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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, (C2×C8).3F5, (C2×C40).9C4, D5⋊C8.3C4, C4.30(C4×F5), (C4×D5).52D4, D10.5(C4⋊C4), C5⋊(C4.10C42), C22.2(C4⋊F5), (C22×D5).7Q8, (C2×Dic5).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), (C2×C52C8).5C4, (C4×D5).44(C2×C4), (C2×C8⋊D5).9C2, (C2×C4).128(C2×F5), (C2×C10).15(C4⋊C4), (C2×C20).143(C2×C4), (C2×C4×D5).283C22, SmallGroup(320,235)

Series: Derived Chief Lower central Upper central

C1C20 — C20.25C42
C1C5C10Dic5C4×D5C2×C4×D5D5⋊M4(2) — C20.25C42
C5C20 — C20.25C42
C1C4C2×C8

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 [×3], C4 [×2], C4 [×2], C22, C22 [×3], C5, C8 [×6], C2×C4, C2×C4 [×5], C23, D5 [×2], C10, C10, C2×C8, C2×C8 [×5], M4(2) [×6], C22×C4, Dic5 [×2], C20 [×2], D10 [×2], D10, C2×C10, C2×M4(2) [×3], C52C8, C40, C5⋊C8 [×4], C4×D5 [×4], C2×Dic5, C2×C20, C22×D5, C4.10C42, C8⋊D5 [×2], C2×C52C8, C2×C40, D5⋊C8 [×4], C4.F5 [×2], C22.F5 [×2], C2×C4×D5, C2×C8⋊D5, D5⋊M4(2) [×2], C20.25C42
Quotients: C1, C2 [×3], C4 [×6], C22, C2×C4 [×3], D4 [×3], Q8, C42, C22⋊C4 [×3], C4⋊C4 [×3], F5, C2.C42, C2×F5, C4.10C42, C4×F5, 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 51 16 46 11 41 6 56)(2 48 5 59 12 58 15 49)(3 45 14 52 13 55 4 42)(7 53 10 44 17 43 20 54)(8 50 19 57 18 60 9 47)(21 66 26 71 31 76 36 61)(22 63 35 64 32 73 25 74)(23 80 24 77 33 70 34 67)(27 68 40 69 37 78 30 79)(28 65 29 62 38 75 39 72)
(1 26 16 21 11 36 6 31)(2 27 17 22 12 37 7 32)(3 28 18 23 13 38 8 33)(4 29 19 24 14 39 9 34)(5 30 20 25 15 40 10 35)(41 76 46 61 51 66 56 71)(42 77 47 62 52 67 57 72)(43 78 48 63 53 68 58 73)(44 79 49 64 54 69 59 74)(45 80 50 65 55 70 60 75)

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

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

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

38 conjugacy classes

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

38 irreducible representations

dim1111112224444444
type+++++-+++
imageC1C2C2C4C4C4D4D4Q8F5C2×F5C4.10C42C4×F5C22⋊F5C4⋊F5C20.25C42
kernelC20.25C42C2×C8⋊D5D5⋊M4(2)C2×C52C8C2×C40D5⋊C8C4×D5C2×Dic5C22×D5C2×C8C2×C4C5C4C4C22C1
# 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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