×D5, C22×C10, C22×C10, C5×D5, C5×C10, C23×D5, C23×C10, D5×C10, C102, D5×C2×C10, C2×C102, D5×C22×C10
Quotients: C1, C2, C22, C5, C23, D5, C10, C24, D10, C2×C10, C22×D5, C22×C10, C5×D5, C23×D5, C23×C10, D5×C10, D5×C2×C10, D5×C22×C10

Smallest permutation representation of D5×C22×C10
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 ··· 2G 2H ··· 2O 5A 5B 5C 5D 5E ··· 5N 10A ··· 10AB 10AC ··· 10CT 10CU ··· 10DZ order 1 2 ··· 2 2 ··· 2 5 5 5 5 5 ··· 5 10 ··· 10 10 ··· 10 10 ··· 10 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 2 2 2 2 type + + + + + image C1 C2 C2 C5 C10 C10 D5 D10 C5×D5 D5×C10 kernel D5×C22×C10 D5×C2×C10 C2×C102 C23×D5 C22×D5 C22×C10 C22×C10 C2×C10 C23 C22 # reps 1 14 1 4 56 4 2 14 8 56

Matrix representation of D5×C22×C10 in GL4(𝔽11) generated by

 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 0 0 0 6 4
,
 1 0 0 0 0 1 0 0 0 0 7 3 0 0 6 4
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] >;

D5×C22×C10 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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