metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C80⋊5C4, C16⋊2F5, D10.2SD16, Dic5.2SD16, C5⋊2C16⋊4C4, C5⋊2(C8.Q8), C4.2(C4⋊F5), C20.9(C4⋊C4), C8.26(C2×F5), C5⋊2C8.2Q8, C40.28(C2×C4), (C4×D5).49D4, C40⋊C4.3C2, C80⋊C2.2C2, C2.5(C40⋊C4), C10.2(C4.Q8), C40.C4.3C2, (C8×D5).32C22, SmallGroup(320,186)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C80⋊5C4
G = < a,b | a80=b4=1, bab-1=a3 >
Smallest permutation representation of C80⋊5C4
►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)
(2 28 10 4)(3 55 19 7)(5 29 37 13)(6 56 46 16)(8 30 64 22)(9 57 73 25)(11 31)(12 58 20 34)(14 32 38 40)(15 59 47 43)(17 33 65 49)(18 60 74 52)(21 61)(23 35 39 67)(24 62 48 70)(26 36 66 76)(27 63 75 79)(42 68 50 44)(45 69 77 53)(51 71)(54 72 78 80)
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), (2,28,10,4)(3,55,19,7)(5,29,37,13)(6,56,46,16)(8,30,64,22)(9,57,73,25)(11,31)(12,58,20,34)(14,32,38,40)(15,59,47,43)(17,33,65,49)(18,60,74,52)(21,61)(23,35,39,67)(24,62,48,70)(26,36,66,76)(27,63,75,79)(42,68,50,44)(45,69,77,53)(51,71)(54,72,78,80)>;
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), (2,28,10,4)(3,55,19,7)(5,29,37,13)(6,56,46,16)(8,30,64,22)(9,57,73,25)(11,31)(12,58,20,34)(14,32,38,40)(15,59,47,43)(17,33,65,49)(18,60,74,52)(21,61)(23,35,39,67)(24,62,48,70)(26,36,66,76)(27,63,75,79)(42,68,50,44)(45,69,77,53)(51,71)(54,72,78,80) );
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)], [(2,28,10,4),(3,55,19,7),(5,29,37,13),(6,56,46,16),(8,30,64,22),(9,57,73,25),(11,31),(12,58,20,34),(14,32,38,40),(15,59,47,43),(17,33,65,49),(18,60,74,52),(21,61),(23,35,39,67),(24,62,48,70),(26,36,66,76),(27,63,75,79),(42,68,50,44),(45,69,77,53),(51,71),(54,72,78,80)]])
32 conjugacy classes
| class | 1 | 2A | 2B | 4A | 4B | 4C | 4D | 5 | 8A | 8B | 8C | 8D | 8E | 10 | 16A | 16B | 16C | 16D | 20A | 20B | 40A | 40B | 40C | 40D | 80A | ··· | 80H |
| order | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 5 | 8 | 8 | 8 | 8 | 8 | 10 | 16 | 16 | 16 | 16 | 20 | 20 | 40 | 40 | 40 | 40 | 80 | ··· | 80 |
| size | 1 | 1 | 10 | 2 | 10 | 40 | 40 | 4 | 2 | 2 | 20 | 40 | 40 | 4 | 4 | 4 | 20 | 20 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | - | + | + | + | ||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | Q8 | D4 | SD16 | SD16 | F5 | C2×F5 | C8.Q8 | C4⋊F5 | C40⋊C4 | C80⋊5C4 |
| kernel | C80⋊5C4 | C80⋊C2 | C40⋊C4 | C40.C4 | C5⋊2C16 | C80 | C5⋊2C8 | C4×D5 | Dic5 | D10 | C16 | C8 | C5 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 2 | 2 | 1 | 1 | 2 | 2 | 4 | 8 |
Matrix representation of C80⋊5C4 ►in GL4(𝔽3) generated by
| 1 | 1 | 1 | 1 |
| 0 | 2 | 2 | 2 |
| 2 | 2 | 1 | 1 |
| 1 | 1 | 1 | 0 |
| 1 | 0 | 1 | 0 |
| 0 | 2 | 1 | 1 |
| 0 | 2 | 0 | 0 |
| 0 | 0 | 1 | 0 |
G:=sub<GL(4,GF(3))| [1,0,2,1,1,2,2,1,1,2,1,1,1,2,1,0],[1,0,0,0,0,2,2,0,1,1,0,1,0,1,0,0] >;
C80⋊5C4 in GAP, Magma, Sage, TeX
C_{80}\rtimes_5C_4 % in TeX
G:=Group("C80:5C4"); // GroupNames label
G:=SmallGroup(320,186);
// by ID
G=gap.SmallGroup(320,186);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,141,288,387,675,80,1684,102,6278,3156]);
// Polycyclic
G:=Group<a,b|a^80=b^4=1,b*a*b^-1=a^3>;
// generators/relations
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