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G = C804C4order 320 = 26·5

4th semidirect product of C80 and C4 acting faithfully

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: C804C4, C161F5, D10.1SD16, Dic5.1SD16, C52C163C4, C51(C8.Q8), C4.1(C4⋊F5), C20.8(C4⋊C4), C8.25(C2×F5), C52C8.1Q8, C40.27(C2×C4), (C4×D5).48D4, C40⋊C4.2C2, C80⋊C2.1C2, C2.4(C40⋊C4), C10.1(C4.Q8), C40.C4.2C2, (C8×D5).31C22, SmallGroup(320,185)

Series: Derived Chief Lower central Upper central

C1C40 — C804C4
C1C5C10C20C4×D5C8×D5C40⋊C4 — C804C4
C5C10C20C40 — C804C4
C1C2C4C8C16

Generators and relations for C804C4
 G = < a,b | a80=b4=1, bab-1=a43 >

10C2
5C4
5C22
40C4
2D5
5C8
5C2×C4
20C8
20C2×C4
8F5
5C2×C8
5C16
10M4(2)
10C4⋊C4
4C2×F5
4C5⋊C8
5C4.Q8
5M5(2)
5C8.C4
2C4.F5
2C4⋊F5
5C8.Q8

Smallest permutation representation of C804C4
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 68 10 44)(3 55 19 7)(4 42 28 50)(5 29 37 13)(6 16 46 56)(8 70 64 62)(9 57 73 25)(11 31)(12 18 20 74)(14 72 38 80)(15 59 47 43)(17 33 65 49)(21 61)(22 48 30 24)(23 35 39 67)(26 76 66 36)(27 63 75 79)(32 78 40 54)(34 52 58 60)(45 69 77 53)(51 71)

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

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

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

32 conjugacy classes

class 1 2A2B4A4B4C4D 5 8A8B8C8D8E 10 16A16B16C16D20A20B40A40B40C40D80A···80H
order1224444588888101616161620204040404080···80
size1110210404042220404044420204444444···4

32 irreducible representations

dim1111112222444444
type++++-+++
imageC1C2C2C2C4C4Q8D4SD16SD16F5C2×F5C8.Q8C4⋊F5C40⋊C4C804C4
kernelC804C4C80⋊C2C40⋊C4C40.C4C52C16C80C52C8C4×D5Dic5D10C16C8C5C4C2C1
# reps1111221122112248

Matrix representation of C804C4 in GL4(𝔽241) generated by

991817836
20563223142
996316281
1601822381
,
1000
0001
0100
240240240240
G:=sub<GL(4,GF(241))| [99,205,99,160,18,63,63,18,178,223,162,223,36,142,81,81],[1,0,0,240,0,0,1,240,0,0,0,240,0,1,0,240] >;

C804C4 in GAP, Magma, Sage, TeX

C_{80}\rtimes_4C_4
% in TeX

G:=Group("C80:4C4");
// GroupNames label

G:=SmallGroup(320,185);
// by ID

G=gap.SmallGroup(320,185);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,141,64,387,675,80,1684,102,6278,3156]);
// Polycyclic

G:=Group<a,b|a^80=b^4=1,b*a*b^-1=a^43>;
// generators/relations

Export

Subgroup lattice of C804C4 in TeX

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