Aliases: C12.9S4, Q8.Dic9, C3.U2(𝔽3), Q8⋊C9⋊2C4, C4○D4.1D9, C6.3(A4⋊C4), C4.5(C3.S4), (C3×Q8).2Dic3, Q8.C18.2C2, C2.3(C6.S4), (C3×C4○D4).1S3, SmallGroup(288,70)
Series: Derived ►Chief ►Lower central ►Upper central
| Q8⋊C9 — C12.9S4 |
Generators and relations for C12.9S4
G = < a,b,c,d,e | a12=1, b2=c2=a6, d3=a4, e2=a9, ab=ba, ac=ca, ad=da, eae-1=a5, cbc-1=a6b, dbd-1=a6bc, ebe-1=bc, dcd-1=b, ece-1=a6c, ede-1=a8d2 >
Character table of C12.9S4
| class | 1 | 2A | 2B | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 6A | 6B | 8A | 8B | 9A | 9B | 9C | 12A | 12B | 12C | 18A | 18B | 18C | 36A | 36B | 36C | 36D | 36E | 36F | |
| size | 1 | 1 | 6 | 2 | 1 | 1 | 6 | 18 | 18 | 18 | 18 | 2 | 12 | 36 | 36 | 8 | 8 | 8 | 2 | 2 | 12 | 8 | 8 | 8 | 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 | 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 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 2 |
| ρ3 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | i | -i | i | -i | 1 | -1 | i | -i | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | linear of order 4 |
| ρ4 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | -i | i | -i | i | 1 | -1 | -i | i | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | linear of order 4 |
| ρ5 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 2 | 2 | 0 | 0 | -1 | -1 | -1 | 2 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ6 | 2 | 2 | 2 | -1 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -1 | -1 | 0 | 0 | ζ95+ζ94 | ζ97+ζ92 | ζ98+ζ9 | -1 | -1 | -1 | ζ95+ζ94 | ζ97+ζ92 | ζ98+ζ9 | ζ98+ζ9 | ζ98+ζ9 | ζ97+ζ92 | ζ97+ζ92 | ζ95+ζ94 | ζ95+ζ94 | orthogonal lifted from D9 |
| ρ7 | 2 | 2 | 2 | -1 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -1 | -1 | 0 | 0 | ζ97+ζ92 | ζ98+ζ9 | ζ95+ζ94 | -1 | -1 | -1 | ζ97+ζ92 | ζ98+ζ9 | ζ95+ζ94 | ζ95+ζ94 | ζ95+ζ94 | ζ98+ζ9 | ζ98+ζ9 | ζ97+ζ92 | ζ97+ζ92 | orthogonal lifted from D9 |
| ρ8 | 2 | 2 | 2 | -1 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -1 | -1 | 0 | 0 | ζ98+ζ9 | ζ95+ζ94 | ζ97+ζ92 | -1 | -1 | -1 | ζ98+ζ9 | ζ95+ζ94 | ζ97+ζ92 | ζ97+ζ92 | ζ97+ζ92 | ζ95+ζ94 | ζ95+ζ94 | ζ98+ζ9 | ζ98+ζ9 | orthogonal lifted from D9 |
| ρ9 | 2 | 2 | -2 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | -1 | -1 | -1 | -2 | -2 | 2 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | symplectic lifted from Dic3, Schur index 2 |
| ρ10 | 2 | 2 | -2 | -1 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | -1 | 1 | 0 | 0 | ζ95+ζ94 | ζ97+ζ92 | ζ98+ζ9 | 1 | 1 | -1 | ζ95+ζ94 | ζ97+ζ92 | ζ98+ζ9 | -ζ98-ζ9 | -ζ98-ζ9 | -ζ97-ζ92 | -ζ97-ζ92 | -ζ95-ζ94 | -ζ95-ζ94 | symplectic lifted from Dic9, Schur index 2 |
