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G = C20.C42order 320 = 26·5

2nd non-split extension by C20 of C42 acting via C42/C2=C2×C4

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C20.2C42, C4⋊C44F5, (C4×F5)⋊3C4, C10.9C4≀C2, C4.F51C4, C4.7(C4×F5), C4⋊Dic59C4, C20.2(C4⋊C4), (C4×D5).4Q8, (C4×D5).19D4, C4.17(C4⋊F5), D10.3(C4⋊C4), C52(C426C4), C2.2(D4⋊F5), C2.2(Q82F5), (C22×D5).56D4, Dic5.22(C4⋊C4), D10.3(C22⋊C4), (C2×Dic5).255D4, C2.8(D10.3Q8), C22.34(C22⋊F5), C10.6(C2.C42), Dic5.30(C22⋊C4), (C5×C4⋊C4)⋊4C4, (C2×C4×F5).1C2, (C2×C4).67(C2×F5), (C2×C4.F5).1C2, (C2×C20).33(C2×C4), (C4×D5).13(C2×C4), C4⋊C47D5.16C2, (C2×C4×D5).185C22, (C2×C10).25(C22⋊C4), SmallGroup(320,213)

Series: Derived Chief Lower central Upper central

C1C20 — C20.C42
C1C5C10Dic5C2×Dic5C2×C4×D5C2×C4.F5 — C20.C42
C5C10C20 — C20.C42
C1C22C2×C4C4⋊C4

Generators and relations for C20.C42
 G = < a,b,c | a20=c4=1, b4=a10, bab-1=a3, cac-1=a11, cbc-1=a5b >

Subgroups: 450 in 110 conjugacy classes, 38 normal (34 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×8], C22, C22 [×4], C5, C8 [×2], C2×C4, C2×C4 [×13], C23, D5 [×2], C10 [×3], C42 [×4], C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, M4(2) [×3], C22×C4 [×2], Dic5 [×2], Dic5, C20 [×2], C20, F5 [×4], D10 [×2], D10 [×2], C2×C10, C2×C42, C42⋊C2, C2×M4(2), C5⋊C8 [×2], C4×D5 [×4], C2×Dic5, C2×Dic5, C2×C20, C2×C20, C2×F5 [×6], C22×D5, C426C4, C4×Dic5, C4⋊Dic5, D10⋊C4, C5×C4⋊C4, C4.F5 [×2], C4.F5, C4×F5 [×2], C4×F5, C2×C5⋊C8, C2×C4×D5, C22×F5, C4⋊C47D5, C2×C4.F5, C2×C4×F5, C20.C42
Quotients: C1, C2 [×3], C4 [×6], C22, C2×C4 [×3], D4 [×3], Q8, C42, C22⋊C4 [×3], C4⋊C4 [×3], F5, C2.C42, C4≀C2 [×2], C2×F5, C426C4, C4×F5, C4⋊F5, C22⋊F5, D10.3Q8, D4⋊F5, Q82F5, C20.C42

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

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

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

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

38 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F4G4H4I···4P4Q4R 5 8A8B8C8D10A10B10C20A···20F
order122222444444444···4445888810101020···20
size111110102244555510···1020204202020204448···8

38 irreducible representations

dim11111111222224444488
type+++++-+++++-+
imageC1C2C2C2C4C4C4C4D4Q8D4D4C4≀C2F5C2×F5C4×F5C4⋊F5C22⋊F5D4⋊F5Q82F5
kernelC20.C42C4⋊C47D5C2×C4.F5C2×C4×F5C4⋊Dic5C5×C4⋊C4C4.F5C4×F5C4×D5C4×D5C2×Dic5C22×D5C10C4⋊C4C2×C4C4C4C22C2C2
# reps11112244111181122211

Matrix representation of C20.C42 in GL8(𝔽41)

320000000
09000000
004000000
000400000
00000001
000040001
000004001
000000401
,
09000000
10000000
00150000
0016400000
0000338190
0000223803
0000303822
0000019383
,
01000000
400000000
00940000
000320000
0000193038
0000022338
0000383220
0000380319

G:=sub<GL(8,GF(41))| [32,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,1,1,1,1],[0,1,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,0,1,16,0,0,0,0,0,0,5,40,0,0,0,0,0,0,0,0,3,22,3,0,0,0,0,0,38,38,0,19,0,0,0,0,19,0,38,38,0,0,0,0,0,3,22,3],[0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,4,32,0,0,0,0,0,0,0,0,19,0,38,38,0,0,0,0,3,22,3,0,0,0,0,0,0,3,22,3,0,0,0,0,38,38,0,19] >;

C20.C42 in GAP, Magma, Sage, TeX

C_{20}.C_4^2
% in TeX

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

G:=SmallGroup(320,213);
// by ID

G=gap.SmallGroup(320,213);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,253,64,184,1684,851,102,6278,3156]);
// Polycyclic

G:=Group<a,b,c|a^20=c^4=1,b^4=a^10,b*a*b^-1=a^3,c*a*c^-1=a^11,c*b*c^-1=a^5*b>;
// generators/relations

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