metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C80⋊3C4, C16⋊6F5, D10.9D8, D5.1SD32, Dic5.6Q16, C5⋊(C16⋊4C4), C5⋊2C16⋊14C4, C4.4(C4⋊F5), C8.21(C2×F5), C5⋊2C8.7Q8, C40.21(C2×C4), (D5×C16).5C2, (C4×D5).75D4, D5.D8.4C2, C20.11(C4⋊C4), C2.5(D5.D8), C10.2(C2.D8), (C8×D5).50C22, SmallGroup(320,188)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C80⋊3C4
G = < a,b | a80=b4=1, bab-1=a23 >
Smallest permutation representation of C80⋊3C4
►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 16 41 56)(2 23 10 79)(3 30 59 22)(4 37 28 45)(5 44 77 68)(6 51 46 11)(7 58 15 34)(8 65 64 57)(9 72 33 80)(12 13 20 69)(14 27 38 35)(17 48 25 24)(18 55 74 47)(19 62 43 70)(21 76 61 36)(26 31 66 71)(29 52 53 60)(32 73 40 49)(39 42 63 50)(54 67 78 75)
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), (1,16,41,56)(2,23,10,79)(3,30,59,22)(4,37,28,45)(5,44,77,68)(6,51,46,11)(7,58,15,34)(8,65,64,57)(9,72,33,80)(12,13,20,69)(14,27,38,35)(17,48,25,24)(18,55,74,47)(19,62,43,70)(21,76,61,36)(26,31,66,71)(29,52,53,60)(32,73,40,49)(39,42,63,50)(54,67,78,75)>;
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), (1,16,41,56)(2,23,10,79)(3,30,59,22)(4,37,28,45)(5,44,77,68)(6,51,46,11)(7,58,15,34)(8,65,64,57)(9,72,33,80)(12,13,20,69)(14,27,38,35)(17,48,25,24)(18,55,74,47)(19,62,43,70)(21,76,61,36)(26,31,66,71)(29,52,53,60)(32,73,40,49)(39,42,63,50)(54,67,78,75) );
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)], [(1,16,41,56),(2,23,10,79),(3,30,59,22),(4,37,28,45),(5,44,77,68),(6,51,46,11),(7,58,15,34),(8,65,64,57),(9,72,33,80),(12,13,20,69),(14,27,38,35),(17,48,25,24),(18,55,74,47),(19,62,43,70),(21,76,61,36),(26,31,66,71),(29,52,53,60),(32,73,40,49),(39,42,63,50),(54,67,78,75)]])
38 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 5 | 8A | 8B | 8C | 8D | 10 | 16A | 16B | 16C | 16D | 16E | 16F | 16G | 16H | 20A | 20B | 40A | 40B | 40C | 40D | 80A | ··· | 80H |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 8 | 8 | 8 | 8 | 10 | 16 | 16 | 16 | 16 | 16 | 16 | 16 | 16 | 20 | 20 | 40 | 40 | 40 | 40 | 80 | ··· | 80 |
| size | 1 | 1 | 5 | 5 | 2 | 10 | 40 | 40 | 40 | 40 | 4 | 2 | 2 | 10 | 10 | 4 | 2 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | - | + | - | + | + | + | ||||||
| image | C1 | C2 | C2 | C4 | C4 | Q8 | D4 | Q16 | D8 | SD32 | F5 | C2×F5 | C4⋊F5 | D5.D8 | C80⋊3C4 |
| kernel | C80⋊3C4 | D5×C16 | D5.D8 | C5⋊2C16 | C80 | C5⋊2C8 | C4×D5 | Dic5 | D10 | D5 | C16 | C8 | C4 | C2 | C1 |
| # reps | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 2 | 2 | 8 | 1 | 1 | 2 | 4 | 8 |
Matrix representation of C80⋊3C4 ►in GL4(𝔽7) generated by
| 2 | 1 | 6 | 2 |
| 2 | 5 | 3 | 2 |
| 5 | 3 | 5 | 4 |
| 3 | 2 | 0 | 5 |
| 1 | 3 | 4 | 5 |
| 0 | 2 | 0 | 6 |
| 0 | 1 | 2 | 0 |
| 0 | 3 | 1 | 2 |
G:=sub<GL(4,GF(7))| [2,2,5,3,1,5,3,2,6,3,5,0,2,2,4,5],[1,0,0,0,3,2,1,3,4,0,2,1,5,6,0,2] >;
C80⋊3C4 in GAP, Magma, Sage, TeX
C_{80}\rtimes_3C_4 % in TeX
G:=Group("C80:3C4"); // GroupNames label
G:=SmallGroup(320,188);
// by ID
G=gap.SmallGroup(320,188);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,141,176,1571,80,1684,102,6278,3156]);
// Polycyclic
G:=Group<a,b|a^80=b^4=1,b*a*b^-1=a^23>;
// generators/relations
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