metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C80⋊6C4, C16⋊2Dic5, C20.48C42, M5(2).3D5, C20.19M4(2), C5⋊2C16⋊6C4, C8.33(C4×D5), C5⋊6(C16⋊C4), C40.118(C2×C4), (C2×C8).151D10, C4.9(C8⋊D5), (C4×Dic5).3C4, C40⋊8C4.11C2, C4.23(C4×Dic5), C8.20(C2×Dic5), C2.4(C40⋊8C4), C20.4C8.8C2, C10.13(C8⋊C4), (C5×M5(2)).2C2, (C2×C40).219C22, C22.6(C8⋊D5), (C2×C10).13M4(2), (C2×C5⋊2C8).1C4, (C2×C4).136(C4×D5), (C2×C20).224(C2×C4), SmallGroup(320,70)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C80⋊C4
G = < a,b | a80=b4=1, bab-1=a29 >
Smallest permutation representation of C80⋊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)
(2 70 42 30)(3 59)(4 48 44 8)(5 37)(6 26 46 66)(7 15)(9 73)(10 62 50 22)(11 51)(12 40 52 80)(13 29)(14 18 54 58)(16 76 56 36)(17 65)(19 43)(20 32 60 72)(23 79)(24 68 64 28)(25 57)(27 35)(31 71)(33 49)(34 38 74 78)(39 63)(45 77)(47 55)(53 69)(67 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), (2,70,42,30)(3,59)(4,48,44,8)(5,37)(6,26,46,66)(7,15)(9,73)(10,62,50,22)(11,51)(12,40,52,80)(13,29)(14,18,54,58)(16,76,56,36)(17,65)(19,43)(20,32,60,72)(23,79)(24,68,64,28)(25,57)(27,35)(31,71)(33,49)(34,38,74,78)(39,63)(45,77)(47,55)(53,69)(67,75)>;
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), (2,70,42,30)(3,59)(4,48,44,8)(5,37)(6,26,46,66)(7,15)(9,73)(10,62,50,22)(11,51)(12,40,52,80)(13,29)(14,18,54,58)(16,76,56,36)(17,65)(19,43)(20,32,60,72)(23,79)(24,68,64,28)(25,57)(27,35)(31,71)(33,49)(34,38,74,78)(39,63)(45,77)(47,55)(53,69)(67,75) );
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)], [(2,70,42,30),(3,59),(4,48,44,8),(5,37),(6,26,46,66),(7,15),(9,73),(10,62,50,22),(11,51),(12,40,52,80),(13,29),(14,18,54,58),(16,76,56,36),(17,65),(19,43),(20,32,60,72),(23,79),(24,68,64,28),(25,57),(27,35),(31,71),(33,49),(34,38,74,78),(39,63),(45,77),(47,55),(53,69),(67,75)]])
62 conjugacy classes
| class | 1 | 2A | 2B | 4A | 4B | 4C | 4D | 4E | 5A | 5B | 8A | 8B | 8C | 8D | 8E | 8F | 10A | 10B | 10C | 10D | 16A | 16B | 16C | 16D | 16E | 16F | 16G | 16H | 20A | 20B | 20C | 20D | 20E | 20F | 40A | ··· | 40H | 40I | 40J | 40K | 40L | 80A | ··· | 80P |
| order | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 10 | 16 | 16 | 16 | 16 | 16 | 16 | 16 | 16 | 20 | 20 | 20 | 20 | 20 | 20 | 40 | ··· | 40 | 40 | 40 | 40 | 40 | 80 | ··· | 80 |
| size | 1 | 1 | 2 | 1 | 1 | 2 | 20 | 20 | 2 | 2 | 2 | 2 | 2 | 2 | 20 | 20 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 20 | 20 | 20 | 20 | 2 | 2 | 2 | 2 | 4 | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
62 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | - | + | ||||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C4 | C4 | D5 | M4(2) | M4(2) | Dic5 | D10 | C4×D5 | C4×D5 | C8⋊D5 | C8⋊D5 | C16⋊C4 | C80⋊C4 |
| kernel | C80⋊C4 | C20.4C8 | C40⋊8C4 | C5×M5(2) | C5⋊2C16 | C80 | C2×C5⋊2C8 | C4×Dic5 | M5(2) | C20 | C2×C10 | C16 | C2×C8 | C8 | C2×C4 | C4 | C22 | C5 | C1 |
| # reps | 1 | 1 | 1 | 1 | 4 | 4 | 2 | 2 | 2 | 2 | 2 | 4 | 2 | 4 | 4 | 8 | 8 | 2 | 8 |
Matrix representation of C80⋊C4 ►in GL4(𝔽241) generated by
| 111 | 229 | 54 | 158 |
| 154 | 99 | 145 | 150 |
| 76 | 221 | 229 | 154 |
| 40 | 36 | 87 | 43 |
| 51 | 191 | 141 | 39 |
| 52 | 190 | 141 | 139 |
| 0 | 0 | 177 | 195 |
| 0 | 0 | 0 | 64 |
G:=sub<GL(4,GF(241))| [111,154,76,40,229,99,221,36,54,145,229,87,158,150,154,43],[51,52,0,0,191,190,0,0,141,141,177,0,39,139,195,64] >;
C80⋊C4 in GAP, Magma, Sage, TeX
C_{80}\rtimes C_4 % in TeX
G:=Group("C80:C4"); // GroupNames label
G:=SmallGroup(320,70);
// by ID
G=gap.SmallGroup(320,70);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,253,64,387,1123,80,102,12550]);
// Polycyclic
G:=Group<a,b|a^80=b^4=1,b*a*b^-1=a^29>;
// generators/relations
Export