Aliases: C3⋊S3.5Q16, C3⋊S3.7SD16, Q8⋊1(C32⋊C4), (Q8×C32)⋊1C4, C32⋊4Q8⋊2C4, C3⋊Dic3.11D4, C32⋊7(Q8⋊C4), C2.8(C62⋊C4), C4.3(C2×C32⋊C4), (Q8×C3⋊S3).1C2, (C3×C12).3(C2×C4), (C2×C3⋊S3).52D4, C3⋊S3⋊3C8.1C2, C4⋊(C32⋊C4).2C2, (C4×C3⋊S3).7C22, (C3×C6).18(C22⋊C4), SmallGroup(288,432)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3⋊S3.5Q16
G = < a,b,c,d,e | a3=b3=c2=d8=1, e2=d4, ab=ba, cac=a-1, dad-1=ab-1, ae=ea, cbc=b-1, dbd-1=a-1b-1, be=eb, cd=dc, ce=ec, ede-1=cd-1 >
Subgroups: 432 in 80 conjugacy classes, 18 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, S3, C6, C8, C2×C4, Q8, Q8, C32, Dic3, C12, D6, C4⋊C4, C2×C8, C2×Q8, C3⋊S3, C3×C6, Dic6, C4×S3, C3×Q8, Q8⋊C4, C3⋊Dic3, C3⋊Dic3, C3×C12, C3×C12, C32⋊C4, C2×C3⋊S3, S3×Q8, C32⋊2C8, C32⋊4Q8, C32⋊4Q8, C4×C3⋊S3, C4×C3⋊S3, Q8×C32, C2×C32⋊C4, C3⋊S3⋊3C8, C4⋊(C32⋊C4), Q8×C3⋊S3, C3⋊S3.5Q16
Quotients: C1, C2, C4, C22, C2×C4, D4, C22⋊C4, SD16, Q16, Q8⋊C4, C32⋊C4, C2×C32⋊C4, C62⋊C4, C3⋊S3.5Q16
Character table of C3⋊S3.5Q16
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 4F | 6A | 6B | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | 12F | |
| size | 1 | 1 | 9 | 9 | 4 | 4 | 2 | 4 | 18 | 36 | 36 | 36 | 4 | 4 | 18 | 18 | 18 | 18 | 8 | 8 | 8 | 8 | 8 | 8 | |
| 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 | -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 | linear of order 2 |
| ρ5 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | i | -1 | -i | 1 | 1 | i | -i | -i | i | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ6 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | i | 1 | -i | 1 | 1 | -i | i | i | -i | -1 | -1 | -1 | 1 | 1 | -1 | linear of order 4 |
| ρ7 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | -i | 1 | i | 1 | 1 | i | -i | -i | i | -1 | -1 | -1 | 1 | 1 | -1 | linear of order 4 |
| ρ8 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -i | -1 | i | 1 | 1 | -i | i | i | -i | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ9 | 2 | 2 | -2 | -2 | 2 | 2 | -2 | 0 | 2 | 0 | 0 | 0 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 0 | orthogonal lifted from D4 |
| ρ10 | 2 | 2 | 2 | 2 | 2 | 2 | -2 | 0 | -2 | 0 | 0 | 0 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 0 | orthogonal lifted from D4 |
| ρ11 | 2 | -2 | -2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | -√2 | -√2 | √2 | √2 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from Q16, Schur index 2 |
| ρ12 | 2 | -2 | -2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | √2 | √2 | -√2 | -√2 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from Q16, Schur index 2 |
| ρ13 | 2 | -2 | 2 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | √-2 | -√-2 | √-2 | -√-2 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from SD16 |
