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G = C4217D10order 320 = 26·5

17th semidirect product of C42 and D10 acting via D10/C5=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C4217D10, C10.212+ 1+4, C4⋊C450D10, (C4×D4)⋊22D5, (D4×C20)⋊24C2, C22⋊D207C2, C207D411C2, (C4×C20)⋊28C22, C22⋊C449D10, (C22×C4)⋊14D10, C23⋊D1021C2, D10⋊D410C2, (C2×D4).221D10, C4.D2028C2, C422D510C2, C4⋊Dic510C22, Dic5⋊D427C2, D10.13D48C2, (C2×C20).162C23, (C2×C10).104C24, (C22×C20)⋊11C22, Dic5.5D49C2, C52(C22.32C24), (C4×Dic5)⋊53C22, (C2×Dic10)⋊7C22, (C2×D20).29C22, C22.6(C4○D20), C23.D510C22, C2.22(D46D10), C2.17(D48D10), D10⋊C431C22, Dic5.14D49C2, (D4×C10).308C22, C10.D433C22, C23.23D102C2, (C2×Dic5).45C23, (C22×D5).38C23, (C23×D5).42C22, C23.101(C22×D5), C22.129(C23×D5), (C22×C10).174C23, (C22×Dic5).99C22, C4⋊C4⋊D58C2, (C4×C5⋊D4)⋊46C2, (C2×C4×D5)⋊49C22, (C5×C4⋊C4)⋊62C22, C10.46(C2×C4○D4), C2.53(C2×C4○D20), (C2×C5⋊D4)⋊5C22, (C2×D10⋊C4)⋊35C2, (C2×C10).17(C4○D4), (C5×C22⋊C4)⋊58C22, (C2×C4).162(C22×D5), SmallGroup(320,1232)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C4217D10
C1C5C10C2×C10C22×D5C23×D5C2×D10⋊C4 — C4217D10
C5C2×C10 — C4217D10
C1C22C4×D4

Generators and relations for C4217D10
 G = < a,b,c,d | a4=b4=c10=d2=1, ab=ba, cac-1=a-1, dad=a-1b2, bc=cb, dbd=a2b, dcd=c-1 >

Subgroups: 1054 in 250 conjugacy classes, 95 normal (91 characteristic)
C1, C2 [×3], C2 [×6], C4 [×10], C22, C22 [×2], C22 [×18], C5, C2×C4 [×5], C2×C4 [×9], D4 [×9], Q8, C23 [×2], C23 [×7], D5 [×3], C10 [×3], C10 [×3], C42, C42, C22⋊C4 [×2], C22⋊C4 [×12], C4⋊C4, C4⋊C4 [×5], C22×C4 [×2], C22×C4 [×2], C2×D4, C2×D4 [×6], C2×Q8, C24, Dic5 [×5], C20 [×5], D10 [×13], C2×C10, C2×C10 [×2], C2×C10 [×5], C2×C22⋊C4, C4×D4, C4×D4, C22≀C2 [×2], C4⋊D4 [×3], C22⋊Q8, C22.D4 [×2], C4.4D4 [×2], C422C2 [×2], Dic10, C4×D5, D20 [×2], C2×Dic5 [×5], C2×Dic5, C5⋊D4 [×5], C2×C20 [×5], C2×C20 [×2], C5×D4 [×2], C22×D5 [×3], C22×D5 [×4], C22×C10 [×2], C22.32C24, C4×Dic5, C10.D4 [×4], C4⋊Dic5, D10⋊C4 [×10], C23.D5 [×2], C4×C20, C5×C22⋊C4 [×2], C5×C4⋊C4, C2×Dic10, C2×C4×D5, C2×D20 [×2], C22×Dic5, C2×C5⋊D4 [×4], C22×C20 [×2], D4×C10, C23×D5, C4.D20, C422D5, Dic5.14D4, C22⋊D20, D10⋊D4, Dic5.5D4, D10.13D4, C4⋊C4⋊D5, C2×D10⋊C4, C4×C5⋊D4, C23.23D10, C207D4, C23⋊D10, Dic5⋊D4, D4×C20, C4217D10
Quotients: C1, C2 [×15], C22 [×35], C23 [×15], D5, C4○D4 [×2], C24, D10 [×7], C2×C4○D4, 2+ 1+4 [×2], C22×D5 [×7], C22.32C24, C4○D20 [×2], C23×D5, C2×C4○D20, D46D10, D48D10, C4217D10

