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G = C802C4order 320 = 26·5

2nd semidirect product of C80 and C4 acting faithfully

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: C802C4, C165F5, D5.1D16, D10.8D8, D5.1Q32, Dic5.5Q16, C5⋊(C163C4), C52C1613C4, C4.3(C4⋊F5), C8.20(C2×F5), C52C8.6Q8, C40.20(C2×C4), (D5×C16).2C2, (C4×D5).74D4, D5.D8.3C2, C20.10(C4⋊C4), C2.4(D5.D8), C10.1(C2.D8), (C8×D5).49C22, SmallGroup(320,187)

Series: Derived Chief Lower central Upper central

C1C40 — C802C4
C1C5C10C20C4×D5C8×D5D5.D8 — C802C4
C5C10C20C40 — C802C4
C1C2C4C8C16

Generators and relations for C802C4
 G = < a,b | a80=b4=1, bab-1=a63 >

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
5C163C4

Smallest permutation representation of C802C4
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)(2 63 50 79)(3 30 19 62)(4 77 68 45)(5 44 37 28)(6 11)(7 58 55 74)(8 25 24 57)(9 72 73 40)(10 39 42 23)(12 53 60 69)(13 20 29 52)(14 67 78 35)(15 34 47 18)(17 48 65 64)(21 76)(22 43 70 59)(26 71)(27 38 75 54)(31 66)(32 33 80 49)(36 61)(41 56)(46 51)

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

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

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

38 conjugacy classes

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

38 irreducible representations

dim1111122222244444
type+++-+-++-++
imageC1C2C2C4C4Q8D4Q16D8D16Q32F5C2×F5C4⋊F5D5.D8C802C4
kernelC802C4D5×C16D5.D8C52C16C80C52C8C4×D5Dic5D10D5D5C16C8C4C2C1
# reps1122211224411248

Matrix representation of C802C4 in GL4(𝔽241) generated by

195621950
019562195
46460108
133179179133
,
019562195
10804646
46460108
195621950
G:=sub<GL(4,GF(241))| [195,0,46,133,62,195,46,179,195,62,0,179,0,195,108,133],[0,108,46,195,195,0,46,62,62,46,0,195,195,46,108,0] >;

C802C4 in GAP, Magma, Sage, TeX

C_{80}\rtimes_2C_4
% in TeX

G:=Group("C80:2C4");
// GroupNames label

G:=SmallGroup(320,187);
// by ID

G=gap.SmallGroup(320,187);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,141,176,675,192,1684,102,6278,3156]);
// Polycyclic

G:=Group<a,b|a^80=b^4=1,b*a*b^-1=a^63>;
// generators/relations

Export

Subgroup lattice of C802C4 in TeX

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