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G = C5×D42order 320 = 26·5

Direct product of C5, D4 and D4

direct product, metabelian, nilpotent (class 2), monomial, 2-elementary

Aliases: C5×D42, C10.1602+ 1+4, C42(D4×C10), C2014(C2×D4), (D4×C20)⋊43C2, (C4×D4)⋊14C10, C41D47C10, C244(C2×C10), C428(C2×C10), C222(D4×C10), C22≀C25C10, C4⋊D410C10, (C4×C20)⋊42C22, (C22×D4)⋊8C10, (D4×C10)⋊64C22, (C23×C10)⋊4C22, (C2×C20).674C23, (C2×C10).365C24, (C22×C20)⋊50C22, C10.193(C22×D4), C22.39(C23×C10), C23.15(C22×C10), C2.12(C5×2+ 1+4), (C22×C10).264C23, (D4×C2×C10)⋊23C2, C2.17(D4×C2×C10), C4⋊C416(C2×C10), (C2×C10)⋊13(C2×D4), (C2×D4)⋊12(C2×C10), (C5×C4⋊D4)⋊37C2, (C5×C41D4)⋊18C2, C22⋊C45(C2×C10), (C5×C4⋊C4)⋊72C22, (C5×C22≀C2)⋊15C2, (C22×C4)⋊10(C2×C10), (C5×C22⋊C4)⋊40C22, (C2×C4).32(C22×C10), SmallGroup(320,1547)

Series: Derived Chief Lower central Upper central

C1C22 — C5×D42
C1C2C22C2×C10C22×C10D4×C10C5×C22≀C2 — C5×D42
C1C22 — C5×D42
C1C2×C10 — C5×D42

Generators and relations for C5×D42
 G = < a,b,c,d,e | a5=b4=c2=d4=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >

Subgroups: 778 in 428 conjugacy classes, 182 normal (14 characteristic)
C1, C2, C2 [×2], C2 [×12], C4 [×4], C4 [×5], C22, C22 [×8], C22 [×36], C5, C2×C4, C2×C4 [×6], C2×C4 [×8], D4 [×8], D4 [×26], C23 [×8], C23 [×20], C10, C10 [×2], C10 [×12], C42, C22⋊C4 [×8], C4⋊C4 [×2], C22×C4 [×4], C2×D4 [×16], C2×D4 [×16], C24 [×4], C20 [×4], C20 [×5], C2×C10, C2×C10 [×8], C2×C10 [×36], C4×D4 [×2], C22≀C2 [×4], C4⋊D4 [×4], C41D4, C22×D4 [×4], C2×C20, C2×C20 [×6], C2×C20 [×8], C5×D4 [×8], C5×D4 [×26], C22×C10 [×8], C22×C10 [×20], D42, C4×C20, C5×C22⋊C4 [×8], C5×C4⋊C4 [×2], C22×C20 [×4], D4×C10 [×16], D4×C10 [×16], C23×C10 [×4], D4×C20 [×2], C5×C22≀C2 [×4], C5×C4⋊D4 [×4], C5×C41D4, D4×C2×C10 [×4], C5×D42
Quotients: C1, C2 [×15], C22 [×35], C5, D4 [×8], C23 [×15], C10 [×15], C2×D4 [×12], C24, C2×C10 [×35], C22×D4 [×2], 2+ 1+4, C5×D4 [×8], C22×C10 [×15], D42, D4×C10 [×12], C23×C10, D4×C2×C10 [×2], C5×2+ 1+4, C5×D42

Smallest permutation representation of C5×D42
On 80 points
Generators in S80
(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 16 30 11)(2 17 26 12)(3 18 27 13)(4 19 28 14)(5 20 29 15)(6 23 76 35)(7 24 77 31)(8 25 78 32)(9 21 79 33)(10 22 80 34)(36 63 50 75)(37 64 46 71)(38 65 47 72)(39 61 48 73)(40 62 49 74)(41 59 53 68)(42 60 54 69)(43 56 55 70)(44 57 51 66)(45 58 52 67)
(1 11)(2 12)(3 13)(4 14)(5 15)(6 23)(7 24)(8 25)(9 21)(10 22)(16 30)(17 26)(18 27)(19 28)(20 29)(31 77)(32 78)(33 79)(34 80)(35 76)(36 75)(37 71)(38 72)(39 73)(40 74)(41 68)(42 69)(43 70)(44 66)(45 67)(46 64)(47 65)(48 61)(49 62)(50 63)(51 57)(52 58)(53 59)(54 60)(55 56)
(1 50 35 43)(2 46 31 44)(3 47 32 45)(4 48 33 41)(5 49 34 42)(6 56 16 75)(7 57 17 71)(8 58 18 72)(9 59 19 73)(10 60 20 74)(11 63 76 70)(12 64 77 66)(13 65 78 67)(14 61 79 68)(15 62 80 69)(21 53 28 39)(22 54 29 40)(23 55 30 36)(24 51 26 37)(25 52 27 38)
(1 30)(2 26)(3 27)(4 28)(5 29)(6 76)(7 77)(8 78)(9 79)(10 80)(11 16)(12 17)(13 18)(14 19)(15 20)(21 33)(22 34)(23 35)(24 31)(25 32)(36 43)(37 44)(38 45)(39 41)(40 42)(46 51)(47 52)(48 53)(49 54)(50 55)(56 63)(57 64)(58 65)(59 61)(60 62)(66 71)(67 72)(68 73)(69 74)(70 75)

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

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

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

125 conjugacy classes

class 1 2A2B2C2D···2K2L2M2N2O4A4B4C4D4E···4I5A5B5C5D10A···10L10M···10AR10AS···10BH20A···20P20Q···20AJ
order12222···2222244444···4555510···1010···1010···1020···2020···20
size11112···2444422224···411111···12···24···42···24···4

125 irreducible representations

dim1111111111112244
type++++++++
imageC1C2C2C2C2C2C5C10C10C10C10C10D4C5×D42+ 1+4C5×2+ 1+4
kernelC5×D42D4×C20C5×C22≀C2C5×C4⋊D4C5×C41D4D4×C2×C10D42C4×D4C22≀C2C4⋊D4C41D4C22×D4C5×D4D4C10C2
# reps12441448161641683214

Matrix representation of C5×D42 in GL5(𝔽41)

370000
01000
00100
00010
00001
,
10000
01000
00100
0003139
0003010
,
400000
01000
00100
0003139
0002910
,
400000
004000
01000
000400
000040
,
400000
01000
004000
000400
000040

G:=sub<GL(5,GF(41))| [37,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,31,30,0,0,0,39,10],[40,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,31,29,0,0,0,39,10],[40,0,0,0,0,0,0,1,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,40],[40,0,0,0,0,0,1,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,40] >;

C5×D42 in GAP, Magma, Sage, TeX

C_5\times D_4^2
% in TeX

G:=Group("C5xD4^2");
// GroupNames label

G:=SmallGroup(320,1547);
// by ID

G=gap.SmallGroup(320,1547);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-5,-2,-2,1149,3446,1242]);
// Polycyclic

G:=Group<a,b,c,d,e|a^5=b^4=c^2=d^4=e^2=1,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=d^-1>;
// generators/relations

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