metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C80⋊5C2, C16⋊3D5, C5⋊3M5(2), D10.1C8, C8.20D10, Dic5.1C8, C40.20C22, C5⋊2C16⋊4C2, C2.3(C8×D5), C5⋊2C8.2C4, (C8×D5).2C2, (C4×D5).2C4, C4.17(C4×D5), C20.43(C2×C4), C10.11(C2×C8), SmallGroup(160,5)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C80⋊C2
G = < a,b | a80=b2=1, bab=a9 >
Smallest permutation representation of C80⋊C2
►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)
(2 10)(3 19)(4 28)(5 37)(6 46)(7 55)(8 64)(9 73)(12 20)(13 29)(14 38)(15 47)(16 56)(17 65)(18 74)(22 30)(23 39)(24 48)(25 57)(26 66)(27 75)(32 40)(33 49)(34 58)(35 67)(36 76)(42 50)(43 59)(44 68)(45 77)(52 60)(53 69)(54 78)(62 70)(63 79)(72 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), (2,10)(3,19)(4,28)(5,37)(6,46)(7,55)(8,64)(9,73)(12,20)(13,29)(14,38)(15,47)(16,56)(17,65)(18,74)(22,30)(23,39)(24,48)(25,57)(26,66)(27,75)(32,40)(33,49)(34,58)(35,67)(36,76)(42,50)(43,59)(44,68)(45,77)(52,60)(53,69)(54,78)(62,70)(63,79)(72,80)>;
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), (2,10)(3,19)(4,28)(5,37)(6,46)(7,55)(8,64)(9,73)(12,20)(13,29)(14,38)(15,47)(16,56)(17,65)(18,74)(22,30)(23,39)(24,48)(25,57)(26,66)(27,75)(32,40)(33,49)(34,58)(35,67)(36,76)(42,50)(43,59)(44,68)(45,77)(52,60)(53,69)(54,78)(62,70)(63,79)(72,80) );
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)], [(2,10),(3,19),(4,28),(5,37),(6,46),(7,55),(8,64),(9,73),(12,20),(13,29),(14,38),(15,47),(16,56),(17,65),(18,74),(22,30),(23,39),(24,48),(25,57),(26,66),(27,75),(32,40),(33,49),(34,58),(35,67),(36,76),(42,50),(43,59),(44,68),(45,77),(52,60),(53,69),(54,78),(62,70),(63,79),(72,80)]])
C80⋊C2 is a maximal subgroup of
C16⋊F5 C16⋊4F5 C80⋊4C4 C80⋊5C4 D20.6C8 D5×M5(2) D20.5C8 D16⋊D5 C16⋊D10 SD32⋊D5 Q32⋊D5 C40.51D6 D30.5C8 C80⋊S3
C80⋊C2 is a maximal quotient of
C40.88D4 C80⋊17C4 D10⋊1C16 C40.51D6 D30.5C8 C80⋊S3
52 conjugacy classes
| class | 1 | 2A | 2B | 4A | 4B | 4C | 5A | 5B | 8A | 8B | 8C | 8D | 8E | 8F | 10A | 10B | 16A | 16B | 16C | 16D | 16E | 16F | 16G | 16H | 20A | 20B | 20C | 20D | 40A | ··· | 40H | 80A | ··· | 80P |
| order | 1 | 2 | 2 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | 10 | 16 | 16 | 16 | 16 | 16 | 16 | 16 | 16 | 20 | 20 | 20 | 20 | 40 | ··· | 40 | 80 | ··· | 80 |
| size | 1 | 1 | 10 | 1 | 1 | 10 | 2 | 2 | 1 | 1 | 1 | 1 | 10 | 10 | 2 | 2 | 2 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
52 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | ||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C8 | C8 | D5 | D10 | M5(2) | C4×D5 | C8×D5 | C80⋊C2 |
| kernel | C80⋊C2 | C5⋊2C16 | C80 | C8×D5 | C5⋊2C8 | C4×D5 | Dic5 | D10 | C16 | C8 | C5 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 2 | 2 | 4 | 4 | 8 | 16 |
Matrix representation of C80⋊C2 ►in GL2(𝔽241) generated by
| 135 | 136 |
| 105 | 52 |
| 189 | 240 |
| 52 | 52 |
G:=sub<GL(2,GF(241))| [135,105,136,52],[189,52,240,52] >;
C80⋊C2 in GAP, Magma, Sage, TeX
C_{80}\rtimes C_2 % in TeX
G:=Group("C80:C2"); // GroupNames label
G:=SmallGroup(160,5);
// by ID
G=gap.SmallGroup(160,5);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-5,217,31,50,69,4613]);
// Polycyclic
G:=Group<a,b|a^80=b^2=1,b*a*b=a^9>;
// generators/relations
Export