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G = C5⋊SD32order 160 = 25·5

The semidirect product of C5 and SD32 acting via SD32/Q16=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C53SD32, Q161D5, C8.6D10, C20.5D4, D40.2C2, C10.10D8, C40.4C22, C52C163C2, (C5×Q16)⋊1C2, C2.6(D4⋊D5), C4.3(C5⋊D4), SmallGroup(160,35)

Series: Derived Chief Lower central Upper central

C1C40 — C5⋊SD32
C1C5C10C20C40D40 — C5⋊SD32
C5C10C20C40 — C5⋊SD32
C1C2C4C8Q16

Generators and relations for C5⋊SD32
 G = < a,b,c | a5=b16=c2=1, bab-1=cac=a-1, cbc=b7 >

40C2
4C4
20C22
8D5
2Q8
10D4
4D10
4C20
5C16
5D8
2D20
2C5×Q8
5SD32

Character table of C5⋊SD32

 class 12A2B4A4B5A5B8A8B10A10B16A16B16C16D20A20B20C20D20E20F40A40B40C40D
 size 114028222222101010104488884444
ρ11111111111111111111111111    trivial
ρ21111-1111111-1-1-1-111-1-1-1-11111    linear of order 2
ρ311-11-1111111111111-1-1-1-11111    linear of order 2
ρ411-111111111-1-1-1-11111111111    linear of order 2
ρ52202022-2-2220000220000-2-2-2-2    orthogonal lifted from D4
ρ62202-2-1+5/2-1-5/222-1+5/2-1-5/20000-1-5/2-1+5/21-5/21-5/21+5/21+5/2-1+5/2-1-5/2-1-5/2-1+5/2    orthogonal lifted from D10
ρ722022-1+5/2-1-5/222-1+5/2-1-5/20000-1-5/2-1+5/2-1+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
ρ822022-1-5/2-1+5/222-1-5/2-1+5/20000-1+5/2-1-5/2-1-5/2-1-5/2-1+5/2-1+5/2-1-5/2-1+5/2-1+5/2-1-5/2    orthogonal lifted from D5
ρ9220-202200222-22-2-2-200000000    orthogonal lifted from D8
ρ102202-2-1-5/2-1+5/222-1-5/2-1+5/20000-1+5/2-1-5/21+5/21+5/21-5/21-5/2-1-5/2-1+5/2-1+5/2-1-5/2    orthogonal lifted from D10
ρ11220-20220022-22-22-2-200000000    orthogonal lifted from D8
ρ1222020-1-5/2-1+5/2-2-2-1-5/2-1+5/20000-1+5/2-1-5/25352ζ5352ζ5455451+5/21-5/21-5/21+5/2    complex lifted from C5⋊D4
ρ1322020-1-5/2-1+5/2-2-2-1-5/2-1+5/20000-1+5/2-1-5/2ζ53525352545ζ5451+5/21-5/21-5/21+5/2    complex lifted from C5⋊D4
ρ1422020-1+5/2-1-5/2-2-2-1+5/2-1-5/20000-1-5/2-1+5/2545ζ5455352ζ53521-5/21+5/21+5/21-5/2    complex lifted from C5⋊D4
ρ1522020-1+5/2-1-5/2-2-2-1+5/2-1-5/20000-1-5/2-1+5/2ζ545545ζ535253521-5/21+5/21+5/21-5/2    complex lifted from C5⋊D4
ρ162-200022-22-2-2ζ1615169ζ16131611ζ16716ζ165163000000-22-22    complex lifted from SD32
ρ172-2000222-2-2-2ζ16131611ζ16716ζ165163ζ16151690000002-22-2    complex lifted from SD32
ρ182-2000222-2-2-2ζ165163ζ1615169ζ16131611ζ167160000002-22-2    complex lifted from SD32
ρ192-200022-22-2-2ζ16716ζ165163ζ1615169ζ16131611000000-22-22    complex lifted from SD32
ρ20440-40-1+5-1-500-1+5-1-500001+51-500000000    orthogonal lifted from D4⋊D5, Schur index 2
ρ21440-40-1-5-1+500-1-5-1+500001-51+500000000    orthogonal lifted from D4⋊D5, Schur index 2
ρ224-4000-1-5-1+5-22221+51-50000000000ζ83ζ5383ζ528ζ538ζ5283ζ5483ζ58ζ548ζ587ζ5487ζ585ζ5485ζ583ζ5383ζ528ζ538ζ52    orthogonal faithful, Schur index 2
ρ234-4000-1-5-1+522-221+51-5000000000083ζ5383ζ528ζ538ζ5287ζ5487ζ585ζ5485ζ583ζ5483ζ58ζ548ζ5ζ83ζ5383ζ528ζ538ζ52    orthogonal faithful, Schur index 2
ρ244-4000-1+5-1-522-221-51+5000000000083ζ5483ζ58ζ548ζ5ζ83ζ5383ζ528ζ538ζ5283ζ5383ζ528ζ538ζ5287ζ5487ζ585ζ5485ζ5    orthogonal faithful, Schur index 2
ρ254-4000-1+5-1-5-22221-51+5000000000087ζ5487ζ585ζ5485ζ583ζ5383ζ528ζ538ζ52ζ83ζ5383ζ528ζ538ζ5283ζ5483ζ58ζ548ζ5    orthogonal faithful, Schur index 2

