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G = C20.33C42order 320 = 26·5

3rd non-split extension by C20 of C42 acting via C42/C22=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C20.33C42, C23.33D20, M4(2)⋊3Dic5, C10.24C4≀C2, C4⋊Dic518C4, (C4×Dic5)⋊9C4, C20.41(C4⋊C4), (C2×C20).10Q8, C4.3(C4×Dic5), C56(C426C4), (C2×C20).492D4, (C5×M4(2))⋊12C4, (C2×C4).25Dic10, (C22×C10).46D4, (C2×M4(2)).6D5, C2.3(D207C4), (C22×C4).325D10, C22.3(C4⋊Dic5), C4.27(C23.D5), C20.141(C22⋊C4), C4.10(C10.D4), (C10×M4(2)).10C2, (C22×C20).124C22, C23.21D10.8C2, C22.42(D10⋊C4), C10.33(C2.C42), C2.14(C10.10C42), (C2×C4).69(C4×D5), (C2×C4×Dic5).2C2, (C2×C10).35(C4⋊C4), (C2×C20).235(C2×C4), (C2×C4).39(C2×Dic5), (C2×C4).179(C5⋊D4), (C2×C10).118(C22⋊C4), SmallGroup(320,113)

Series: Derived Chief Lower central Upper central

C1C20 — C20.33C42
C1C5C10C2×C10C2×C20C22×C20C23.21D10 — C20.33C42
C5C10C20 — C20.33C42
C1C2×C4C22×C4C2×M4(2)

Generators and relations for C20.33C42
 G = < a,b,c | a20=b4=1, c4=a10, bab-1=a-1, cac-1=a11, cbc-1=a5b >

Subgroups: 358 in 110 conjugacy classes, 51 normal (39 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×4], C4 [×6], C22 [×3], C22 [×2], C5, C8 [×2], C2×C4 [×6], C2×C4 [×8], C23, C10, C10 [×2], C10 [×2], C42 [×4], C22⋊C4, C4⋊C4 [×2], C2×C8, M4(2) [×2], M4(2), C22×C4, C22×C4, Dic5 [×6], C20 [×4], C2×C10 [×3], C2×C10 [×2], C2×C42, C42⋊C2, C2×M4(2), C40 [×2], C2×Dic5 [×8], C2×C20 [×6], C22×C10, C426C4, C4×Dic5 [×2], C4×Dic5 [×2], C4⋊Dic5 [×2], C23.D5, C2×C40, C5×M4(2) [×2], C5×M4(2), C22×Dic5, C22×C20, C2×C4×Dic5, C23.21D10, C10×M4(2), C20.33C42
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, C4≀C2 [×2], Dic10, C4×D5 [×2], D20, C2×Dic5, C5⋊D4 [×2], C426C4, C4×Dic5, C10.D4 [×2], C4⋊Dic5, D10⋊C4 [×2], C23.D5, D207C4 [×2], C10.10C42, C20.33C42

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

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

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

68 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F4G···4N4O4P4Q4R5A5B8A8B8C8D10A···10F10G10H10I10J20A···20H20I20J20K20L40A···40P
order1222224444444···4444455888810···101010101020···202020202040···40
size11112211112210···10202020202244442···244442···244444···4

68 irreducible representations

dim1111111222222222224
type+++++-++-+-+
imageC1C2C2C2C4C4C4D4Q8D4D5Dic5D10C4≀C2Dic10C4×D5C5⋊D4D20D207C4
kernelC20.33C42C2×C4×Dic5C23.21D10C10×M4(2)C4×Dic5C4⋊Dic5C5×M4(2)C2×C20C2×C20C22×C10C2×M4(2)M4(2)C22×C4C10C2×C4C2×C4C2×C4C23C2
# reps1111444211242848848

Matrix representation of C20.33C42 in GL4(𝔽41) generated by

35500
14000
00320
00409
,
28400
191300
00123
00040
,
32000
03200
003239
00369
G:=sub<GL(4,GF(41))| [35,1,0,0,5,40,0,0,0,0,32,40,0,0,0,9],[28,19,0,0,4,13,0,0,0,0,1,0,0,0,23,40],[32,0,0,0,0,32,0,0,0,0,32,36,0,0,39,9] >;

C20.33C42 in GAP, Magma, Sage, TeX

C_{20}._{33}C_4^2
% in TeX

G:=Group("C20.33C4^2");
// GroupNames label

G:=SmallGroup(320,113);
// by ID

G=gap.SmallGroup(320,113);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,253,64,136,1684,851,102,12550]);
// Polycyclic

G:=Group<a,b,c|a^20=b^4=1,c^4=a^10,b*a*b^-1=a^-1,c*a*c^-1=a^11,c*b*c^-1=a^5*b>;
// generators/relations

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