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## G = C5⋊SD32order 160 = 25·5

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

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

 Derived series C1 — C40 — C5⋊SD32
 Chief series C1 — C5 — C10 — C20 — C40 — D40 — C5⋊SD32
 Lower central C5 — C10 — C20 — C40 — C5⋊SD32
 Upper central C1 — C2 — C4 — C8 — Q16

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

Character table of C5⋊SD32

 class 1 2A 2B 4A 4B 5A 5B 8A 8B 10A 10B 16A 16B 16C 16D 20A 20B 20C 20D 20E 20F 40A 40B 40C 40D size 1 1 40 2 8 2 2 2 2 2 2 10 10 10 10 4 4 8 8 8 8 4 4 4 4 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 1 1 -1 1 1 1 1 1 1 -1 -1 -1 -1 1 1 -1 -1 -1 -1 1 1 1 1 linear of order 2 ρ3 1 1 -1 1 -1 1 1 1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 1 1 1 1 linear of order 2 ρ4 1 1 -1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 1 1 1 1 1 1 1 1 1 1 linear of order 2 ρ5 2 2 0 2 0 2 2 -2 -2 2 2 0 0 0 0 2 2 0 0 0 0 -2 -2 -2 -2 orthogonal lifted from D4 ρ6 2 2 0 2 -2 -1+√5/2 -1-√5/2 2 2 -1+√5/2 -1-√5/2 0 0 0 0 -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 D10 ρ7 2 2 0 2 2 -1+√5/2 -1-√5/2 2 2 -1+√5/2 -1-√5/2 0 0 0 0 -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 ρ8 2 2 0 2 2 -1-√5/2 -1+√5/2 2 2 -1-√5/2 -1+√5/2 0 0 0 0 -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 ρ9 2 2 0 -2 0 2 2 0 0 2 2 √2 -√2 √2 -√2 -2 -2 0 0 0 0 0 0 0 0 orthogonal lifted from D8 ρ10 2 2 0 2 -2 -1-√5/2 -1+√5/2 2 2 -1-√5/2 -1+√5/2 0 0 0 0 -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 D10 ρ11 2 2 0 -2 0 2 2 0 0 2 2 -√2 √2 -√2 √2 -2 -2 0 0 0 0 0 0 0 0 orthogonal lifted from D8 ρ12 2 2 0 2 0 -1-√5/2 -1+√5/2 -2 -2 -1-√5/2 -1+√5/2 0 0 0 0 -1+√5/2 -1-√5/2 -ζ53+ζ52 ζ53-ζ52 ζ54-ζ5 -ζ54+ζ5 1+√5/2 1-√5/2 1-√5/2 1+√5/2 complex lifted from C5⋊D4 ρ13 2 2 0 2 0 -1-√5/2 -1+√5/2 -2 -2 -1-√5/2 -1+√5/2 0 0 0 0 -1+√5/2 -1-√5/2 ζ53-ζ52 -ζ53+ζ52 -ζ54+ζ5 ζ54-ζ5 1+√5/2 1-√5/2 1-√5/2 1+√5/2 complex lifted from C5⋊D4 ρ14 2 2 0 2 0 -1+√5/2 -1-√5/2 -2 -2 -1+√5/2 -1-√5/2 0 0 0 0 -1-√5/2 -1+√5/2 -ζ54+ζ5 