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G = C20.25C42order 320 = 26·5

18th non-split extension by C20 of C42 acting via C42/C4=C4

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C20.25C42, (C2xC8).3F5, (C2xC40).9C4, D5:C8.3C4, C4.30(C4xF5), (C4xD5).52D4, D10.5(C4:C4), C5:(C4.10C42), C22.2(C4:F5), (C22xD5).7Q8, (C2xDic5).99D4, C4.27(C22:F5), C20.27(C22:C4), D5:M4(2).10C2, Dic5.4(C22:C4), C2.13(D10.3Q8), C10.12(C2.C42), (C2xC5:2C8).5C4, (C4xD5).44(C2xC4), (C2xC8:D5).9C2, (C2xC4).128(C2xF5), (C2xC10).15(C4:C4), (C2xC20).143(C2xC4), (C2xC4xD5).283C22, SmallGroup(320,235)

Series: Derived Chief Lower central Upper central

C1C20 — C20.25C42
C1C5C10Dic5C4xD5C2xC4xD5D5:M4(2) — C20.25C42
C5C20 — C20.25C42
C1C4C2xC8

Generators and relations for C20.25C42
 G = < a,b,c | a20=1, b4=c4=a10, bab-1=a13, ac=ca, cbc-1=a5b >

Subgroups: 322 in 86 conjugacy classes, 34 normal (20 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2xC4, C2xC4, C23, D5, C10, C10, C2xC8, C2xC8, M4(2), C22xC4, Dic5, C20, D10, D10, C2xC10, C2xM4(2), C5:2C8, C40, C5:C8, C4xD5, C2xDic5, C2xC20, C22xD5, C4.10C42, C8:D5, C2xC5:2C8, C2xC40, D5:C8, C4.F5, C22.F5, C2xC4xD5, C2xC8:D5, D5:M4(2), C20.25C42
Quotients: C1, C2, C4, C22, C2xC4, D4, Q8, C42, C22:C4, C4:C4, F5, C2.C42, C2xF5, C4.10C42, C4xF5, C4:F5, C22:F5, D10.3Q8, C20.25C42

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

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

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

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

38 conjugacy classes

class 1 2A2B2C2D4A4B4C4D4E 5 8A8B8C···8L10A10B10C20A20B20C20D40A···40H
order12222444445888···81010102020202040···40
size1121010112101044420···2044444444···4

38 irreducible representations

dim1111112224444444
type+++++-+++
imageC1C2C2C4C4C4D4D4Q8F5C2xF5C4.10C42C4xF5C22:F5C4:F5C20.25C42
kernelC20.25C42C2xC8:D5D5:M4(2)C2xC5:2C8C2xC40D5:C8C4xD5C2xDic5C22xD5C2xC8C2xC4C5C4C4C22C1
# reps1122282111122228

Matrix representation of C20.25C42 in GL4(F41) generated by

133200
9000
0262828
26401332
,
3834390
1431039
12537
18172710
,
63900
23500
2701323
27141828
G:=sub<GL(4,GF(41))| [13,9,0,26,32,0,26,40,0,0,28,13,0,0,28,32],[38,14,1,18,34,31,25,17,39,0,3,27,0,39,7,10],[6,2,27,27,39,35,0,14,0,0,13,18,0,0,23,28] >;

C20.25C42 in GAP, Magma, Sage, TeX

C_{20}._{25}C_4^2
% in TeX

G:=Group("C20.25C4^2");
// GroupNames label

G:=SmallGroup(320,235);
// by ID

G=gap.SmallGroup(320,235);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,253,64,387,184,1123,136,102,6278,3156]);
// Polycyclic

G:=Group<a,b,c|a^20=1,b^4=c^4=a^10,b*a*b^-1=a^13,a*c=c*a,c*b*c^-1=a^5*b>;
// generators/relations

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