| ρ14 | 2 | -2 | 2 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | -√-2 | √-2 | -√-2 | √-2 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from SD16 |
| ρ15 | 4 | 4 | 0 | 0 | 1 | -2 | 4 | 4 | 0 | 0 | 0 | 0 | -2 | 1 | 0 | 0 | 0 | 0 | -2 | 1 | 1 | 1 | -2 | -2 | orthogonal lifted from C32⋊C4 |
| ρ16 | 4 | 4 | 0 | 0 | 1 | -2 | -4 | 0 | 0 | 0 | 0 | 0 | -2 | 1 | 0 | 0 | 0 | 0 | 0 | 3 | -3 | -1 | 2 | 0 | orthogonal lifted from C62⋊C4 |
| ρ17 | 4 | 4 | 0 | 0 | 1 | -2 | -4 | 0 | 0 | 0 | 0 | 0 | -2 | 1 | 0 | 0 | 0 | 0 | 0 | -3 | 3 | -1 | 2 | 0 | orthogonal lifted from C62⋊C4 |
| ρ18 | 4 | 4 | 0 | 0 | -2 | 1 | 4 | -4 | 0 | 0 | 0 | 0 | 1 | -2 | 0 | 0 | 0 | 0 | -1 | 2 | 2 | -2 | 1 | -1 | orthogonal lifted from C2×C32⋊C4 |
| ρ19 | 4 | 4 | 0 | 0 | -2 | 1 | -4 | 0 | 0 | 0 | 0 | 0 | 1 | -2 | 0 | 0 | 0 | 0 | 3 | 0 | 0 | 2 | -1 | -3 | orthogonal lifted from C62⋊C4 |
| ρ20 | 4 | 4 | 0 | 0 | -2 | 1 | 4 | 4 | 0 | 0 | 0 | 0 | 1 | -2 | 0 | 0 | 0 | 0 | 1 | -2 | -2 | -2 | 1 | 1 | orthogonal lifted from C32⋊C4 |
| ρ21 | 4 | 4 | 0 | 0 | -2 | 1 | -4 | 0 | 0 | 0 | 0 | 0 | 1 | -2 | 0 | 0 | 0 | 0 | -3 | 0 | 0 | 2 | -1 | 3 | orthogonal lifted from C62⋊C4 |
| ρ22 | 4 | 4 | 0 | 0 | 1 | -2 | 4 | -4 | 0 | 0 | 0 | 0 | -2 | 1 | 0 | 0 | 0 | 0 | 2 | -1 | -1 | 1 | -2 | 2 | orthogonal lifted from C2×C32⋊C4 |
| ρ23 | 8 | -8 | 0 | 0 | -4 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic faithful, Schur index 2 |
| ρ24 | 8 | -8 | 0 | 0 | 2 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 4 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic faithful, Schur index 2 |
►On 48 points
Generators in S48
(1 37 46)(2 38 47)(3 48 39)(4 41 40)(5 33 42)(6 34 43)(7 44 35)(8 45 36)(9 30 19)(10 31 20)(11 21 32)(12 22 25)(13 26 23)(14 27 24)(15 17 28)(16 18 29)
(2 47 38)(4 40 41)(6 43 34)(8 36 45)(10 20 31)(12 25 22)(14 24 27)(16 29 18)
(9 30)(10 31)(11 32)(12 25)(13 26)(14 27)(15 28)(16 29)(33 42)(34 43)(35 44)(36 45)(37 46)(38 47)(39 48)(40 41)
(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)
(1 23 5 19)(2 22 6 18)(3 21 7 17)(4 20 8 24)(9 37 13 33)(10 45 14 41)(11 35 15 39)(12 43 16 47)(25 34 29 38)(26 42 30 46)(27 40 31 36)(28 48 32 44)
G:=sub<Sym(48)| (1,37,46)(2,38,47)(3,48,39)(4,41,40)(5,33,42)(6,34,43)(7,44,35)(8,45,36)(9,30,19)(10,31,20)(11,21,32)(12,22,25)(13,26,23)(14,27,24)(15,17,28)(16,18,29), (2,47,38)(4,40,41)(6,43,34)(8,36,45)(10,20,31)(12,25,22)(14,24,27)(16,29,18), (9,30)(10,31)(11,32)(12,25)(13,26)(14,27)(15,28)(16,29)(33,42)(34,43)(35,44)(36,45)(37,46)(38,47)(39,48)(40,41), (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), (1,23,5,19)(2,22,6,18)(3,21,7,17)(4,20,8,24)(9,37,13,33)(10,45,14,41)(11,35,15,39)(12,43,16,47)(25,34,29,38)(26,42,30,46)(27,40,31,36)(28,48,32,44)>;
