metabelian, supersoluble, monomial, A-group
Aliases: C32⋊4C8, C12.6S3, C6.3Dic3, C3⋊(C3⋊C8), (C3×C6).3C4, C4.2(C3⋊S3), (C3×C12).4C2, C2.(C3⋊Dic3), SmallGroup(72,13)
Series: Derived ►Chief ►Lower central ►Upper central
| C32 — C32⋊4C8 |
Generators and relations for C32⋊4C8
G = < a,b,c | a3=b3=c8=1, ab=ba, cac-1=a-1, cbc-1=b-1 >
9C8
Character table of C32⋊4C8
| class | 1 | 2 | 3A | 3B | 3C | 3D | 4A | 4B | 6A | 6B | 6C | 6D | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | |
| size | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 2 | 9 | 9 | 9 | 9 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | |
| 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 | -i | i | -i | i | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | linear of order 4 |
| ρ4 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | i | -i | i | -i | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | linear of order 4 |
| ρ5 | 1 | -1 | 1 | 1 | 1 | 1 | -i | i | -1 | -1 | -1 | -1 | ζ83 | ζ8 | ζ87 | ζ85 | -i | -i | -i | -i | i | i | i | i | linear of order 8 |
| ρ6 | 1 | -1 | 1 | 1 | 1 | 1 | i | -i | -1 | -1 | -1 | -1 | ζ85 | ζ87 | ζ8 | ζ83 | i | i | i | i | -i | -i | -i | -i | linear of order 8 |
| ρ7 | 1 | -1 | 1 | 1 | 1 | 1 | -i | i | -1 | -1 | -1 | -1 | ζ87 | ζ85 | ζ83 | ζ8 | -i | -i | -i | -i | i | i | i | i | linear of order 8 |
| ρ8 | 1 | -1 | 1 | 1 | 1 | 1 | i | -i | -1 | -1 | -1 | -1 | ζ8 | ζ83 | ζ85 | ζ87 | i | i | i | i | -i | -i | -i | -i | linear of order 8 |
| ρ9 | 2 | 2 | -1 | 2 | -1 | -1 | 2 | 2 | 2 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | -1 | -1 | 2 | -1 | -1 | 2 | -1 | -1 | orthogonal lifted from S3 |
| ρ10 | 2 | 2 | 2 | -1 | -1 | -1 | 2 | 2 | -1 | -1 | -1 | 2 | 0 | 0 | 0 | 0 | -1 | 2 | -1 | -1 | 2 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ11 | 2 | 2 | -1 | -1 | -1 | 2 | 2 | 2 | -1 | -1 | 2 | -1 | 0 | 0 | 0 | 0 | 2 | -1 | -1 | -1 | -1 | -1 | -1 | 2 | orthogonal lifted from S3 |
| ρ12 | 2 | 2 | -1 | -1 | 2 | -1 | 2 | 2 | -1 | 2 | -1 | -1 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 2 | -1 | -1 | 2 | -1 | orthogonal lifted from S3 |
| ρ13 | 2 | 2 | -1 | -1 | 2 | -1 | -2 | -2 | -1 | 2 | -1 | -1 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | -2 | 1 | 1 | -2 | 1 | symplectic lifted from Dic3, Schur index 2 |
| ρ14 | 2 | 2 | 2 | -1 | -1 | -1 | -2 | -2 | -1 | -1 | -1 | 2 | 0 | 0 | 0 | 0 | 1 | -2 | 1 | 1 | -2 | 1 | 1 | 1 | symplectic lifted from Dic3, Schur index 2 |
