metabelian, soluble, monomial, A-group
Aliases: C52⋊C16, (C5×C10).C8, C2.(C52⋊C8), C52⋊4C8.C2, C52⋊6C4.1C4, SmallGroup(400,116)
Series: Derived ►Chief ►Lower central ►Upper central
| C52 — C52⋊C16 |
Generators and relations for C52⋊C16
G = < a,b,c | a5=b5=c16=1, ab=ba, cac-1=ab2, cbc-1=ab-1 >
Character table of C52⋊C16
| class | 1 | 2 | 4A | 4B | 5A | 5B | 5C | 8A | 8B | 8C | 8D | 10A | 10B | 10C | 16A | 16B | 16C | 16D | 16E | 16F | 16G | 16H | |
| size | 1 | 1 | 25 | 25 | 8 | 8 | 8 | 25 | 25 | 25 | 25 | 8 | 8 | 8 | 25 | 25 | 25 | 25 | 25 | 25 | 25 | 25 | |
| ρ1 | 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 | linear of order 2 |
| ρ3 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | -i | i | i | i | -i | -i | -i | i | linear of order 4 |
| ρ4 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | i | -i | -i | -i | i | i | i | -i | linear of order 4 |
| ρ5 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | -i | -i | i | i | 1 | 1 | 1 | ζ8 | ζ87 | ζ83 | ζ83 | ζ85 | ζ85 | ζ8 | ζ87 | linear of order 8 |
| ρ6 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | -i | -i | i | i | 1 | 1 | 1 | ζ85 | ζ83 | ζ87 | ζ87 | ζ8 | ζ8 | ζ85 | ζ83 | linear of order 8 |
| ρ7 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | i | i | -i | -i | 1 | 1 | 1 | ζ87 | ζ8 | ζ85 | ζ85 | ζ83 | ζ83 | ζ87 | ζ8 | linear of order 8 |
| ρ8 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | i | i | -i | -i | 1 | 1 | 1 | ζ83 | ζ85 | ζ8 | ζ8 | ζ87 | ζ87 | ζ83 | ζ85 | linear of order 8 |
| ρ9 | 1 | -1 | i | -i | 1 | 1 | 1 | ζ166 | ζ1614 | ζ1610 | ζ162 | -1 | -1 | -1 | ζ16 | ζ167 | ζ1611 | ζ163 | ζ1613 | ζ165 | ζ169 | ζ1615 | linear of order 16 |
| ρ10 | 1 | -1 | i | -i | 1 | 1 | 1 | ζ166 | ζ1614 | ζ1610 | ζ162 | -1 | -1 | -1 | ζ169 | ζ1615 | ζ163 | ζ1611 | ζ165 | ζ1613 | ζ16 | ζ167 | linear of order 16 |
| ρ11 | 1 | -1 | i | -i | 1 | 1 | 1 | ζ1614 | ζ166 | ζ162 | ζ1610 | -1 | -1 | -1 | ζ165 | ζ163 | ζ167 | ζ1615 | ζ16 | ζ169 | ζ1613 | ζ1611 | linear of order 16 |
| ρ12 | 1 | -1 | i | -i | 1 | 1 | 1 | ζ1614 | ζ166 | ζ162 | ζ1610 | -1 | -1 | -1 | ζ1613 | ζ1611 | ζ1615 | ζ167 | ζ169 | ζ16 | ζ165 | ζ163 | linear of order 16 |
| ρ13 | 1 | -1 | -i | i | 1 | 1 | 1 | ζ1610 | ζ162 | ζ166 | ζ1614 | -1 | -1 | -1 | ζ167 | ζ16 | ζ1613 | ζ165 | ζ1611 | ζ163 | ζ1615 | ζ169 | linear of order 16 |
| ρ14 | 1 | -1 | -i | i | 1 | 1 | 1 | ζ1610 | ζ162 | ζ166 | ζ1614 | -1 | -1 | -1 | ζ1615 | ζ169 | ζ165 | ζ1613 | ζ163 | ζ1611 | ζ167 | ζ16 | linear of order 16 |
| ρ15 | 1 | -1 | -i | i | 1 | 1 | 1 | ζ162 | ζ1610 | ζ1614 | ζ166 | -1 | -1 | -1 | ζ1611 | ζ1613 | ζ169 | ζ16 | ζ1615 | ζ167 | ζ163 | ζ165 | linear of order 16 |
| ρ16 | 1 | -1 | -i | i | 1 | 1 | 1 | ζ162 | ζ1610 | ζ1614 | ζ166 | -1 | -1 | -1 | ζ163 | ζ165 | ζ16 | ζ169 | ζ167 | ζ1615 | ζ1611 | ζ1613 | linear of order 16 |
