metabelian, soluble, monomial, A-group
Aliases: C32⋊2C16, (C3×C6).2C8, (C3×C12).1C4, C2.(C32⋊2C8), C4.2(C32⋊C4), C32⋊4C8.3C2, SmallGroup(144,51)
Series: Derived ►Chief ►Lower central ►Upper central
| C32 — C32⋊2C16 |
Generators and relations for C32⋊2C16
G = < a,b,c | a3=b3=c16=1, cbc-1=ab=ba, cac-1=a-1b >
Character table of C32⋊2C16
| class | 1 | 2 | 3A | 3B | 4A | 4B | 6A | 6B | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 16A | 16B | 16C | 16D | 16E | 16F | 16G | 16H | |
| size | 1 | 1 | 4 | 4 | 1 | 1 | 4 | 4 | 9 | 9 | 9 | 9 | 4 | 4 | 4 | 4 | 9 | 9 | 9 | 9 | 9 | 9 | 9 | 9 | |
| 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 | 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 | linear of order 2 |
| ρ3 | 1 | 1 | 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 | 1 | 1 | -i | i | i | i | -i | -i | -i | i | linear of order 4 |
| ρ5 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | i | i | -i | -i | -1 | -1 | -1 | -1 | ζ83 | ζ85 | ζ8 | ζ8 | ζ87 | ζ87 | ζ83 | ζ85 | linear of order 8 |
| ρ6 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | i | i | -i | -i | -1 | -1 | -1 | -1 | ζ87 | ζ8 | ζ85 | ζ85 | ζ83 | ζ83 | ζ87 | ζ8 | linear of order 8 |
| ρ7 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | -i | -i | i | i | -1 | -1 | -1 | -1 | ζ85 | ζ83 | ζ87 | ζ87 | ζ8 | ζ8 | ζ85 | ζ83 | linear of order 8 |
| ρ8 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | -i | -i | i | i | -1 | -1 | -1 | -1 | ζ8 | ζ87 | ζ83 | ζ83 | ζ85 | ζ85 | ζ8 | ζ87 | linear of order 8 |
| ρ9 | 1 | -1 | 1 | 1 | i | -i | -1 | -1 | ζ1614 | ζ166 | ζ1610 | ζ162 | -i | -i | i | i | ζ169 | ζ1615 | ζ163 | ζ1611 | ζ165 | ζ1613 | ζ16 | ζ167 | linear of order 16 |
| ρ10 | 1 | -1 | 1 | 1 | -i | i | -1 | -1 | ζ1610 | ζ162 | ζ1614 | ζ166 | i | i | -i | -i | ζ163 | ζ165 | ζ16 | ζ169 | ζ167 | ζ1615 | ζ1611 | ζ1613 | linear of order 16 |
| ρ11 | 1 | -1 | 1 | 1 | -i | i | -1 | -1 | ζ1610 | ζ162 | ζ1614 | ζ166 | i | i | -i | -i | ζ1611 | ζ1613 | ζ169 | ζ16 | ζ1615 | ζ167 | ζ163 | ζ165 | linear of order 16 |
| ρ12 | 1 | -1 | 1 | 1 | i | -i | -1 | -1 | ζ1614 | ζ166 | ζ1610 | ζ162 | -i | -i | i | i | ζ16 | ζ167 | ζ1611 | ζ163 | ζ1613 | ζ165 | ζ169 | ζ1615 | linear of order 16 |
| ρ13 | 1 | -1 | 1 | 1 | -i | i | -1 | -1 | ζ162 | ζ1610 | ζ166 | ζ1614 | i | i | -i | -i | ζ167 | ζ16 | ζ1613 | ζ165 | ζ1611 | ζ163 | ζ1615 | ζ169 | linear of order 16 |
| ρ14 | 1 | -1 | 1 | 1 | i | -i | -1 | -1 | ζ166 | ζ1614 | ζ162 | ζ1610 | -i | -i | i | i | ζ165 | ζ163 | ζ167 | ζ1615 | ζ16 | ζ169 | ζ1613 | ζ1611 | linear of order 16 |
| ρ15 | 1 | -1 | 1 | 1 | -i | i | -1 | -1 | ζ162 | ζ1610 | ζ166 | ζ1614 | i | i | -i | -i | ζ1615 | ζ169 | ζ165 | ζ1613 | ζ163 | ζ1611 | ζ167 | ζ16 | linear of order 16 |
| ρ16 | 1 | -1 | 1 | 1 | i | -i | -1 | -1 | ζ166 | ζ1614 | ζ162 | ζ1610 | -i | -i | i | i | ζ1613 | ζ1611 | ζ1615 | ζ167 | ζ169 | ζ16 | ζ165 | ζ163 | linear of order 16 |
| ρ17 | 4 | 4 | -2 | 1 | 4 | 4 | 1 | -2 | 0 | 0 | 0 | 0 | -2 | 1 | 1 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C32⋊C4 |
| ρ18 | 4 | 4 | 1 | -2 | 4 | 4 | -2 | 1 | 0 | 0 | 0 | 0 | 1 | -2 | -2 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C32⋊C4 |
