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G = C16⋊F5order 320 = 26·5

3rd semidirect product of C16 and F5 acting via F5/C5=C4

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: C809C4, C163F5, C20.15C42, D10.1M4(2), Dic5.1M4(2), C5⋊C161C4, C52C167C4, D5⋊C8.1C4, C51(C16⋊C4), (C4×F5).1C4, C4.24(C4×F5), C8.30(C2×F5), C40.34(C2×C4), C8⋊F5.2C2, C80⋊C2.3C2, C2.4(C8⋊F5), C10.1(C8⋊C4), C8.F5.2C2, (C8×D5).33C22, C52C8.16(C2×C4), (C4×D5).40(C2×C4), SmallGroup(320,183)

Series: Derived Chief Lower central Upper central

C1C20 — C16⋊F5
C1C5C10C20C4×D5C8×D5C8⋊F5 — C16⋊F5
C5C20 — C16⋊F5
C1C4C16

Generators and relations for C16⋊F5
 G = < a,b,c | a16=b5=c4=1, ab=ba, cac-1=a13, 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 C16⋊F5
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 27 63 67 38)(2 28 64 68 39)(3 29 49 69 40)(4 30 50 70 41)(5 31 51 71 42)(6 32 52 72 43)(7 17 53 73 44)(8 18 54 74 45)(9 19 55 75 46)(10 20 56 76 47)(11 21 57 77 48)(12 22 58 78 33)(13 23 59 79 34)(14 24 60 80 35)(15 25 61 65 36)(16 26 62 66 37)
(2 6 10 14)(3 11)(4 16 12 8)(7 15)(17 61 44 65)(18 50 37 78)(19 55 46 75)(20 60 39 72)(21 49 48 69)(22 54 41 66)(23 59 34 79)(24 64 43 76)(25 53 36 73)(26 58 45 70)(27 63 38 67)(28 52 47 80)(29 57 40 77)(30 62 33 74)(31 51 42 71)(32 56 35 68)

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

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

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

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⋊F5C16⋊F5
kernelC16⋊F5C80⋊C2C8.F5C8⋊F5C52C16C80C5⋊C16D5⋊C8C4×F5Dic5D10C16C8C5C4C2C1
# reps11112242222112248

Matrix representation of C16⋊F5 in GL4(𝔽241) generated by

23723322211
230226222211
30191511
2301984
,
240240240240
1000
0100
0010
,
1000
0001
0100
240240240240
G:=sub<GL(4,GF(241))| [237,230,30,230,233,226,19,19,222,222,15,8,11,211,11,4],[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] >;

C16⋊F5 in GAP, Magma, Sage, TeX

C_{16}\rtimes F_5
% in TeX

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

G:=SmallGroup(320,183);
// by ID

G=gap.SmallGroup(320,183);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,253,64,387,1123,80,102,6278,3156]);
// Polycyclic

G:=Group<a,b,c|a^16=b^5=c^4=1,a*b=b*a,c*a*c^-1=a^13,c*b*c^-1=b^3>;
// generators/relations

Export

Subgroup lattice of C16⋊F5 in TeX

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