| ρ15 | 2 | 2 | -1 | -1 | -1 | 2 | -2 | -2 | -1 | -1 | 2 | -1 | 0 | 0 | 0 | 0 | -2 | 1 | 1 | 1 | 1 | 1 | 1 | -2 | symplectic lifted from Dic3, Schur index 2 |
| ρ16 | 2 | 2 | -1 | 2 | -1 | -1 | -2 | -2 | 2 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 1 | 1 | -2 | 1 | 1 | -2 | 1 | 1 | symplectic lifted from Dic3, Schur index 2 |
| ρ17 | 2 | -2 | 2 | -1 | -1 | -1 | 2i | -2i | 1 | 1 | 1 | -2 | 0 | 0 | 0 | 0 | -i | 2i | -i | -i | -2i | i | i | i | complex lifted from C3⋊C8 |
| ρ18 | 2 | -2 | -1 | -1 | 2 | -1 | -2i | 2i | 1 | -2 | 1 | 1 | 0 | 0 | 0 | 0 | i | i | i | -2i | -i | -i | 2i | -i | complex lifted from C3⋊C8 |
| ρ19 | 2 | -2 | -1 | 2 | -1 | -1 | 2i | -2i | -2 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | -i | -i | 2i | -i | i | -2i | i | i | complex lifted from C3⋊C8 |
| ρ20 | 2 | -2 | -1 | 2 | -1 | -1 | -2i | 2i | -2 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | i | i | -2i | i | -i | 2i | -i | -i | complex lifted from C3⋊C8 |
| ρ21 | 2 | -2 | -1 | -1 | 2 | -1 | 2i | -2i | 1 | -2 | 1 | 1 | 0 | 0 | 0 | 0 | -i | -i | -i | 2i | i | i | -2i | i | complex lifted from C3⋊C8 |
| ρ22 | 2 | -2 | 2 | -1 | -1 | -1 | -2i | 2i | 1 | 1 | 1 | -2 | 0 | 0 | 0 | 0 | i | -2i | i | i | 2i | -i | -i | -i | complex lifted from C3⋊C8 |
| ρ23 | 2 | -2 | -1 | -1 | -1 | 2 | -2i | 2i | 1 | 1 | -2 | 1 | 0 | 0 | 0 | 0 | -2i | i | i | i | -i | -i | -i | 2i | complex lifted from C3⋊C8 |
| ρ24 | 2 | -2 | -1 | -1 | -1 | 2 | 2i | -2i | 1 | 1 | -2 | 1 | 0 | 0 | 0 | 0 | 2i | -i | -i | -i | i | i | i | -2i | complex lifted from C3⋊C8 |
►Regular action on 72 points
Generators in S72
(1 68 11)(2 12 69)(3 70 13)(4 14 71)(5 72 15)(6 16 65)(7 66 9)(8 10 67)(17 30 43)(18 44 31)(19 32 45)(20 46 25)(21 26 47)(22 48 27)(23 28 41)(24 42 29)(33 56 62)(34 63 49)(35 50 64)(36 57 51)(37 52 58)(38 59 53)(39 54 60)(40 61 55)
(1 34 31)(2 32 35)(3 36 25)(4 26 37)(5 38 27)(6 28 39)(7 40 29)(8 30 33)(9 55 42)(10 43 56)(11 49 44)(12 45 50)(13 51 46)(14 47 52)(15 53 48)(16 41 54)(17 62 67)(18 68 63)(19 64 69)(20 70 57)(21 58 71)(22 72 59)(23 60 65)(24 66 61)
(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)
G:=sub<Sym(72)| (1,68,11)(2,12,69)(3,70,13)(4,14,71)(5,72,15)(6,16,65)(7,66,9)(8,10,67)(17,30,43)(18,44,31)(19,32,45)(20,46,25)(21,26,47)(22,48,27)(23,28,41)(24,42,29)(33,56,62)(34,63,49)(35,50,64)(36,57,51)(37,52,58)(38,59,53)(39,54,60)(40,61,55), (1,34,31)(2,32,35)(3,36,25)(4,26,37)(5,38,27)(6,28,39)(7,40,29)(8,30,33)(9,55,42)(10,43,56)(11,49,44)(12,45,50)(13,51,46)(14,47,52)(15,53,48)(16,41,54)(17,62,67)(18,68,63)(19,64,69)(20,70,57)(21,58,71)(22,72,59)(23,60,65)(24,66,61), (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)>;