| ρ19 | 4 | 4 | 1 | -2 | -4 | -4 | -2 | 1 | 0 | 0 | 0 | 0 | -1 | 2 | 2 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from C32⋊2C8, Schur index 2 |
| ρ20 | 4 | 4 | -2 | 1 | -4 | -4 | 1 | -2 | 0 | 0 | 0 | 0 | 2 | -1 | -1 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from C32⋊2C8, Schur index 2 |
| ρ21 | 4 | -4 | -2 | 1 | 4i | -4i | -1 | 2 | 0 | 0 | 0 | 0 | 2i | -i | i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
| ρ22 | 4 | -4 | -2 | 1 | -4i | 4i | -1 | 2 | 0 | 0 | 0 | 0 | -2i | i | -i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
| ρ23 | 4 | -4 | 1 | -2 | -4i | 4i | 2 | -1 | 0 | 0 | 0 | 0 | i | -2i | 2i | -i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
| ρ24 | 4 | -4 | 1 | -2 | 4i | -4i | 2 | -1 | 0 | 0 | 0 | 0 | -i | 2i | -2i | i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
►On 48 points
Generators in S48
(2 30 48)(4 34 32)(6 18 36)(8 38 20)(10 22 40)(12 42 24)(14 26 44)(16 46 28)
(1 29 47)(2 30 48)(3 33 31)(4 34 32)(5 17 35)(6 18 36)(7 37 19)(8 38 20)(9 21 39)(10 22 40)(11 41 23)(12 42 24)(13 25 43)(14 26 44)(15 45 27)(16 46 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)
G:=sub<Sym(48)| (2,30,48)(4,34,32)(6,18,36)(8,38,20)(10,22,40)(12,42,24)(14,26,44)(16,46,28), (1,29,47)(2,30,48)(3,33,31)(4,34,32)(5,17,35)(6,18,36)(7,37,19)(8,38,20)(9,21,39)(10,22,40)(11,41,23)(12,42,24)(13,25,43)(14,26,44)(15,45,27)(16,46,28), (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)>;
G:=Group( (2,30,48)(4,34,32)(6,18,36)(8,38,20)(10,22,40)(12,42,24)(14,26,44)(16,46,28), (1,29,47)(2,30,48)(3,33,31)(4,34,32)(5,17,35)(6,18,36)(7,37,19)(8,38,20)(9,21,39)(10,22,40)(11,41,23)(12,42,24)(13,25,43)(14,26,44)(15,45,27)(16,46,28), (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) );
G=PermutationGroup([[(2,30,48),(4,34,32),(6,18,36),(8,38,20),(10,22,40),(12,42,24),(14,26,44),(16,46,28)], [(1,29,47),(2,30,48),(3,33,31),(4,34,32),(5,17,35),(6,18,36),(7,37,19),(8,38,20),(9,21,39),(10,22,40),(11,41,23),(12,42,24),(13,25,43),(14,26,44),(15,45,27),(16,46,28)], [(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)]])
C32⋊2C16 is a maximal subgroup of
C32⋊C32 C32⋊D16 C32⋊SD32 C32⋊Q32 C3⋊S3⋊3C16 C32⋊3M5(2) C62.4C8 C33⋊4C16
C32⋊2C16 is a maximal quotient of
C32⋊2C32 He3⋊2C16 C33⋊4C16
Matrix representation of C32⋊2C16 ►in GL4(𝔽5) generated by
| 2 | 0 | 2 | 0 |
| 0 | 1 | 0 | 4 |
| 4 | 0 | 2 | 0 |
| 0 | 3 | 0 | 3 |
| 1 | 0 | 0 | 0 |
| 0 | 3 | 0 | 1 |
| 0 | 0 | 1 | 0 |
| 0 | 2 | 0 | 1 |
| 0 | 3 | 0 | 3 |
| 1 | 0 | 4 | 0 |
| 0 | 3 | 0 | 0 |
| 0 | 0 | 3 | 0 |
G:=sub<GL(4,GF(5))| [2,0,4,0,0,1,0,3,2,0,2,0,0,4,0,3],[1,0,0,0,0,3,0,2,0,0,1,0,0,1,0,1],[0,1,0,0,3,0,3,0,0,4,0,3,3,0,0,0] >;
C32⋊2C16 in GAP, Magma, Sage, TeX
C_3^2\rtimes_2C_{16} % in TeX
G:=Group("C3^2:2C16"); // GroupNames label
G:=SmallGroup(144,51);
// by ID
G=gap.SmallGroup(144,51);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-3,3,12,31,50,3364,490,4613,1739]);
// Polycyclic
G:=Group<a,b,c|a^3=b^3=c^16=1,c*b*c^-1=a*b=b*a,c*a*c^-1=a^-1*b>;
// generators/relations
Export
Subgroup lattice of C32⋊2C16 in TeX
Character table of C32⋊2C16 in TeX