Smallest permutation representation of C4217D10
On 80 points
Generators in S80
(1 41 23 46)(2 47 24 42)(3 43 25 48)(4 49 21 44)(5 45 22 50)(6 63 35 68)(7 69 31 64)(8 65 32 70)(9 61 33 66)(10 67 34 62)(11 51 16 56)(12 57 17 52)(13 53 18 58)(14 59 19 54)(15 55 20 60)(26 79 38 74)(27 75 39 80)(28 71 40 76)(29 77 36 72)(30 73 37 78)
(1 34 40 15)(2 35 36 11)(3 31 37 12)(4 32 38 13)(5 33 39 14)(6 29 16 24)(7 30 17 25)(8 26 18 21)(9 27 19 22)(10 28 20 23)(41 62 76 55)(42 63 77 56)(43 64 78 57)(44 65 79 58)(45 66 80 59)(46 67 71 60)(47 68 72 51)(48 69 73 52)(49 70 74 53)(50 61 75 54)
(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 23)(2 22)(3 21)(4 25)(5 24)(6 9)(7 8)(11 14)(12 13)(16 19)(17 18)(26 37)(27 36)(28 40)(29 39)(30 38)(31 32)(33 35)(41 76)(42 75)(43 74)(44 73)(45 72)(46 71)(47 80)(48 79)(49 78)(50 77)(51 61)(52 70)(53 69)(54 68)(55 67)(56 66)(57 65)(58 64)(59 63)(60 62)

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

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

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

62 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I4A4B4C4D4E4F4G4H···4L5A5B10A···10F10G···10N20A···20H20I···20X
order122222222244444444···45510···1010···1020···2020···20
size1111224202020222244420···20222···24···42···24···4

62 irreducible representations

dim111111111111111122222222444
type++++++++++++++++++++++++
imageC1C2C2C2C2C2C2C2C2C2C2C2C2C2C2C2D5C4○D4D10D10D10D10D10C4○D202+ 1+4D46D10D48D10
kernelC4217D10C4.D20C422D5Dic5.14D4C22⋊D20D10⋊D4Dic5.5D4D10.13D4C4⋊C4⋊D5C2×D10⋊C4C4×C5⋊D4C23.23D10C207D4C23⋊D10Dic5⋊D4D4×C20C4×D4C2×C10C42C22⋊C4C4⋊C4C22×C4C2×D4C22C10C2C2
# reps1111111111111111242424216244

Matrix representation of C4217D10 in GL6(𝔽41)

25310000
5160000
0017352121
007244038
003540186
00213523
,
3200000
0320000
00111300
00193000
00003913
0000282
,
4000000
0400000
00343500
007000
00273916
0020356
,
1310000
0400000
007100
00343400
001412400
00393961

G:=sub<GL(6,GF(41))| [25,5,0,0,0,0,31,16,0,0,0,0,0,0,17,7,35,2,0,0,35,24,40,1,0,0,21,40,18,35,0,0,21,38,6,23],[32,0,0,0,0,0,0,32,0,0,0,0,0,0,11,19,0,0,0,0,13,30,0,0,0,0,0,0,39,28,0,0,0,0,13,2],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,34,7,27,2,0,0,35,0,39,0,0,0,0,0,1,35,0,0,0,0,6,6],[1,0,0,0,0,0,31,40,0,0,0,0,0,0,7,34,14,39,0,0,1,34,12,39,0,0,0,0,40,6,0,0,0,0,0,1] >;

C4217D10 in GAP, Magma, Sage, TeX

C_4^2\rtimes_{17}D_{10}
% in TeX

G:=Group("C4^2:17D10");
// GroupNames label

G:=SmallGroup(320,1232);
// by ID

G=gap.SmallGroup(320,1232);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,758,100,675,570,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^4=c^10=d^2=1,a*b=b*a,c*a*c^-1=a^-1,d*a*d=a^-1*b^2,b*c=c*b,d*b*d=a^2*b,d*c*d=c^-1>;
// generators/relations

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