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G = C5⋊Q16order 80 = 24·5

The semidirect product of C5 and Q16 acting via Q16/Q8=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C52Q16, Q8.D5, C4.4D10, C10.10D4, C20.4C22, Dic10.2C2, C52C8.1C2, (C5×Q8).1C2, C2.7(C5⋊D4), SmallGroup(80,18)

Series: Derived Chief Lower central Upper central

C1C20 — C5⋊Q16
C1C5C10C20Dic10 — C5⋊Q16
C5C10C20 — C5⋊Q16
C1C2C4Q8

Generators and relations for C5⋊Q16
 G = < a,b,c | a5=b8=1, c2=b4, bab-1=a-1, ac=ca, cbc-1=b-1 >

2C4
10C4
5C8
5Q8
2Dic5
2C20
5Q16

Character table of C5⋊Q16

 class 124A4B4C5A5B8A8B10A10B20A20B20C20D20E20F
 size 11242022101022444444
ρ111111111111111111    trivial
ρ21111-111-1-111111111    linear of order 2
ρ3111-1-11111111-1-1-11-1    linear of order 2
ρ4111-1111-1-1111-1-1-11-1    linear of order 2
ρ522-200220022-2000-20    orthogonal lifted from D4
ρ6222-20-1+5/2-1-5/200-1-5/2-1+5/2-1-5/21+5/21-5/21-5/2-1+5/21+5/2    orthogonal lifted from D10
ρ722220-1+5/2-1-5/200-1-5/2-1+5/2-1-5/2-1-5/2-1+5/2-1+5/2-1+5/2-1-5/2    orthogonal lifted from D5
ρ8222-20-1-5/2-1+5/200-1+5/2-1-5/2-1+5/21-5/21+5/21+5/2-1-5/21-5/2    orthogonal lifted from D10
ρ922220-1-5/2-1+5/200-1+5/2-1-5/2-1+5/2-1+5/2-1-5/2-1-5/2-1-5/2-1+5/2    orthogonal lifted from D5
ρ102-200022-22-2-2000000    symplectic lifted from Q16, Schur index 2
ρ112-2000222-2-2-2000000    symplectic lifted from Q16, Schur index 2
ρ1222-200-1-5/2-1+5/200-1+5/2-1-5/21-5/2545ζ535253521+5/2ζ545    complex lifted from C5⋊D4
ρ1322-200-1-5/2-1+5/200-1+5/2-1-5/21-5/2ζ5455352ζ53521+5/2545    complex lifted from C5⋊D4
ρ1422-200-1+5/2-1-5/200-1-5/2-1+5/21+5/25352545ζ5451-5/2ζ5352    complex lifted from C5⋊D4
ρ1522-200-1+5/2-1-5/200-1-5/2-1+5/21+5/2ζ5352ζ5455451-5/25352    complex lifted from C5⋊D4
ρ164-4000-1-5-1+5001-51+5000000    symplectic faithful, Schur index 2
ρ174-4000-1+5-1-5001+51-5000000    symplectic faithful, Schur index 2

Smallest permutation representation of C5⋊Q16
Regular action on 80 points
Generators in S80
(1 66 12 41 75)(2 76 42 13 67)(3 68 14 43 77)(4 78 44 15 69)(5 70 16 45 79)(6 80 46 9 71)(7 72 10 47 73)(8 74 48 11 65)(17 29 60 50 40)(18 33 51 61 30)(19 31 62 52 34)(20 35 53 63 32)(21 25 64 54 36)(22 37 55 57 26)(23 27 58 56 38)(24 39 49 59 28)
(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 60 5 64)(2 59 6 63)(3 58 7 62)(4 57 8 61)(9 35 13 39)(10 34 14 38)(11 33 15 37)(12 40 16 36)(17 45 21 41)(18 44 22 48)(19 43 23 47)(20 42 24 46)(25 75 29 79)(26 74 30 78)(27 73 31 77)(28 80 32 76)(49 71 53 67)(50 70 54 66)(51 69 55 65)(52 68 56 72)

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

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

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

C5⋊Q16 is a maximal subgroup of
SD16⋊D5  SD163D5  D5×Q16  Q16⋊D5  C20.C23  D4.8D10  D4.9D10  C15⋊Q16  C5⋊Dic12  C157Q16  Q8.D15  C25⋊Q16  C522Q16  C523Q16  C527Q16
C5⋊Q16 is a maximal quotient of
C10.D8  C10.Q16  Q8⋊Dic5  C15⋊Q16  C5⋊Dic12  C157Q16  C25⋊Q16  C522Q16  C523Q16  C527Q16

Matrix representation of C5⋊Q16 in GL4(𝔽41) generated by

6100
40000
0010
0001
,
6100
63500
001229
001212
,
1000
0100
003427
00277
G:=sub<GL(4,GF(41))| [6,40,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[6,6,0,0,1,35,0,0,0,0,12,12,0,0,29,12],[1,0,0,0,0,1,0,0,0,0,34,27,0,0,27,7] >;

C5⋊Q16 in GAP, Magma, Sage, TeX

C_5\rtimes Q_{16}
% in TeX

G:=Group("C5:Q16");
// GroupNames label

G:=SmallGroup(80,18);
// by ID

G=gap.SmallGroup(80,18);
# by ID

G:=PCGroup([5,-2,-2,-2,-2,-5,40,61,46,182,97,42,1604]);
// Polycyclic

G:=Group<a,b,c|a^5=b^8=1,c^2=b^4,b*a*b^-1=a^-1,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

Export

Subgroup lattice of C5⋊Q16 in TeX
Character table of C5⋊Q16 in TeX

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