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G = C20.60(C4⋊C4)  order 320 = 26·5

7th non-split extension by C20 of C4⋊C4 acting via C4⋊C4/C2×C4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: (C2×C20).8Q8, C20.60(C4⋊C4), (C2×C20).105D4, (C2×C4).127D20, C4.Dic5⋊10C4, (C2×C4).3Dic10, (C2×C10).24C42, C42⋊C2.3D5, (C22×C4).58D10, C22⋊C4.1Dic5, C23.6(C2×Dic5), C22.2(C4×Dic5), C20.57(C22⋊C4), C5⋊4(M4(2)⋊4C4), C4.9(C10.D4), C22.2(C4⋊Dic5), C4.47(D10⋊C4), (C22×C20).122C22, C22.11(C23.D5), C10.28(C2.C42), C2.10(C10.10C42), (C2×C5⋊2C8)⋊2C4, (C2×C4).19(C4×D5), (C5×C22⋊C4).9C4, (C2×C10).33(C4⋊C4), (C2×C20).231(C2×C4), (C2×C4.Dic5).9C2, (C2×C4).232(C5⋊D4), (C5×C42⋊C2).3C2, (C22×C10).98(C2×C4), (C2×C10).155(C22⋊C4), SmallGroup(320,91)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2×C10 — C20.60(C4⋊C4)
Chief series
C1 — C5 — C10 — C20 — C2×C20 — C22×C20 — C2×C4.Dic5 — C20.60(C4⋊C4)
Lower central
C5 — C10 — C2×C10 — C20.60(C4⋊C4)
Upper central
C1 — C4 — C22×C4 — C42⋊C2

Generators and relations for C20.60(C4⋊C4)
 G = < a,b,c | a20=c4=1, b4=a10, bab-1=a-1, ac=ca, cbc-1=a5b3 >

Subgroups: 230 in 90 conjugacy classes, 47 normal (39 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, C23, C10, C10, C42, C22⋊C4, C4⋊C4, C2×C8, M4(2), C22×C4, C20, C20, C2×C10, C2×C10, C42⋊C2, C2×M4(2), C5⋊2C8, C2×C20, C2×C20, C22×C10, M4(2)⋊4C4, C2×C5⋊2C8, C2×C5⋊2C8, C4.Dic5, C4.Dic5, C4×C20, C5×C22⋊C4, C5×C4⋊C4, C22×C20, C2×C4.Dic5, C5×C42⋊C2, C20.60(C4⋊C4)
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, D5, C42, C22⋊C4, C4⋊C4, Dic5, D10, C2.C42, Dic10, C4×D5, D20, C2×Dic5, C5⋊D4, M4(2)⋊4C4, C4×Dic5, C10.D4, C4⋊Dic5, D10⋊C4, C23.D5, C10.10C42, C20.60(C4⋊C4)

Smallest permutation representation of C20.60(C4⋊C4)
►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 23 74 60 11 33 64 50)(2 22 75 59 12 32 65 49)(3 21 76 58 13 31 66 48)(4 40 77 57 14 30 67 47)(5 39 78 56 15 29 68 46)(6 38 79 55 16 28 69 45)(7 37 80 54 17 27 70 44)(8 36 61 53 18 26 71 43)(9 35 62 52 19 25 72 42)(10 34 63 51 20 24 73 41)

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

62 conjugacy classes

class 1 2A2B2C2D4A4B4C4D4E4F4G4H4I5A5B8A···8H10A···10F10G10H10I10J20A···20H20I···20AB
order12222444444444558···810···101010101020···2020···20
size112221122244442220···202···244442···24···4

62 irreducible representations

dim11111122222222244
type++++-+-+-+
imageC1C2C2C4C4C4D4Q8D5Dic5D10Dic10C4×D5D20C5⋊D4M4(2)⋊4C4C20.60(C4⋊C4)
kernelC20.60(C4⋊C4)C2×C4.Dic5C5×C42⋊C2C2×C5⋊2C8C4.Dic5C5×C22⋊C4C2×C20C2×C20C42⋊C2C22⋊C4C22×C4C2×C4C2×C4C2×C4C2×C4C5C1
# reps12144431242484828

Matrix representation of C20.60(C4⋊C4) ►in GL6(𝔽41)

2300000
2250000
0032000
0003200
000090
000009
,
110000
39400000
00003237
000009
00403600
0025100
,
3200000
0320000
00403600
000100
00004036
0000251

G:=sub<GL(6,GF(41))| [23,2,0,0,0,0,0,25,0,0,0,0,0,0,32,0,0,0,0,0,0,32,0,0,0,0,0,0,9,0,0,0,0,0,0,9],[1,39,0,0,0,0,1,40,0,0,0,0,0,0,0,0,40,25,0,0,0,0,36,1,0,0,32,0,0,0,0,0,37,9,0,0],[32,0,0,0,0,0,0,32,0,0,0,0,0,0,40,0,0,0,0,0,36,1,0,0,0,0,0,0,40,25,0,0,0,0,36,1] >;
 

C20.60(C4⋊C4) in GAP, Magma, Sage, TeX 

C_{20}._{60}(C_4\rtimes C_4)
 
% in TeX
 
G:=Group("C20.60(C4:C4)");
 
// GroupNames label
 
G:=SmallGroup(320,91);
 
// by ID
 
G=gap.SmallGroup(320,91);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,253,64,184,1123,851,12550]);
 
// Polycyclic
 
G:=Group<a,b,c|a^20=c^4=1,b^4=a^10,b*a*b^-1=a^-1,a*c=c*a,c*b*c^-1=a^5*b^3>;
 
// generators/relations
 

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