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G = C164F5order 320 = 26·5

4th semidirect product of C16 and F5 acting via F5/C5=C4

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: C164F5, C8010C4, C20.16C42, D10.2M4(2), Dic5.2M4(2), C5⋊C162C4, C52C168C4, D5⋊C8.2C4, C52(C16⋊C4), (C4×F5).2C4, C4.25(C4×F5), C8.31(C2×F5), C40.35(C2×C4), C8⋊F5.3C2, C80⋊C2.4C2, C2.5(C8⋊F5), C10.2(C8⋊C4), C8.F5.3C2, (C8×D5).34C22, C52C8.17(C2×C4), (C4×D5).41(C2×C4), SmallGroup(320,184)

Series: Derived Chief Lower central Upper central

C1C20 — C164F5
C1C5C10C20C4×D5C8×D5C8⋊F5 — C164F5
C5C20 — C164F5
C1C4C16

Generators and relations for C164F5
 G = < a,b,c | a16=b5=c4=1, ab=ba, cac-1=a5, cbc-1=b3 >

10C2
5C22
5C4
20C4
2D5
5C8
5C2×C4
10C8
10C2×C4
4F5
5C16
5C16
5C2×C8
5C16
5C2×C8
5C42
2C5⋊C8
2C2×F5
5C8⋊C4
5M5(2)
5M5(2)
5C16⋊C4

Smallest permutation representation of C164F5
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)
(1 51 36 77 22)(2 52 37 78 23)(3 53 38 79 24)(4 54 39 80 25)(5 55 40 65 26)(6 56 41 66 27)(7 57 42 67 28)(8 58 43 68 29)(9 59 44 69 30)(10 60 45 70 31)(11 61 46 71 32)(12 62 47 72 17)(13 63 48 73 18)(14 64 33 74 19)(15 49 34 75 20)(16 50 35 76 21)
(2 14 10 6)(3 11)(4 8 12 16)(7 15)(17 76 54 43)(18 73 63 48)(19 70 56 37)(20 67 49 42)(21 80 58 47)(22 77 51 36)(23 74 60 41)(24 71 53 46)(25 68 62 35)(26 65 55 40)(27 78 64 45)(28 75 57 34)(29 72 50 39)(30 69 59 44)(31 66 52 33)(32 79 61 38)

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), (1,51,36,77,22)(2,52,37,78,23)(3,53,38,79,24)(4,54,39,80,25)(5,55,40,65,26)(6,56,41,66,27)(7,57,42,67,28)(8,58,43,68,29)(9,59,44,69,30)(10,60,45,70,31)(11,61,46,71,32)(12,62,47,72,17)(13,63,48,73,18)(14,64,33,74,19)(15,49,34,75,20)(16,50,35,76,21), (2,14,10,6)(3,11)(4,8,12,16)(7,15)(17,76,54,43)(18,73,63,48)(19,70,56,37)(20,67,49,42)(21,80,58,47)(22,77,51,36)(23,74,60,41)(24,71,53,46)(25,68,62,35)(26,65,55,40)(27,78,64,45)(28,75,57,34)(29,72,50,39)(30,69,59,44)(31,66,52,33)(32,79,61,38)>;

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), (1,51,36,77,22)(2,52,37,78,23)(3,53,38,79,24)(4,54,39,80,25)(5,55,40,65,26)(6,56,41,66,27)(7,57,42,67,28)(8,58,43,68,29)(9,59,44,69,30)(10,60,45,70,31)(11,61,46,71,32)(12,62,47,72,17)(13,63,48,73,18)(14,64,33,74,19)(15,49,34,75,20)(16,50,35,76,21), (2,14,10,6)(3,11)(4,8,12,16)(7,15)(17,76,54,43)(18,73,63,48)(19,70,56,37)(20,67,49,42)(21,80,58,47)(22,77,51,36)(23,74,60,41)(24,71,53,46)(25,68,62,35)(26,65,55,40)(27,78,64,45)(28,75,57,34)(29,72,50,39)(30,69,59,44)(31,66,52,33)(32,79,61,38) );

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)], [(1,51,36,77,22),(2,52,37,78,23),(3,53,38,79,24),(4,54,39,80,25),(5,55,40,65,26),(6,56,41,66,27),(7,57,42,67,28),(8,58,43,68,29),(9,59,44,69,30),(10,60,45,70,31),(11,61,46,71,32),(12,62,47,72,17),(13,63,48,73,18),(14,64,33,74,19),(15,49,34,75,20),(16,50,35,76,21)], [(2,14,10,6),(3,11),(4,8,12,16),(7,15),(17,76,54,43),(18,73,63,48),(19,70,56,37),(20,67,49,42),(21,80,58,47),(22,77,51,36),(23,74,60,41),(24,71,53,46),(25,68,62,35),(26,65,55,40),(27,78,64,45),(28,75,57,34),(29,72,50,39),(30,69,59,44),(31,66,52,33),(32,79,61,38)])

38 conjugacy classes

class 1 2A2B4A4B4C4D4E 5 8A8B8C8D8E8F 10 16A16B16C···16H20A20B40A40B40C40D80A···80H
order12244444588888810161616···1620204040404080···80
size1110111020204221010202044420···204444444···4

38 irreducible representations

dim11111111122444444
type++++++
imageC1C2C2C2C4C4C4C4C4M4(2)M4(2)F5C2×F5C16⋊C4C4×F5C8⋊F5C164F5
kernelC164F5C80⋊C2C8.F5C8⋊F5C52C16C80C5⋊C16D5⋊C8C4×F5Dic5D10C16C8C5C4C2C1
# reps11112242222112248

Matrix representation of C164F5 in GL4(𝔽241) generated by

12024088152
8920988177
6415332152
891531121
,
240240240240
1000
0100
0010
,
1000
0001
0100
240240240240
G:=sub<GL(4,GF(241))| [120,89,64,89,240,209,153,153,88,88,32,1,152,177,152,121],[240,1,0,0,240,0,1,0,240,0,0,1,240,0,0,0],[1,0,0,240,0,0,1,240,0,0,0,240,0,1,0,240] >;

C164F5 in GAP, Magma, Sage, TeX

C_{16}\rtimes_4F_5
% in TeX

G:=Group("C16:4F5");
// GroupNames label

G:=SmallGroup(320,184);
// by ID

G=gap.SmallGroup(320,184);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,253,64,387,1123,192,102,6278,3156]);
// Polycyclic

G:=Group<a,b,c|a^16=b^5=c^4=1,a*b=b*a,c*a*c^-1=a^5,c*b*c^-1=b^3>;
// generators/relations

Export

Subgroup lattice of C164F5 in TeX

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