metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C5⋊3SD32, Q16⋊1D5, C8.6D10, C20.5D4, D40.2C2, C10.10D8, C40.4C22, C5⋊2C16⋊3C2, (C5×Q16)⋊1C2, C2.6(D4⋊D5), C4.3(C5⋊D4), SmallGroup(160,35)
Series: Derived ►Chief ►Lower central ►Upper central
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 |
►On 80 points
Generators in S80
(1 40 70 23 49)(2 50 24 71 41)(3 42 72 25 51)(4 52 26 73 43)(5 44 74 27 53)(6 54 28 75 45)(7 46 76 29 55)(8 56 30 77 47)(9 48 78 31 57)(10 58 32 79 33)(11 34 80 17 59)(12 60 18 65 35)(13 36 66 19 61)(14 62 20 67 37)(15 38 68 21 63)(16 64 22 69 39)
(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 76)(18 67)(19 74)(20 65)(21 72)(22 79)(23 70)(24 77)(25 68)(26 75)(27 66)(28 73)(29 80)(30 71)(31 78)(32 69)(33 64)(34 55)(35 62)(36 53)(37 60)(38 51)(39 58)(40 49)(41 56)(42 63)(43 54)(44 61)(45 52)(46 59)(47 50)(48 57)
G:=sub<Sym(80)| (1,40,70,23,49)(2,50,24,71,41)(3,42,72,25,51)(4,52,26,73,43)(5,44,74,27,53)(6,54,28,75,45)(7,46,76,29,55)(8,56,30,77,47)(9,48,78,31,57)(10,58,32,79,33)(11,34,80,17,59)(12,60,18,65,35)(13,36,66,19,61)(14,62,20,67,37)(15,38,68,21,63)(16,64,22,69,39), (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)(17,76)(18,67)(19,74)(20,65)(21,72)(22,79)(23,70)(24,77)(25,68)(26,75)(27,66)(28,73)(29,80)(30,71)(31,78)(32,69)(33,64)(34,55)(35,62)(36,53)(37,60)(38,51)(39,58)(40,49)(41,56)(42,63)(43,54)(44,61)(45,52)(46,59)(47,50)(48,57)>;
G:=Group( (1,40,70,23,49)(2,50,24,71,41)(3,42,72,25,51)(4,52,26,73,43)(5,44,74,27,53)(6,54,28,75,45)(7,46,76,29,55)(8,56,30,77,47)(9,48,78,31,57)(10,58,32,79,33)(11,34,80,17,59)(12,60,18,65,35)(13,36,66,19,61)(14,62,20,67,37)(15,38,68,21,63)(16,64,22,69,39), (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)(17,76)(18,67)(19,74)(20,65)(21,72)(22,79)(23,70)(24,77)(25,68)(26,75)(27,66)(28,73)(29,80)(30,71)(31,78)(32,69)(33,64)(34,55)(35,62)(36,53)(37,60)(38,51)(39,58)(40,49)(41,56)(42,63)(43,54)(44,61)(45,52)(46,59)(47,50)(48,57) );
G=PermutationGroup([[(1,40,70,23,49),(2,50,24,71,41),(3,42,72,25,51),(4,52,26,73,43),(5,44,74,27,53),(6,54,28,75,45),(7,46,76,29,55),(8,56,30,77,47),(9,48,78,31,57),(10,58,32,79,33),(11,34,80,17,59),(12,60,18,65,35),(13,36,66,19,61),(14,62,20,67,37),(15,38,68,21,63),(16,64,22,69,39)], [(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),(17,76),(18,67),(19,74),(20,65),(21,72),(22,79),(23,70),(24,77),(25,68),(26,75),(27,66),(28,73),(29,80),(30,71),(31,78),(32,69),(33,64),(34,55),(35,62),(36,53),(37,60),(38,51),(39,58),(40,49),(41,56),(42,63),(43,54),(44,61),(45,52),(46,59),(47,50),(48,57)]])
C5⋊SD32 is a maximal subgroup of
D5×SD32 C16⋊D10 Q32⋊D5 D80⋊5C2 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
Export
Subgroup lattice of C5⋊SD32 in TeX
Character table of C5⋊SD32 in TeX