metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C20.24C42, (C2×C8)⋊2F5, (C2×C40)⋊4C4, (C4×F5)⋊2C4, C4.29(C4×F5), (C4×D5).51D4, C5⋊2(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×C5⋊2C8)⋊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
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, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, C23, D5, C10, C10, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), C22×C4, Dic5, C20, F5, D10, D10, C2×C10, C42⋊C2, C2×M4(2), C5⋊2C8, C40, C4×D5, C2×Dic5, C2×C20, C2×F5, C22×D5, C4.9C42, C8⋊D5, C2×C5⋊2C8, C2×C40, C4×F5, C4⋊F5, C22⋊F5, C2×C4×D5, C2×C8⋊D5, D10.C23, C20.24C42
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C42, C22⋊C4, C4⋊C4, F5, C2.C42, C2×F5, C4.9C42, C4×F5, C4⋊F5, C22⋊F5, D10.3Q8, C20.24C42
►On 80 points
(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 61 51 71)(42 78 60 64)(43 75 49 77)(44 72 58 70)(45 69 47 63)(46 66 56 76)(48 80 54 62)(50 74 52 68)(53 65 59 67)(55 79 57 73)
(1 76 26 51 11 66 36 41)(2 77 27 52 12 67 37 42)(3 78 28 53 13 68 38 43)(4 79 29 54 14 69 39 44)(5 80 30 55 15 70 40 45)(6 61 31 56 16 71 21 46)(7 62 32 57 17 72 22 47)(8 63 33 58 18 73 23 48)(9 64 34 59 19 74 24 49)(10 65 35 60 20 75 25 50)
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)(41,61,51,71)(42,78,60,64)(43,75,49,77)(44,72,58,70)(45,69,47,63)(46,66,56,76)(48,80,54,62)(50,74,52,68)(53,65,59,67)(55,79,57,73), (1,76,26,51,11,66,36,41)(2,77,27,52,12,67,37,42)(3,78,28,53,13,68,38,43)(4,79,29,54,14,69,39,44)(5,80,30,55,15,70,40,45)(6,61,31,56,16,71,21,46)(7,62,32,57,17,72,22,47)(8,63,33,58,18,73,23,48)(9,64,34,59,19,74,24,49)(10,65,35,60,20,75,25,50)>;
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)(41,61,51,71)(42,78,60,64)(43,75,49,77)(44,72,58,70)(45,69,47,63)(46,66,56,76)(48,80,54,62)(50,74,52,68)(53,65,59,67)(55,79,57,73), (1,76,26,51,11,66,36,41)(2,77,27,52,12,67,37,42)(3,78,28,53,13,68,38,43)(4,79,29,54,14,69,39,44)(5,80,30,55,15,70,40,45)(6,61,31,56,16,71,21,46)(7,62,32,57,17,72,22,47)(8,63,33,58,18,73,23,48)(9,64,34,59,19,74,24,49)(10,65,35,60,20,75,25,50) );
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),(41,61,51,71),(42,78,60,64),(43,75,49,77),(44,72,58,70),(45,69,47,63),(46,66,56,76),(48,80,54,62),(50,74,52,68),(53,65,59,67),(55,79,57,73)], [(1,76,26,51,11,66,36,41),(2,77,27,52,12,67,37,42),(3,78,28,53,13,68,38,43),(4,79,29,54,14,69,39,44),(5,80,30,55,15,70,40,45),(6,61,31,56,16,71,21,46),(7,62,32,57,17,72,22,47),(8,63,33,58,18,73,23,48),(9,64,34,59,19,74,24,49),(10,65,35,60,20,75,25,50)]])
38 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 4F | ··· | 4M | 5 | 8A | 8B | 8C | 8D | 10A | 10B | 10C | 20A | 20B | 20C | 20D | 40A | ··· | 40H |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 20 | 20 | 20 | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 2 | 10 | 10 | 1 | 1 | 2 | 10 | 10 | 20 | ··· | 20 | 4 | 4 | 4 | 20 | 20 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | - | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C4 | C4 | C4 | D4 | Q8 | D4 | F5 | C2×F5 | C4.9C42 | C4×F5 | C22⋊F5 | C4⋊F5 | C20.24C42 |
| kernel | C20.24C42 | C2×C8⋊D5 | D10.C23 | C2×C5⋊2C8 | C2×C40 | C4×F5 | C4×D5 | C2×Dic5 | C22×D5 | C2×C8 | C2×C4 | C5 | C4 | C4 | C22 | C1 |
| # reps | 1 | 1 | 2 | 2 | 2 | 8 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 8 |
Matrix representation of C20.24C42 ►in GL4(𝔽41) generated by
| 32 | 32 | 32 | 32 |
| 9 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 |
| 0 | 0 | 9 | 0 |
| 34 | 0 | 27 | 27 |
| 27 | 27 | 0 | 34 |
| 14 | 7 | 14 | 0 |
| 7 | 34 | 34 | 7 |
| 30 | 19 | 6 | 13 |
| 28 | 17 | 6 | 34 |
| 7 | 35 | 24 | 13 |
| 28 | 35 | 22 | 11 |
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