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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.Dic510C4, (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), C54(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×C52C8)⋊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

C1C2×C10 — C20.60(C4⋊C4)
C1C5C10C20C2×C20C22×C20C2×C4.Dic5 — C20.60(C4⋊C4)
C5C10C2×C10 — C20.60(C4⋊C4)
C1C4C22×C4C42⋊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 [×3], C4 [×4], C4 [×2], C22 [×3], C22, C5, C8 [×4], C2×C4 [×6], C2×C4 [×2], C23, C10, C10 [×3], C42, C22⋊C4 [×2], C4⋊C4, C2×C8 [×3], M4(2) [×5], C22×C4, C20 [×4], C20 [×2], C2×C10 [×3], C2×C10, C42⋊C2, C2×M4(2) [×2], C52C8 [×4], C2×C20 [×6], C2×C20 [×2], C22×C10, M4(2)⋊4C4, C2×C52C8 [×2], C2×C52C8, C4.Dic5 [×2], C4.Dic5 [×3], C4×C20, C5×C22⋊C4 [×2], C5×C4⋊C4, C22×C20, C2×C4.Dic5 [×2], C5×C42⋊C2, C20.60(C4⋊C4)
Quotients: C1, C2 [×3], C4 [×6], C22, C2×C4 [×3], D4 [×3], Q8, D5, C42, C22⋊C4 [×3], C4⋊C4 [×3], Dic5 [×2], D10, C2.C42, Dic10, C4×D5 [×2], D20, C2×Dic5, C5⋊D4 [×2], M4(2)⋊4C4, C4×Dic5, C10.D4 [×2], C4⋊Dic5, D10⋊C4 [×2], 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 27 51 77 11 37 41 67)(2 26 52 76 12 36 42 66)(3 25 53 75 13 35 43 65)(4 24 54 74 14 34 44 64)(5 23 55 73 15 33 45 63)(6 22 56 72 16 32 46 62)(7 21 57 71 17 31 47 61)(8 40 58 70 18 30 48 80)(9 39 59 69 19 29 49 79)(10 38 60 68 20 28 50 78)
(1 11)(2 12)(3 13)(4 14)(5 15)(6 16)(7 17)(8 18)(9 19)(10 20)(21 66 31 76)(22 67 32 77)(23 68 33 78)(24 69 34 79)(25 70 35 80)(26 71 36 61)(27 72 37 62)(28 73 38 63)(29 74 39 64)(30 75 40 65)

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

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

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

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×C52C8C4.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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