G:=Group( (1,68,11)(2,12,69)(3,70,13)(4,14,71)(5,72,15)(6,16,65)(7,66,9)(8,10,67)(17,30,43)(18,44,31)(19,32,45)(20,46,25)(21,26,47)(22,48,27)(23,28,41)(24,42,29)(33,56,62)(34,63,49)(35,50,64)(36,57,51)(37,52,58)(38,59,53)(39,54,60)(40,61,55), (1,34,31)(2,32,35)(3,36,25)(4,26,37)(5,38,27)(6,28,39)(7,40,29)(8,30,33)(9,55,42)(10,43,56)(11,49,44)(12,45,50)(13,51,46)(14,47,52)(15,53,48)(16,41,54)(17,62,67)(18,68,63)(19,64,69)(20,70,57)(21,58,71)(22,72,59)(23,60,65)(24,66,61), (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) );
G=PermutationGroup([[(1,68,11),(2,12,69),(3,70,13),(4,14,71),(5,72,15),(6,16,65),(7,66,9),(8,10,67),(17,30,43),(18,44,31),(19,32,45),(20,46,25),(21,26,47),(22,48,27),(23,28,41),(24,42,29),(33,56,62),(34,63,49),(35,50,64),(36,57,51),(37,52,58),(38,59,53),(39,54,60),(40,61,55)], [(1,34,31),(2,32,35),(3,36,25),(4,26,37),(5,38,27),(6,28,39),(7,40,29),(8,30,33),(9,55,42),(10,43,56),(11,49,44),(12,45,50),(13,51,46),(14,47,52),(15,53,48),(16,41,54),(17,62,67),(18,68,63),(19,64,69),(20,70,57),(21,58,71),(22,72,59),(23,60,65),(24,66,61)], [(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)]])
C32⋊4C8 is a maximal subgroup of
C32⋊2C16 S3×C3⋊C8 D6.Dic3 C32⋊2D8 Dic6⋊S3 C32⋊2Q16 C8×C3⋊S3 C24⋊S3 C12.58D6 C32⋊7D8 C32⋊9SD16 C32⋊11SD16 C32⋊7Q16 He3⋊3C8 C36.S3 C33⋊7C8 C12.12S4 C3⋊U2(𝔽3) C60.S3 C30.Dic3
C32⋊4C8 is a maximal quotient of
C24.S3 C36.S3 He3⋊4C8 C33⋊7C8 C12.12S4 C60.S3 C30.Dic3
Matrix representation of C32⋊4C8 ►in GL5(𝔽73)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 72 | 72 | 0 | 0 |
| 0 | 0 | 0 | 1 | 3 |
| 0 | 0 | 0 | 72 | 71 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 72 | 72 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 63 | 0 | 0 | 0 | 0 |
| 0 | 60 | 62 | 0 | 0 |
| 0 | 2 | 13 | 0 | 0 |
| 0 | 0 | 0 | 49 | 34 |
| 0 | 0 | 0 | 11 | 24 |
G:=sub<GL(5,GF(73))| [1,0,0,0,0,0,0,72,0,0,0,1,72,0,0,0,0,0,1,72,0,0,0,3,71],[1,0,0,0,0,0,0,72,0,0,0,1,72,0,0,0,0,0,1,0,0,0,0,0,1],[63,0,0,0,0,0,60,2,0,0,0,62,13,0,0,0,0,0,49,11,0,0,0,34,24] >;
C32⋊4C8 in GAP, Magma, Sage, TeX
C_3^2\rtimes_4C_8
% in TeX
G:=Group("C3^2:4C8"); // GroupNames label
G:=SmallGroup(72,13);
// by ID
G=gap.SmallGroup(72,13);
# by ID
G:=PCGroup([5,-2,-2,-2,-3,-3,10,26,323,1204]);
// Polycyclic
G:=Group<a,b,c|a^3=b^3=c^8=1,a*b=b*a,c*a*c^-1=a^-1,c*b*c^-1=b^-1>;
// generators/relations
Export
Subgroup lattice of C32⋊4C8 in TeX
Character table of C32⋊4C8 in TeX