ζ54-ζ5 -ζ53+ζ52 ζ53-ζ52 1-√5/2 1+√5/2 1+√5/2 1-√5/2 complex lifted from C5⋊D4 ρ15 2 2 0 2 0 -1+√5/2 -1-√5/2 -2 -2 -1+√5/2 -1-√5/2 0 0 0 0 -1-√5/2 -1+√5/2 ζ54-ζ5 -ζ54+ζ5 ζ53-ζ52 -ζ53+ζ52 1-√5/2 1+√5/2 1+√5/2 1-√5/2 complex lifted from C5⋊D4 ρ16 2 -2 0 0 0 2 2 -√2 √2 -2 -2 ζ1615+ζ169 ζ1613+ζ1611 ζ167+ζ16 ζ165+ζ163 0 0 0 0 0 0 -√2 √2 -√2 √2 complex lifted from SD32 ρ17 2 -2 0 0 0 2 2 √2 -√2 -2 -2 ζ1613+ζ1611 ζ167+ζ16 ζ165+ζ163 ζ1615+ζ169 0 0 0 0 0 0 √2 -√2 √2 -√2 complex lifted from SD32 ρ18 2 -2 0 0 0 2 2 √2 -√2 -2 -2 ζ165+ζ163 ζ1615+ζ169 ζ1613+ζ1611 ζ167+ζ16 0 0 0 0 0 0 √2 -√2 √2 -√2 complex lifted from SD32 ρ19 2 -2 0 0 0 2 2 -√2 √2 -2 -2 ζ167+ζ16 ζ165+ζ163 ζ1615+ζ169 ζ1613+ζ1611 0 0 0 0 0 0 -√2 √2 -√2 √2 complex lifted from SD32 ρ20 4 4 0 -4 0 -1+√5 -1-√5 0 0 -1+√5 -1-√5 0 0 0 0 1+√5 1-√5 0 0 0 0 0 0 0 0 orthogonal lifted from D4⋊D5, Schur index 2 ρ21 4 4 0 -4 0 -1-√5 -1+√5 0 0 -1-√5 -1+√5 0 0 0 0 1-√5 1+√5 0 0 0 0 0 0 0 0 orthogonal lifted from D4⋊D5, Schur index 2 ρ22 4 -4 0 0 0 -1-√5 -1+√5 -2√2 2√2 1+√5 1-√5 0 0 0 0 0 0 0 0 0 0 ζ83ζ53+ζ83ζ52-ζ8ζ53-ζ8ζ52 -ζ83ζ54-ζ83ζ5+ζ8ζ54+ζ8ζ5 -ζ87ζ54-ζ87ζ5+ζ85ζ54+ζ85ζ5 -ζ83ζ53-ζ83ζ52+ζ8ζ53+ζ8ζ52 orthogonal faithful, Schur index 2 ρ23 4 -4 0 0 0 -1-√5 -1+√5 2√2 -2√2 1+√5 1-√5 0 0 0 0 0 0 0 0 0 0 -ζ83ζ53-ζ83ζ52+ζ8ζ53+ζ8ζ52 -ζ87ζ54-ζ87ζ5+ζ85ζ54+ζ85ζ5 -ζ83ζ54-ζ83ζ5+ζ8ζ54+ζ8ζ5 ζ83ζ53+ζ83ζ52-ζ8ζ53-ζ8ζ52 orthogonal faithful, Schur index 2 ρ24 4 -4 0 0 0 -1+√5 -1-√5 2√2 -2√2 1-√5 1+√5 0 0 0 0 0 0 0 0 0 0 -ζ83ζ54-ζ83ζ5+ζ8ζ54+ζ8ζ5 ζ83ζ53+ζ83ζ52-ζ8ζ53-ζ8ζ52 -ζ83ζ53-ζ83ζ52+ζ8ζ53+ζ8ζ52 -ζ87ζ54-ζ87ζ5+ζ85ζ54+ζ85ζ5 orthogonal faithful, Schur index 2 ρ25 4 -4 0 0 0 -1+√5 -1-√5 -2√2 2√2 1-√5 1+√5 0 0 0 0 0 0 0 0 0 0 -ζ87ζ54-ζ87ζ5+ζ85ζ54+ζ85ζ5 -ζ83ζ53-ζ83ζ52+ζ8ζ53+ζ8ζ52 ζ83ζ53+ζ83ζ52-ζ8ζ53-ζ8ζ52 -ζ83ζ54-ζ83ζ5+ζ8ζ54+ζ8ζ5 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

 1 0 0 0 0 1 0 0 0 0 51 1 0 0 240 0
,
 179 14 0 0 137 144 0 0 0 0 51 190 0 0 240 190
,
 1 0 0 0 118 240 0 0 0 0 51 190 0 0 240 190
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

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