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G = C421Dic5order 320 = 26·5

1st semidirect product of C42 and Dic5 acting via Dic5/C5=C4

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C421Dic5, C20.49C42, (C4×C20)⋊3C4, (C2×C20).6Q8, (C4×Dic5)⋊3C4, (C2×C4).13D20, C20.59(C4⋊C4), C53(C4.9C42), (C2×C20).104D4, (C2×C4).2Dic10, C4.24(C4×Dic5), C4.12(C4⋊Dic5), (C22×C10).42D4, (C22×C4).57D10, C42⋊C2.1D5, C20.56(C22⋊C4), C4.8(C10.D4), C23.16(C5⋊D4), C4.33(D10⋊C4), (C22×C20).120C22, C23.21D10.7C2, C22.3(C10.D4), C22.10(C23.D5), C22.17(D10⋊C4), C2.8(C10.10C42), C10.26(C2.C42), (C2×C52C8)⋊1C4, (C2×C4).139(C4×D5), (C2×C10).31(C4⋊C4), (C2×C20).229(C2×C4), (C2×C4).20(C5⋊D4), (C2×C4).72(C2×Dic5), (C2×C4.Dic5).7C2, (C5×C42⋊C2).1C2, (C2×C10).153(C22⋊C4), SmallGroup(320,89)

Series: Derived Chief Lower central Upper central

C1C20 — C421Dic5
C1C5C10C2×C10C22×C10C22×C20C23.21D10 — C421Dic5
C5C20 — C421Dic5
C1C4C42⋊C2

Generators and relations for C421Dic5
 G = < a,b,c,d | a4=b4=c10=1, d2=c5, ab=ba, cac-1=ab2, dad-1=ab-1, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 278 in 94 conjugacy classes, 47 normal (39 characteristic)
C1, C2, C2 [×3], C4 [×4], C4 [×4], C22 [×3], C22, C5, C8 [×2], C2×C4 [×6], C2×C4 [×4], C23, C10, C10 [×3], C42 [×2], C42 [×2], C22⋊C4 [×2], C4⋊C4 [×2], C2×C8 [×2], M4(2) [×2], C22×C4, Dic5 [×2], C20 [×4], C20 [×2], C2×C10 [×3], C2×C10, C42⋊C2, C42⋊C2, C2×M4(2), C52C8 [×2], C2×Dic5 [×2], C2×C20 [×6], C2×C20 [×2], C22×C10, C4.9C42, C2×C52C8 [×2], C4.Dic5 [×2], C4×Dic5 [×2], C4⋊Dic5, C23.D5, C4×C20 [×2], C5×C22⋊C4, C5×C4⋊C4, C22×C20, C2×C4.Dic5, C23.21D10, C5×C42⋊C2, C421Dic5
Quotients: C1, C2 [×3], C4 [×6], C22, C2×C4 [×3], D4 [×3], Q8, D5, C42, C22⋊C4 [×3], C4⋊C4 [×3], Dic5 [×2], D10, C2.C42, Dic10, C4×D5 [×2], D20, C2×Dic5, C5⋊D4 [×2], C4.9C42, C4×Dic5, C10.D4 [×2], C4⋊Dic5, D10⋊C4 [×2], C23.D5, C10.10C42, C421Dic5

Smallest permutation representation of C421Dic5
On 80 points
Generators in S80
(1 59 19 61)(2 55 20 67)(3 51 16 63)(4 57 17 69)(5 53 18 65)(6 60 14 62)(7 56 15 68)(8 52 11 64)(9 58 12 70)(10 54 13 66)(21 46 36 76)(22 42 37 72)(23 48 38 78)(24 44 39 74)(25 50 40 80)(26 47 31 77)(27 43 32 73)(28 49 33 79)(29 45 34 75)(30 41 35 71)
(1 33 10 39)(2 34 6 40)(3 35 7 36)(4 31 8 37)(5 32 9 38)(11 22 17 26)(12 23 18 27)(13 24 19 28)(14 25 20 29)(15 21 16 30)(41 68 46 63)(42 69 47 64)(43 70 48 65)(44 61 49 66)(45 62 50 67)(51 71 56 76)(52 72 57 77)(53 73 58 78)(54 74 59 79)(55 75 60 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 26)(2 30)(3 29)(4 28)(5 27)(6 21)(7 25)(8 24)(9 23)(10 22)(11 33)(12 32)(13 31)(14 35)(15 34)(16 40)(17 39)(18 38)(19 37)(20 36)(41 75 46 80)(42 74 47 79)(43 73 48 78)(44 72 49 77)(45 71 50 76)(51 62 56 67)(52 61 57 66)(53 70 58 65)(54 69 59 64)(55 68 60 63)

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

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

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

62 conjugacy classes

class 1 2A2B2C2D4A4B4C4D4E4F4G4H4I4J4K4L4M5A5B8A8B8C8D10A···10F10G10H10I10J20A···20H20I···20AB
order12222444444444444455888810···101010101020···2020···20
size112221122244442020202022202020202···244442···24···4

62 irreducible representations

dim11111112222222222244
type+++++-++-+-+
imageC1C2C2C2C4C4C4D4Q8D4D5Dic5D10Dic10C4×D5D20C5⋊D4C5⋊D4C4.9C42C421Dic5
kernelC421Dic5C2×C4.Dic5C23.21D10C5×C42⋊C2C2×C52C8C4×Dic5C4×C20C2×C20C2×C20C22×C10C42⋊C2C42C22×C4C2×C4C2×C4C2×C4C2×C4C23C5C1
# reps11114442112424844428

Matrix representation of C421Dic5 in GL4(𝔽41) generated by

901138
09329
529320
1236032
,
32000
03200
00320
00032
,
3540252
103913
0061
00400
,
18203717
352304
003916
00282
G:=sub<GL(4,GF(41))| [9,0,5,12,0,9,29,36,11,3,32,0,38,29,0,32],[32,0,0,0,0,32,0,0,0,0,32,0,0,0,0,32],[35,1,0,0,40,0,0,0,25,39,6,40,2,13,1,0],[18,35,0,0,20,23,0,0,37,0,39,28,17,4,16,2] >;

C421Dic5 in GAP, Magma, Sage, TeX

C_4^2\rtimes_1{\rm Dic}_5
% in TeX

G:=Group("C4^2:1Dic5");
// GroupNames label

G:=SmallGroup(320,89);
// by ID

G=gap.SmallGroup(320,89);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,253,64,387,1123,102,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^4=c^10=1,d^2=c^5,a*b=b*a,c*a*c^-1=a*b^2,d*a*d^-1=a*b^-1,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

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