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G = C425D10order 320 = 26·5

5th semidirect product of C42 and D10 acting via D10/C5=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C425D10, D20.13D4, Dic10.13D4, (C2×Q8)⋊2D10, C4.51(D4×D5), C4.4D44D5, C20.28(C2×D4), (C4×C20)⋊13C22, (C2×D4).51D10, C20.D45C2, C53(D4.9D4), D204C411C2, (Q8×C10)⋊2C22, C10.51C22≀C2, D46D10.4C2, C20.C232C2, (C22×C10).22D4, C4.Dic56C22, (C2×C20).379C23, C4○D20.19C22, (D4×C10).67C22, C23.10(C5⋊D4), C2.19(C23⋊D10), (C5×C4.4D4)⋊4C2, (C2×C10).510(C2×D4), C22.31(C2×C5⋊D4), (C2×C4).116(C22×D5), SmallGroup(320,688)

Series: Derived Chief Lower central Upper central

C1C2×C20 — C425D10
C1C5C10C20C2×C20C4○D20D46D10 — C425D10
C5C10C2×C20 — C425D10
C1C2C2×C4C4.4D4

Generators and relations for C425D10
 G = < a,b,c,d | a4=b4=c10=d2=1, ab=ba, cac-1=a-1b2, dad=a-1b, cbc-1=b-1, bd=db, dcd=c-1 >

Subgroups: 686 in 152 conjugacy classes, 39 normal (19 characteristic)
C1, C2, C2 [×5], C4 [×2], C4 [×4], C22, C22 [×8], C5, C8 [×2], C2×C4, C2×C4 [×6], D4 [×10], Q8 [×4], C23 [×2], C23 [×2], D5 [×2], C10, C10 [×3], C42, C22⋊C4 [×2], M4(2) [×2], SD16 [×2], Q16 [×2], C2×D4, C2×D4 [×4], C2×Q8, C4○D4 [×4], Dic5 [×2], C20 [×2], C20 [×2], D10 [×4], C2×C10, C2×C10 [×4], C4.D4, C4≀C2 [×2], C4.4D4, C8.C22 [×2], 2+ 1+4, C52C8 [×2], Dic10 [×2], C4×D5 [×2], D20 [×2], C2×Dic5 [×2], C5⋊D4 [×6], C2×C20, C2×C20 [×2], C5×D4 [×2], C5×Q8 [×2], C22×D5 [×2], C22×C10 [×2], D4.9D4, C4.Dic5 [×2], Q8⋊D5 [×2], C5⋊Q16 [×2], C4×C20, C5×C22⋊C4 [×2], C4○D20 [×2], D4×D5 [×2], D42D5 [×2], C2×C5⋊D4 [×2], D4×C10, Q8×C10, D204C4 [×2], C20.D4, C20.C23 [×2], C5×C4.4D4, D46D10, C425D10
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, D5, C2×D4 [×3], D10 [×3], C22≀C2, C5⋊D4 [×2], C22×D5, D4.9D4, D4×D5 [×2], C2×C5⋊D4, C23⋊D10, C425D10

Smallest permutation representation of C425D10
On 80 points
Generators in S80
(1 31 23 69)(2 79 24 59)(3 33 25 61)(4 71 26 51)(5 35 27 63)(6 73 28 53)(7 37 29 65)(8 75 30 55)(9 39 21 67)(10 77 22 57)(11 72 44 52)(12 36 45 64)(13 74 46 54)(14 38 47 66)(15 76 48 56)(16 40 49 68)(17 78 50 58)(18 32 41 70)(19 80 42 60)(20 34 43 62)
(1 28 50 12)(2 13 41 29)(3 30 42 14)(4 15 43 21)(5 22 44 16)(6 17 45 23)(7 24 46 18)(8 19 47 25)(9 26 48 20)(10 11 49 27)(31 53 58 36)(32 37 59 54)(33 55 60 38)(34 39 51 56)(35 57 52 40)(61 75 80 66)(62 67 71 76)(63 77 72 68)(64 69 73 78)(65 79 74 70)
(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 10)(2 9)(3 8)(4 7)(5 6)(11 28)(12 27)(13 26)(14 25)(15 24)(16 23)(17 22)(18 21)(19 30)(20 29)(31 52)(32 51)(33 60)(34 59)(35 58)(36 57)(37 56)(38 55)(39 54)(40 53)(41 48)(42 47)(43 46)(44 45)(49 50)(62 70)(63 69)(64 68)(65 67)(71 79)(72 78)(73 77)(74 76)

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

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

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

44 conjugacy classes

class 1 2A2B2C2D2E2F4A4B4C4D4E4F4G5A5B8A8B10A···10F10G10H10I10J20A···20L20M20N20O20P
order12222224444444558810···101010101020···2020202020
size1124420202244820202240402···288884···48888

44 irreducible representations

dim11111122222222444
type++++++++++++++
imageC1C2C2C2C2C2D4D4D4D5D10D10D10C5⋊D4D4.9D4D4×D5C425D10
kernelC425D10D204C4C20.D4C20.C23C5×C4.4D4D46D10Dic10D20C22×C10C4.4D4C42C2×D4C2×Q8C23C5C4C1
# reps12121122222228248

Matrix representation of C425D10 in GL6(𝔽41)

17400000
1240000
0016162516
0016161625
0016251616
0025161616
,
4000000
0400000
000010
000001
0040000
0004000
,
770000
34400000
000001
000010
000100
001000
,
770000
40340000
000001
0000400
0004000
001000

G:=sub<GL(6,GF(41))| [17,1,0,0,0,0,40,24,0,0,0,0,0,0,16,16,16,25,0,0,16,16,25,16,0,0,25,16,16,16,0,0,16,25,16,16],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,1,0,0,0,0,0,0,1,0,0],[7,34,0,0,0,0,7,40,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1,0,0,0],[7,40,0,0,0,0,7,34,0,0,0,0,0,0,0,0,0,1,0,0,0,0,40,0,0,0,0,40,0,0,0,0,1,0,0,0] >;

C425D10 in GAP, Magma, Sage, TeX

C_4^2\rtimes_5D_{10}
% in TeX

G:=Group("C4^2:5D10");
// GroupNames label

G:=SmallGroup(320,688);
// by ID

G=gap.SmallGroup(320,688);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,254,219,184,1123,570,297,136,1684,12550]);
// 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*b^2,d*a*d=a^-1*b,c*b*c^-1=b^-1,b*d=d*b,d*c*d=c^-1>;
// generators/relations

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