G:=Group( (1,37,46)(2,38,47)(3,48,39)(4,41,40)(5,33,42)(6,34,43)(7,44,35)(8,45,36)(9,30,19)(10,31,20)(11,21,32)(12,22,25)(13,26,23)(14,27,24)(15,17,28)(16,18,29), (2,47,38)(4,40,41)(6,43,34)(8,36,45)(10,20,31)(12,25,22)(14,24,27)(16,29,18), (9,30)(10,31)(11,32)(12,25)(13,26)(14,27)(15,28)(16,29)(33,42)(34,43)(35,44)(36,45)(37,46)(38,47)(39,48)(40,41), (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), (1,23,5,19)(2,22,6,18)(3,21,7,17)(4,20,8,24)(9,37,13,33)(10,45,14,41)(11,35,15,39)(12,43,16,47)(25,34,29,38)(26,42,30,46)(27,40,31,36)(28,48,32,44) );
G=PermutationGroup([[(1,37,46),(2,38,47),(3,48,39),(4,41,40),(5,33,42),(6,34,43),(7,44,35),(8,45,36),(9,30,19),(10,31,20),(11,21,32),(12,22,25),(13,26,23),(14,27,24),(15,17,28),(16,18,29)], [(2,47,38),(4,40,41),(6,43,34),(8,36,45),(10,20,31),(12,25,22),(14,24,27),(16,29,18)], [(9,30),(10,31),(11,32),(12,25),(13,26),(14,27),(15,28),(16,29),(33,42),(34,43),(35,44),(36,45),(37,46),(38,47),(39,48),(40,41)], [(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)], [(1,23,5,19),(2,22,6,18),(3,21,7,17),(4,20,8,24),(9,37,13,33),(10,45,14,41),(11,35,15,39),(12,43,16,47),(25,34,29,38),(26,42,30,46),(27,40,31,36),(28,48,32,44)]])
Matrix representation of C3⋊S3.5Q16 ►in GL6(𝔽73)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 1 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 72 | 72 |
| 0 | 0 | 72 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 1 | 72 | 72 |
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 1 | 1 | 72 | 72 |
| 6 | 67 | 0 | 0 | 0 | 0 |
| 6 | 6 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 1 |
| 0 | 0 | 1 | 1 | 71 | 72 |
| 0 | 0 | 25 | 25 | 72 | 0 |
| 0 | 0 | 25 | 24 | 72 | 0 |
| 13 | 66 | 0 | 0 | 0 | 0 |
| 66 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 25 | 25 | 72 | 0 |
| 0 | 0 | 25 | 25 | 0 | 72 |
G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,72,0,72,0,0,1,0,1,0,0,0,0,0,72,1,0,0,0,0,72,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,1,0,1,0,0,0,0,0,72,0,0,0,0,1,72],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,1,0,1,0,0,1,0,0,1,0,0,0,0,1,72,0,0,0,0,0,72],[6,6,0,0,0,0,67,6,0,0,0,0,0,0,0,1,25,25,0,0,0,1,25,24,0,0,72,71,72,72,0,0,1,72,0,0],[13,66,0,0,0,0,66,60,0,0,0,0,0,0,1,0,25,25,0,0,0,1,25,25,0,0,0,0,72,0,0,0,0,0,0,72] >;
C3⋊S3.5Q16 in GAP, Magma, Sage, TeX
C_3\rtimes S_3._5Q_{16} % in TeX
G:=Group("C3:S3.5Q16"); // GroupNames label
G:=SmallGroup(288,432);
// by ID
G=gap.SmallGroup(288,432);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,28,141,120,675,346,80,9413,691,12550,2372]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^3=c^2=d^8=1,e^2=d^4,a*b=b*a,c*a*c=a^-1,d*a*d^-1=a*b^-1,a*e=e*a,c*b*c=b^-1,d*b*d^-1=a^-1*b^-1,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=c*d^-1>;
// generators/relations
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