metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C3⋊2Q32, C8.7D6, Q16.S3, C12.6D4, C6.11D8, C24.5C22, Dic12.2C2, C3⋊C16.C2, C2.7(D4⋊S3), C4.4(C3⋊D4), (C3×Q16).1C2, SmallGroup(96,36)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3⋊Q32
G = < a,b,c | a3=b16=1, c2=b8, bab-1=a-1, ac=ca, cbc-1=b-1 >
Character table of C3⋊Q32
| class | 1 | 2 | 3 | 4A | 4B | 4C | 6 | 8A | 8B | 12A | 12B | 12C | 16A | 16B | 16C | 16D | 24A | 24B | |
| size | 1 | 1 | 2 | 2 | 8 | 24 | 2 | 2 | 2 | 4 | 8 | 8 | 6 | 6 | 6 | 6 | 4 | 4 | |
| ρ1 | 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 | linear of order 2 |
| ρ3 | 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 | linear of order 2 |
| ρ5 | 2 | 2 | -1 | 2 | -2 | 0 | -1 | 2 | 2 | -1 | 1 | 1 | 0 | 0 | 0 | 0 | -1 | -1 | orthogonal lifted from D6 |
| ρ6 | 2 | 2 | -1 | 2 | 2 | 0 | -1 | 2 | 2 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | -1 | -1 | orthogonal lifted from S3 |
| ρ7 | 2 | 2 | 2 | 2 | 0 | 0 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | orthogonal lifted from D4 |
| ρ8 | 2 | 2 | 2 | -2 | 0 | 0 | 2 | 0 | 0 | -2 | 0 | 0 | -√2 | √2 | -√2 | √2 | 0 | 0 | orthogonal lifted from D8 |
| ρ9 | 2 | 2 | 2 | -2 | 0 | 0 | 2 | 0 | 0 | -2 | 0 | 0 | √2 | -√2 | √2 | -√2 | 0 | 0 | orthogonal lifted from D8 |
| ρ10 | 2 | -2 | 2 | 0 | 0 | 0 | -2 | -√2 | √2 | 0 | 0 | 0 | -ζ165+ζ163 | ζ1615-ζ169 | ζ165-ζ163 | -ζ1615+ζ169 | √2 | -√2 | symplectic lifted from Q32, Schur index 2 |
| ρ11 | 2 | -2 | 2 | 0 | 0 | 0 | -2 | √2 | -√2 | 0 | 0 | 0 | -ζ1615+ζ169 | -ζ165+ζ163 | ζ1615-ζ169 | ζ165-ζ163 | -√2 | √2 | symplectic lifted from Q32, Schur index 2 |
| ρ12 | 2 | -2 | 2 | 0 | 0 | 0 | -2 | -√2 | √2 | 0 | 0 | 0 | ζ165-ζ163 | -ζ1615+ζ169 | -ζ165+ζ163 | ζ1615-ζ169 | √2 | -√2 | symplectic lifted from Q32, Schur index 2 |
| ρ13 | 2 | -2 | 2 | 0 | 0 | 0 | -2 | √2 | -√2 | 0 | 0 | 0 | ζ1615-ζ169 | ζ165-ζ163 | -ζ1615+ζ169 | -ζ165+ζ163 | -√2 | √2 | symplectic lifted from Q32, Schur index 2 |
| ρ14 | 2 | 2 | -1 | 2 | 0 | 0 | -1 | -2 | -2 | -1 | -√-3 | √-3 | 0 | 0 | 0 | 0 | 1 | 1 | complex lifted from C3⋊D4 |
| ρ15 | 2 | 2 | -1 | 2 | 0 | 0 | -1 | -2 | -2 | -1 | √-3 | -√-3 | 0 | 0 | 0 | 0 | 1 | 1 | complex lifted from C3⋊D4 |
| ρ16 | 4 | 4 | -2 | -4 | 0 | 0 | -2 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4⋊S3, Schur index 2 |
| ρ17 | 4 | -4 | -2 | 0 | 0 | 0 | 2 | 2√2 | -2√2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | √2 | -√2 | symplectic faithful, Schur index 2 |
| ρ18 | 4 | -4 | -2 | 0 | 0 | 0 | 2 | -2√2 | 2√2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -√2 | √2 | symplectic faithful, Schur index 2 |
►Regular action on 96 points
Generators in S96
(1 30 49)(2 50 31)(3 32 51)(4 52 17)(5 18 53)(6 54 19)(7 20 55)(8 56 21)(9 22 57)(10 58 23)(11 24 59)(12 60 25)(13 26 61)(14 62 27)(15 28 63)(16 64 29)(33 84 73)(34 74 85)(35 86 75)(36 76 87)(37 88 77)(38 78 89)(39 90 79)(40 80 91)(41 92 65)(42 66 93)(43 94 67)(44 68 95)(45 96 69)(46 70 81)(47 82 71)(48 72 83)
(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)(81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96)
(1 36 9 44)(2 35 10 43)(3 34 11 42)(4 33 12 41)(5 48 13 40)(6 47 14 39)(7 46 15 38)(8 45 16 37)(17 73 25 65)(18 72 26 80)(19 71 27 79)(20 70 28 78)(21 69 29 77)(22 68 30 76)(23 67 31 75)(24 66 32 74)(49 87 57 95)(50 86 58 94)(51 85 59 93)(52 84 60 92)(53 83 61 91)(54 82 62 90)(55 81 63 89)(56 96 64 88)
