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G = C20.24C42order 320 = 26·5

17th non-split extension by C20 of C42 acting via C42/C4=C4

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C20.24C42, (C2×C8)⋊2F5, (C2×C40)⋊4C4, (C4×F5)⋊2C4, C4.29(C4×F5), (C4×D5).51D4, C52(C4.9C42), C22.1(C4⋊F5), Dic5.5(C4⋊C4), (C2×Dic5).10Q8, (C22×D5).57D4, C4.25(C22⋊F5), C20.25(C22⋊C4), D10.5(C22⋊C4), C2.11(D10.3Q8), C10.10(C2.C42), D10.C23.10C2, (C2×C52C8)⋊5C4, (C4×D5).43(C2×C4), (C2×C8⋊D5).8C2, (C2×C4).126(C2×F5), (C2×C10).13(C4⋊C4), (C2×C20).142(C2×C4), (C2×C4×D5).282C22, SmallGroup(320,233)

Series: Derived Chief Lower central Upper central

C1C20 — C20.24C42
C1C5C10D10C4×D5C2×C4×D5D10.C23 — C20.24C42
C5C20 — C20.24C42
C1C4C2×C8

Generators and relations for C20.24C42
 G = < a,b,c | a20=b4=1, c4=a10, bab-1=a13, ac=ca, cbc-1=a5b >

Subgroups: 418 in 94 conjugacy classes, 34 normal (20 characteristic)
C1, C2, C2 [×3], C4 [×2], C4 [×6], C22, C22 [×3], C5, C8 [×2], C2×C4, C2×C4 [×9], C23, D5 [×2], C10, C10, C42 [×4], C22⋊C4 [×2], C4⋊C4 [×2], C2×C8, C2×C8, M4(2) [×2], C22×C4, Dic5 [×2], C20 [×2], F5 [×4], D10 [×2], D10, C2×C10, C42⋊C2 [×2], C2×M4(2), C52C8, C40, C4×D5 [×4], C2×Dic5, C2×C20, C2×F5 [×4], C22×D5, C4.9C42, C8⋊D5 [×2], C2×C52C8, C2×C40, C4×F5 [×4], C4⋊F5 [×2], C22⋊F5 [×2], C2×C4×D5, C2×C8⋊D5, D10.C23 [×2], C20.24C42
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, C4.9C42, C4×F5, C4⋊F5, C22⋊F5, D10.3Q8, C20.24C42

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

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

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

38 conjugacy classes

class 1 2A2B2C2D4A4B4C4D4E4F···4M 5 8A8B8C8D10A10B10C20A20B20C20D40A···40H
order12222444444···4588881010102020202040···40
size1121010112101020···20444202044444444···4

38 irreducible representations

dim1111112224444444
type++++-++++
imageC1C2C2C4C4C4D4Q8D4F5C2×F5C4.9C42C4×F5C22⋊F5C4⋊F5C20.24C42
kernelC20.24C42C2×C8⋊D5D10.C23C2×C52C8C2×C40C4×F5C4×D5C2×Dic5C22×D5C2×C8C2×C4C5C4C4C22C1
# reps1122282111122228

Matrix representation of C20.24C42 in GL4(𝔽41) generated by

32323232
9000
0900
0090
,
3402727
2727034
147140
734347
,
3019613
2817634
7352413
28352211
G:=sub<GL(4,GF(41))| [32,9,0,0,32,0,9,0,32,0,0,9,32,0,0,0],[34,27,14,7,0,27,7,34,27,0,14,34,27,34,0,7],[30,28,7,28,19,17,35,35,6,6,24,22,13,34,13,11] >;

C20.24C42 in GAP, Magma, Sage, TeX

C_{20}._{24}C_4^2
% in TeX

G:=Group("C20.24C4^2");
// GroupNames label

G:=SmallGroup(320,233);
// by ID

G=gap.SmallGroup(320,233);
# by ID

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

G:=Group<a,b,c|a^20=b^4=1,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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