| ρ17 | 8 | 8 | 0 | 0 | 3 | -2 | -2 | 0 | 0 | 0 | 0 | 3 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C52⋊C8 |
| ρ18 | 8 | 8 | 0 | 0 | -2 | -2 | 3 | 0 | 0 | 0 | 0 | -2 | -2 | 3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C52⋊C8 |
| ρ19 | 8 | 8 | 0 | 0 | -2 | 3 | -2 | 0 | 0 | 0 | 0 | -2 | 3 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C52⋊C8 |
| ρ20 | 8 | -8 | 0 | 0 | -2 | 3 | -2 | 0 | 0 | 0 | 0 | 2 | -3 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic faithful, Schur index 2 |
| ρ21 | 8 | -8 | 0 | 0 | 3 | -2 | -2 | 0 | 0 | 0 | 0 | -3 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic faithful, Schur index 2 |
| ρ22 | 8 | -8 | 0 | 0 | -2 | -2 | 3 | 0 | 0 | 0 | 0 | 2 | 2 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic faithful, Schur index 2 |
►On 80 points
Generators in S80
(1 34 75 60 17)(2 76 18 35 61)(3 62 36 19 77)(4 37 78 63 20)(5 21 64 79 38)(6 49 39 22 80)(7 65 23 40 50)(8 24 51 66 41)(9 42 67 52 25)(10 68 26 43 53)(11 54 44 27 69)(12 45 70 55 28)(13 29 56 71 46)(14 57 47 30 72)(15 73 31 48 58)(16 32 59 74 33)
(1 60 34 17 75)(2 61 35 18 76)(3 19 62 77 36)(4 20 63 78 37)(5 79 21 38 64)(6 80 22 39 49)(7 40 65 50 23)(8 41 66 51 24)(9 52 42 25 67)(10 53 43 26 68)(11 27 54 69 44)(12 28 55 70 45)(13 71 29 46 56)(14 72 30 47 57)(15 48 73 58 31)(16 33 74 59 32)
(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)
G:=sub<Sym(80)| (1,34,75,60,17)(2,76,18,35,61)(3,62,36,19,77)(4,37,78,63,20)(5,21,64,79,38)(6,49,39,22,80)(7,65,23,40,50)(8,24,51,66,41)(9,42,67,52,25)(10,68,26,43,53)(11,54,44,27,69)(12,45,70,55,28)(13,29,56,71,46)(14,57,47,30,72)(15,73,31,48,58)(16,32,59,74,33), (1,60,34,17,75)(2,61,35,18,76)(3,19,62,77,36)(4,20,63,78,37)(5,79,21,38,64)(6,80,22,39,49)(7,40,65,50,23)(8,41,66,51,24)(9,52,42,25,67)(10,53,43,26,68)(11,27,54,69,44)(12,28,55,70,45)(13,71,29,46,56)(14,72,30,47,57)(15,48,73,58,31)(16,33,74,59,32), (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)>;
G:=Group( (1,34,75,60,17)(2,76,18,35,61)(3,62,36,19,77)(4,37,78,63,20)(5,21,64,79,38)(6,49,39,22,80)(7,65,23,40,50)(8,24,51,66,41)(9,42,67,52,25)(10,68,26,43,53)(11,54,44,27,69)(12,45,70,55,28)(13,29,56,71,46)(14,57,47,30,72)(15,73,31,48,58)(16,32,59,74,33), (1,60,34,17,75)(2,61,35,18,76)(3,19,62,77,36)(4,20,63,78,37)(5,79,21,38,64)(6,80,22,39,49)(7,40,65,50,23)(8,41,66,51,24)(9,52,42,25,67)(10,53,43,26,68)(11,27,54,69,44)(12,28,55,70,45)(13,71,29,46,56)(14,72,30,47,57)(15,48,73,58,31)(16,33,74,59,32), (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) );
