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G = C803C4order 320 = 26·5

3rd semidirect product of C80 and C4 acting faithfully

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: C803C4, C166F5, D10.9D8, D5.1SD32, Dic5.6Q16, C5⋊(C164C4), C52C1614C4, C4.4(C4⋊F5), C8.21(C2×F5), C52C8.7Q8, C40.21(C2×C4), (D5×C16).5C2, (C4×D5).75D4, D5.D8.4C2, C20.11(C4⋊C4), C2.5(D5.D8), C10.2(C2.D8), (C8×D5).50C22, SmallGroup(320,188)

Series: Derived Chief Lower central Upper central

C1C40 — C803C4
C1C5C10C20C4×D5C8×D5D5.D8 — C803C4
C5C10C20C40 — C803C4
C1C2C4C8C16

Generators and relations for C803C4
 G = < a,b | a80=b4=1, bab-1=a23 >

5C2
5C2
5C22
5C4
40C4
40C4
5C8
5C2×C4
20C2×C4
20C2×C4
8F5
8F5
5C16
5C2×C8
10C4⋊C4
10C4⋊C4
4C2×F5
4C2×F5
5C2.D8
5C2×C16
5C2.D8
2C4⋊F5
2C4⋊F5
5C164C4

Smallest permutation representation of C803C4
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 16 41 56)(2 23 10 79)(3 30 59 22)(4 37 28 45)(5 44 77 68)(6 51 46 11)(7 58 15 34)(8 65 64 57)(9 72 33 80)(12 13 20 69)(14 27 38 35)(17 48 25 24)(18 55 74 47)(19 62 43 70)(21 76 61 36)(26 31 66 71)(29 52 53 60)(32 73 40 49)(39 42 63 50)(54 67 78 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), (1,16,41,56)(2,23,10,79)(3,30,59,22)(4,37,28,45)(5,44,77,68)(6,51,46,11)(7,58,15,34)(8,65,64,57)(9,72,33,80)(12,13,20,69)(14,27,38,35)(17,48,25,24)(18,55,74,47)(19,62,43,70)(21,76,61,36)(26,31,66,71)(29,52,53,60)(32,73,40,49)(39,42,63,50)(54,67,78,75)>;

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

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

38 conjugacy classes

class 1 2A2B2C4A4B4C4D4E4F 5 8A8B8C8D 10 16A16B16C16D16E16F16G16H20A20B40A40B40C40D80A···80H
order12224444445888810161616161616161620204040404080···80
size115521040404040422101042222101010104444444···4

38 irreducible representations

dim111112222244444
type+++-+-+++
imageC1C2C2C4C4Q8D4Q16D8SD32F5C2×F5C4⋊F5D5.D8C803C4
kernelC803C4D5×C16D5.D8C52C16C80C52C8C4×D5Dic5D10D5C16C8C4C2C1
# reps112221122811248

Matrix representation of C803C4 in GL4(𝔽7) generated by

2162
2532
5354
3205
,
1345
0206
0120
0312
G:=sub<GL(4,GF(7))| [2,2,5,3,1,5,3,2,6,3,5,0,2,2,4,5],[1,0,0,0,3,2,1,3,4,0,2,1,5,6,0,2] >;

C803C4 in GAP, Magma, Sage, TeX

C_{80}\rtimes_3C_4
% in TeX

G:=Group("C80:3C4");
// GroupNames label

G:=SmallGroup(320,188);
// by ID

G=gap.SmallGroup(320,188);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,141,176,1571,80,1684,102,6278,3156]);
// Polycyclic

G:=Group<a,b|a^80=b^4=1,b*a*b^-1=a^23>;
// generators/relations

Export

Subgroup lattice of C803C4 in TeX

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