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G = C80⋊C4order 320 = 26·5

6th semidirect product of C80 and C4 acting faithfully

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: C806C4, C162Dic5, C20.48C42, M5(2).3D5, C20.19M4(2), C52C166C4, C8.33(C4×D5), C56(C16⋊C4), C40.118(C2×C4), (C2×C8).151D10, C4.9(C8⋊D5), (C4×Dic5).3C4, C408C4.11C2, C4.23(C4×Dic5), C8.20(C2×Dic5), C2.4(C408C4), C20.4C8.8C2, C10.13(C8⋊C4), (C5×M5(2)).2C2, (C2×C40).219C22, C22.6(C8⋊D5), (C2×C10).13M4(2), (C2×C52C8).1C4, (C2×C4).136(C4×D5), (C2×C20).224(C2×C4), SmallGroup(320,70)

Series: Derived Chief Lower central Upper central

C1C20 — C80⋊C4
C1C5C10C20C2×C20C2×C40C408C4 — C80⋊C4
C5C20 — C80⋊C4
C1C4M5(2)

Generators and relations for C80⋊C4
 G = < a,b | a80=b4=1, bab-1=a29 >

2C2
20C4
2C10
10C8
10C2×C4
4Dic5
5C16
5C16
5C2×C8
5C42
2C52C8
2C2×Dic5
5C8⋊C4
5M5(2)
5C16⋊C4

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 2A2B4A4B4C4D4E5A5B8A8B8C8D8E8F10A10B10C10D16A16B16C16D16E16F16G16H20A20B20C20D20E20F40A···40H40I40J40K40L80A···80P
order122444445588888810101010161616161616161620202020202040···404040404080···80
size1121122020222222202022444444202020202222442···244444···4

62 irreducible representations

dim1111111122222222244
type+++++-+
imageC1C2C2C2C4C4C4C4D5M4(2)M4(2)Dic5D10C4×D5C4×D5C8⋊D5C8⋊D5C16⋊C4C80⋊C4
kernelC80⋊C4C20.4C8C408C4C5×M5(2)C52C16C80C2×C52C8C4×Dic5M5(2)C20C2×C10C16C2×C8C8C2×C4C4C22C5C1
# reps1111442222242448828

Matrix representation of C80⋊C4 in GL4(𝔽241) generated by

11122954158
15499145150
76221229154
40368743
,
5119114139
52190141139
00177195
00064
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

Subgroup lattice of C80⋊C4 in TeX

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