G=PermutationGroup([[(1,34,75,60,17),(2,76,18,35,61),(3,62,36,19,77),(4,37,78,63,20),(5,21,64,79,38),(6,49,39,22,80),(7,65,23,40,50),(8,24,51,66,41),(9,42,67,52,25),(10,68,26,43,53),(11,54,44,27,69),(12,45,70,55,28),(13,29,56,71,46),(14,57,47,30,72),(15,73,31,48,58),(16,32,59,74,33)], [(1,60,34,17,75),(2,61,35,18,76),(3,19,62,77,36),(4,20,63,78,37),(5,79,21,38,64),(6,80,22,39,49),(7,40,65,50,23),(8,41,66,51,24),(9,52,42,25,67),(10,53,43,26,68),(11,27,54,69,44),(12,28,55,70,45),(13,71,29,46,56),(14,72,30,47,57),(15,48,73,58,31),(16,33,74,59,32)], [(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)]])
Matrix representation of C52⋊C16 ►in GL8(𝔽241)
| 0 | 240 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 240 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 240 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 240 | 0 | 0 | 0 | 0 | 0 | 0 |
| 97 | 34 | 104 | 240 | 0 | 0 | 0 | 1 |
| 119 | 54 | 88 | 24 | 240 | 240 | 240 | 240 |
| 97 | 34 | 104 | 240 | 1 | 0 | 0 | 0 |
| 97 | 34 | 104 | 240 | 0 | 1 | 0 | 0 |
| 240 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 240 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 240 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 240 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 130 | 145 | 200 | 40 | 0 | 0 | 1 | 0 |
| 130 | 145 | 200 | 40 | 0 | 0 | 0 | 1 |
| 152 | 165 | 184 | 65 | 240 | 240 | 240 | 240 |
| 130 | 145 | 200 | 40 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 240 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 240 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 240 | 0 | 0 | 1 |
| 22 | 20 | 225 | 25 | 239 | 240 | 240 | 240 |
| 125 | 42 | 53 | 4 | 33 | 144 | 240 | 40 |
| 2 | 220 | 192 | 140 | 33 | 144 | 240 | 40 |
| 141 | 115 | 103 | 17 | 33 | 144 | 240 | 40 |
| 52 | 233 | 40 | 156 | 33 | 144 | 240 | 40 |
G:=sub<GL(8,GF(241))| [0,0,0,1,97,119,97,97,240,240,240,240,34,54,34,34,1,0,0,0,104,88,104,104,0,1,0,0,240,24,240,240,0,0,0,0,0,240,1,0,0,0,0,0,0,240,0,1,0,0,0,0,0,240,0,0,0,0,0,0,1,240,0,0],[240,240,240,240,130,130,152,130,1,0,0,0,145,145,165,145,0,1,0,0,200,200,184,200,0,0,1,0,40,40,65,40,0,0,0,0,0,0,240,1,0,0,0,0,0,0,240,0,0,0,0,0,1,0,240,0,0,0,0,0,0,1,240,0],[0,0,0,22,125,2,141,52,0,0,0,20,42,220,115,233,0,0,0,225,53,192,103,40,0,0,0,25,4,140,17,156,240,240,240,239,33,33,33,33,1,0,0,240,144,144,144,144,0,1,0,240,240,240,240,240,0,0,1,240,40,40,40,40] >;
C52⋊C16 in GAP, Magma, Sage, TeX
C_5^2\rtimes C_{16} % in TeX
G:=Group("C5^2:C16"); // GroupNames label
G:=SmallGroup(400,116);
// by ID
G=gap.SmallGroup(400,116);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-5,5,12,31,50,10564,490,496,9797,2891,2897]);
// Polycyclic
G:=Group<a,b,c|a^5=b^5=c^16=1,a*b=b*a,c*a*c^-1=a*b^2,c*b*c^-1=a*b^-1>;
// generators/relations
Export
Subgroup lattice of C52⋊C16 in TeX
Character table of C52⋊C16 in TeX