Smallest permutation representation of C5⋊SD32
On 80 points
Generators in S80
(1 67 53 31 47)(2 48 32 54 68)(3 69 55 17 33)(4 34 18 56 70)(5 71 57 19 35)(6 36 20 58 72)(7 73 59 21 37)(8 38 22 60 74)(9 75 61 23 39)(10 40 24 62 76)(11 77 63 25 41)(12 42 26 64 78)(13 79 49 27 43)(14 44 28 50 80)(15 65 51 29 45)(16 46 30 52 66)
(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 8)(3 15)(4 6)(5 13)(7 11)(10 16)(12 14)(17 51)(18 58)(19 49)(20 56)(21 63)(22 54)(23 61)(24 52)(25 59)(26 50)(27 57)(28 64)(29 55)(30 62)(31 53)(32 60)(33 65)(34 72)(35 79)(36 70)(37 77)(38 68)(39 75)(40 66)(41 73)(42 80)(43 71)(44 78)(45 69)(46 76)(47 67)(48 74)

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

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

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

C5⋊SD32 is a maximal subgroup of
D5×SD32  C16⋊D10  Q32⋊D5  D805C2  Q16.D10  D8⋊D10  C40.30C23  C40.D6  Dic12⋊D5  C8.6D30
C5⋊SD32 is a maximal quotient of
C10.SD32  C40.5D4  C40.15D4  C40.D6  Dic12⋊D5  C8.6D30

Matrix representation of C5⋊SD32 in GL4(𝔽241) generated by

1000
0100
00511
002400
,
1791400
13714400
0051190
00240190
,
1000
11824000
0051190
00240190
G:=sub<GL(4,GF(241))| [1,0,0,0,0,1,0,0,0,0,51,240,0,0,1,0],[179,137,0,0,14,144,0,0,0,0,51,240,0,0,190,190],[1,118,0,0,0,240,0,0,0,0,51,240,0,0,190,190] >;

C5⋊SD32 in GAP, Magma, Sage, TeX

C_5\rtimes {\rm SD}_{32}
% in TeX

G:=Group("C5:SD32");
// GroupNames label

G:=SmallGroup(160,35);
// by ID

G=gap.SmallGroup(160,35);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-5,73,103,218,116,122,579,297,69,4613]);
// Polycyclic

G:=Group<a,b,c|a^5=b^16=c^2=1,b*a*b^-1=c*a*c=a^-1,c*b*c=b^7>;
// generators/relations

Export

Subgroup lattice of C5⋊SD32 in TeX
Character table of C5⋊SD32 in TeX

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