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G = C20.32C42order 320 = 26·5

2nd non-split extension by C20 of C42 acting via C42/C22=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C20.32C42, C4⋊C44Dic5, C10.25C4≀C2, (C2×C20).7Q8, (C4×Dic5)⋊8C4, C4.Dic59C4, C20.39(C4⋊C4), C4.2(C4×Dic5), C55(C426C4), (C2×C4).126D20, (C2×C20).488D4, C4.2(C4⋊Dic5), (C2×C4).24Dic10, (C22×C10).43D4, C42⋊C2.2D5, (C22×C4).324D10, C23.47(C5⋊D4), C4.46(D10⋊C4), C20.108(C22⋊C4), C2.1(D42Dic5), C4.30(C10.D4), (C22×C20).121C22, C22.4(C10.D4), C22.28(C23.D5), C22.18(D10⋊C4), C10.27(C2.C42), C2.9(C10.10C42), (C5×C4⋊C4)⋊13C4, (C2×C4).68(C4×D5), (C2×C4×Dic5).1C2, (C2×C10).32(C4⋊C4), (C2×C20).230(C2×C4), (C2×C4).36(C2×Dic5), (C2×C4.Dic5).8C2, (C2×C4).268(C5⋊D4), (C5×C42⋊C2).2C2, (C2×C10).154(C22⋊C4), SmallGroup(320,90)

Series: Derived Chief Lower central Upper central

C1C20 — C20.32C42
C1C5C10C20C2×C20C22×C20C2×C4.Dic5 — C20.32C42
C5C10C20 — C20.32C42
C1C2×C4C22×C4C42⋊C2

Generators and relations for C20.32C42
 G = < a,b,c | a20=c4=1, b4=a10, bab-1=a-1, cac-1=a11, cbc-1=a5b >

Subgroups: 326 in 110 conjugacy classes, 51 normal (39 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×4], C4 [×6], C22 [×3], C22 [×2], C5, C8 [×2], C2×C4 [×6], C2×C4 [×8], C23, C10, C10 [×2], C10 [×2], C42 [×4], C22⋊C4, C4⋊C4 [×2], C2×C8, M4(2) [×3], C22×C4, C22×C4, Dic5 [×4], C20 [×4], C20 [×2], C2×C10 [×3], C2×C10 [×2], C2×C42, C42⋊C2, C2×M4(2), C52C8 [×2], C2×Dic5 [×6], C2×C20 [×6], C2×C20 [×2], C22×C10, C426C4, C2×C52C8, C4.Dic5 [×2], C4.Dic5, C4×Dic5 [×2], C4×Dic5, C4×C20, C5×C22⋊C4, C5×C4⋊C4 [×2], C22×Dic5, C22×C20, C2×C4.Dic5, C2×C4×Dic5, C5×C42⋊C2, C20.32C42
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, C4≀C2 [×2], Dic10, C4×D5 [×2], D20, C2×Dic5, C5⋊D4 [×2], C426C4, C4×Dic5, C10.D4 [×2], C4⋊Dic5, D10⋊C4 [×2], C23.D5, C10.10C42, D42Dic5 [×2], C20.32C42

Smallest permutation representation of C20.32C42
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 76 26 51 11 66 36 41)(2 75 27 50 12 65 37 60)(3 74 28 49 13 64 38 59)(4 73 29 48 14 63 39 58)(5 72 30 47 15 62 40 57)(6 71 31 46 16 61 21 56)(7 70 32 45 17 80 22 55)(8 69 33 44 18 79 23 54)(9 68 34 43 19 78 24 53)(10 67 35 42 20 77 25 52)
(1 66 21 46)(2 77 22 57)(3 68 23 48)(4 79 24 59)(5 70 25 50)(6 61 26 41)(7 72 27 52)(8 63 28 43)(9 74 29 54)(10 65 30 45)(11 76 31 56)(12 67 32 47)(13 78 33 58)(14 69 34 49)(15 80 35 60)(16 71 36 51)(17 62 37 42)(18 73 38 53)(19 64 39 44)(20 75 40 55)

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

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

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

68 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F4G4H4I4J4K···4R5A5B8A8B8C8D10A···10F10G10H10I10J20A···20H20I···20AB
order12222244444444444···455888810···101010101020···2020···20
size111122111122444410···1022202020202···244442···24···4

68 irreducible representations

dim11111112222222222224
type+++++-++-+-+
imageC1C2C2C2C4C4C4D4Q8D4D5Dic5D10C4≀C2Dic10C4×D5D20C5⋊D4C5⋊D4D42Dic5
kernelC20.32C42C2×C4.Dic5C2×C4×Dic5C5×C42⋊C2C4.Dic5C4×Dic5C5×C4⋊C4C2×C20C2×C20C22×C10C42⋊C2C4⋊C4C22×C4C10C2×C4C2×C4C2×C4C2×C4C23C2
# reps11114442112428484448

Matrix representation of C20.32C42 in GL4(𝔽41) generated by

9600
03200
00351
00400
,
153800
372600
003928
00162
,
26500
41500
00213
002839
G:=sub<GL(4,GF(41))| [9,0,0,0,6,32,0,0,0,0,35,40,0,0,1,0],[15,37,0,0,38,26,0,0,0,0,39,16,0,0,28,2],[26,4,0,0,5,15,0,0,0,0,2,28,0,0,13,39] >;

C20.32C42 in GAP, Magma, Sage, TeX

C_{20}._{32}C_4^2
% in TeX

G:=Group("C20.32C4^2");
// GroupNames label

G:=SmallGroup(320,90);
// by ID

G=gap.SmallGroup(320,90);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,253,64,184,1684,851,102,12550]);
// Polycyclic

G:=Group<a,b,c|a^20=c^4=1,b^4=a^10,b*a*b^-1=a^-1,c*a*c^-1=a^11,c*b*c^-1=a^5*b>;
// generators/relations

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