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G = C80⋊C2order 160 = 25·5

5th semidirect product of C80 and C2 acting faithfully

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: C805C2, C163D5, C53M5(2), D10.1C8, C8.20D10, Dic5.1C8, C40.20C22, C52C164C2, C2.3(C8×D5), C52C8.2C4, (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

C1C10 — C80⋊C2
C1C5C10C20C40C8×D5 — C80⋊C2
C5C10 — C80⋊C2
C1C8C16

Generators and relations for C80⋊C2
 G = < a,b | a80=b2=1, bab=a9 >

10C2
5C4
5C22
2D5
5C2×C4
5C8
5C2×C8
5C16
5M5(2)

Smallest permutation representation of C80⋊C2
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 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  C164F5  C804C4  C805C4  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  C8017C4  D101C16  C40.51D6  D30.5C8  C80⋊S3

52 conjugacy classes

class 1 2A2B4A4B4C5A5B8A8B8C8D8E8F10A10B16A16B16C16D16E16F16G16H20A20B20C20D40A···40H80A···80P
order12244455888888101016161616161616162020202040···4080···80
size1110111022111110102222221010101022222···22···2

52 irreducible representations

dim11111111222222
type++++++
imageC1C2C2C2C4C4C8C8D5D10M5(2)C4×D5C8×D5C80⋊C2
kernelC80⋊C2C52C16C80C8×D5C52C8C4×D5Dic5D10C16C8C5C4C2C1
# reps111122442244816

Matrix representation of C80⋊C2 in GL2(𝔽241) generated by

135136
10552
,
189240
5252
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

Subgroup lattice of C80⋊C2 in TeX

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