G:=sub<Sym(96)| (1,30,49)(2,50,31)(3,32,51)(4,52,17)(5,18,53)(6,54,19)(7,20,55)(8,56,21)(9,22,57)(10,58,23)(11,24,59)(12,60,25)(13,26,61)(14,62,27)(15,28,63)(16,64,29)(33,84,73)(34,74,85)(35,86,75)(36,76,87)(37,88,77)(38,78,89)(39,90,79)(40,80,91)(41,92,65)(42,66,93)(43,94,67)(44,68,95)(45,96,69)(46,70,81)(47,82,71)(48,72,83), (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)(81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96), (1,36,9,44)(2,35,10,43)(3,34,11,42)(4,33,12,41)(5,48,13,40)(6,47,14,39)(7,46,15,38)(8,45,16,37)(17,73,25,65)(18,72,26,80)(19,71,27,79)(20,70,28,78)(21,69,29,77)(22,68,30,76)(23,67,31,75)(24,66,32,74)(49,87,57,95)(50,86,58,94)(51,85,59,93)(52,84,60,92)(53,83,61,91)(54,82,62,90)(55,81,63,89)(56,96,64,88)>;
G:=Group( (1,30,49)(2,50,31)(3,32,51)(4,52,17)(5,18,53)(6,54,19)(7,20,55)(8,56,21)(9,22,57)(10,58,23)(11,24,59)(12,60,25)(13,26,61)(14,62,27)(15,28,63)(16,64,29)(33,84,73)(34,74,85)(35,86,75)(36,76,87)(37,88,77)(38,78,89)(39,90,79)(40,80,91)(41,92,65)(42,66,93)(43,94,67)(44,68,95)(45,96,69)(46,70,81)(47,82,71)(48,72,83), (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)(81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96), (1,36,9,44)(2,35,10,43)(3,34,11,42)(4,33,12,41)(5,48,13,40)(6,47,14,39)(7,46,15,38)(8,45,16,37)(17,73,25,65)(18,72,26,80)(19,71,27,79)(20,70,28,78)(21,69,29,77)(22,68,30,76)(23,67,31,75)(24,66,32,74)(49,87,57,95)(50,86,58,94)(51,85,59,93)(52,84,60,92)(53,83,61,91)(54,82,62,90)(55,81,63,89)(56,96,64,88) );
G=PermutationGroup([[(1,30,49),(2,50,31),(3,32,51),(4,52,17),(5,18,53),(6,54,19),(7,20,55),(8,56,21),(9,22,57),(10,58,23),(11,24,59),(12,60,25),(13,26,61),(14,62,27),(15,28,63),(16,64,29),(33,84,73),(34,74,85),(35,86,75),(36,76,87),(37,88,77),(38,78,89),(39,90,79),(40,80,91),(41,92,65),(42,66,93),(43,94,67),(44,68,95),(45,96,69),(46,70,81),(47,82,71),(48,72,83)], [(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),(81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96)], [(1,36,9,44),(2,35,10,43),(3,34,11,42),(4,33,12,41),(5,48,13,40),(6,47,14,39),(7,46,15,38),(8,45,16,37),(17,73,25,65),(18,72,26,80),(19,71,27,79),(20,70,28,78),(21,69,29,77),(22,68,30,76),(23,67,31,75),(24,66,32,74),(49,87,57,95),(50,86,58,94),(51,85,59,93),(52,84,60,92),(53,83,61,91),(54,82,62,90),(55,81,63,89),(56,96,64,88)]])
C3⋊Q32 is a maximal subgroup of
SD32⋊S3 D6.2D8 S3×Q32 Q32⋊S3 C24.27C23 Q16.D6 D8.9D6 C9⋊Q32 C32⋊2Q32 C32⋊3Q32 C32⋊7Q32 C15⋊Q32 C3⋊Dic40 C15⋊7Q32
C3⋊Q32 is a maximal quotient of
C6.6D16 C6.Q32 C6.5Q32 C9⋊Q32 C32⋊2Q32 C32⋊3Q32 C32⋊7Q32 C15⋊Q32 C3⋊Dic40 C15⋊7Q32
Matrix representation of C3⋊Q32 ►in GL4(𝔽7) generated by
| 0 | 0 | 5 | 2 |
| 5 | 3 | 0 | 3 |
| 2 | 5 | 0 | 3 |
| 5 | 5 | 1 | 2 |
| 4 | 2 | 3 | 2 |
| 1 | 0 | 0 | 5 |
| 1 | 6 | 3 | 4 |
| 0 | 0 | 1 | 0 |
| 3 | 0 | 1 | 2 |
| 1 | 0 | 6 | 6 |
| 2 | 2 | 5 | 3 |
| 1 | 6 | 3 | 6 |
G:=sub<GL(4,GF(7))| [0,5,2,5,0,3,5,5,5,0,0,1,2,3,3,2],[4,1,1,0,2,0,6,0,3,0,3,1,2,5,4,0],[3,1,2,1,0,0,2,6,1,6,5,3,2,6,3,6] >;
C3⋊Q32 in GAP, Magma, Sage, TeX
C_3\rtimes Q_{32} % in TeX
G:=Group("C3:Q32"); // GroupNames label
G:=SmallGroup(96,36);
// by ID
G=gap.SmallGroup(96,36);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-3,96,73,103,218,116,122,579,297,69,2309]);
// Polycyclic
G:=Group<a,b,c|a^3=b^16=1,c^2=b^8,b*a*b^-1=a^-1,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
Export
Subgroup lattice of C3⋊Q32 in TeX
Character table of C3⋊Q32 in TeX