| ρ11 | 2 | 2 | -2 | -1 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | -1 | 1 | 0 | 0 | ζ98+ζ9 | ζ95+ζ94 | ζ97+ζ92 | 1 | 1 | -1 | ζ98+ζ9 | ζ95+ζ94 | ζ97+ζ92 | -ζ97-ζ92 | -ζ97-ζ92 | -ζ95-ζ94 | -ζ95-ζ94 | -ζ98-ζ9 | -ζ98-ζ9 | symplectic lifted from Dic9, Schur index 2 |
| ρ12 | 2 | 2 | -2 | -1 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | -1 | 1 | 0 | 0 | ζ97+ζ92 | ζ98+ζ9 | ζ95+ζ94 | 1 | 1 | -1 | ζ97+ζ92 | ζ98+ζ9 | ζ95+ζ94 | -ζ95-ζ94 | -ζ95-ζ94 | -ζ98-ζ9 | -ζ98-ζ9 | -ζ97-ζ92 | -ζ97-ζ92 | symplectic lifted from Dic9, Schur index 2 |
| ρ13 | 2 | -2 | 0 | 2 | 2i | -2i | 0 | -1-i | -1+i | 1+i | 1-i | -2 | 0 | 0 | 0 | -1 | -1 | -1 | 2i | -2i | 0 | 1 | 1 | 1 | -i | i | -i | i | -i | i | complex lifted from U2(𝔽3) |
| ρ14 | 2 | -2 | 0 | 2 | -2i | 2i | 0 | 1-i | 1+i | -1+i | -1-i | -2 | 0 | 0 | 0 | -1 | -1 | -1 | -2i | 2i | 0 | 1 | 1 | 1 | i | -i | i | -i | i | -i | complex lifted from U2(𝔽3) |
| ρ15 | 2 | -2 | 0 | 2 | 2i | -2i | 0 | 1+i | 1-i | -1-i | -1+i | -2 | 0 | 0 | 0 | -1 | -1 | -1 | 2i | -2i | 0 | 1 | 1 | 1 | -i | i | -i | i | -i | i | complex lifted from U2(𝔽3) |
| ρ16 | 2 | -2 | 0 | 2 | -2i | 2i | 0 | -1+i | -1-i | 1-i | 1+i | -2 | 0 | 0 | 0 | -1 | -1 | -1 | -2i | 2i | 0 | 1 | 1 | 1 | i | -i | i | -i | i | -i | complex lifted from U2(𝔽3) |
| ρ17 | 3 | 3 | -1 | 3 | 3 | 3 | -1 | -1 | -1 | -1 | -1 | 3 | -1 | 1 | 1 | 0 | 0 | 0 | 3 | 3 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S4 |
| ρ18 | 3 | 3 | -1 | 3 | 3 | 3 | -1 | 1 | 1 | 1 | 1 | 3 | -1 | -1 | -1 | 0 | 0 | 0 | 3 | 3 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S4 |
| ρ19 | 3 | 3 | 1 | 3 | -3 | -3 | -1 | i | -i | i | -i | 3 | 1 | -i | i | 0 | 0 | 0 | -3 | -3 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from A4⋊C4 |
| ρ20 | 3 | 3 | 1 | 3 | -3 | -3 | -1 | -i | i | -i | i | 3 | 1 | i | -i | 0 | 0 | 0 | -3 | -3 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from A4⋊C4 |
| ρ21 | 4 | -4 | 0 | 4 | 4i | -4i | 0 | 0 | 0 | 0 | 0 | -4 | 0 | 0 | 0 | 1 | 1 | 1 | 4i | -4i | 0 | -1 | -1 | -1 | i | -i | i | -i | i | -i | complex lifted from U2(𝔽3) |
| ρ22 | 4 | -4 | 0 | 4 | -4i | 4i | 0 | 0 | 0 | 0 | 0 | -4 | 0 | 0 | 0 | 1 | 1 | 1 | -4i | 4i | 0 | -1 | -1 | -1 | -i | i | -i | i | -i | i | complex lifted from U2(𝔽3) |
| ρ23 | 4 | -4 | 0 | -2 | -4i | 4i | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | -ζ97-ζ92 | -ζ98-ζ9 | -ζ95-ζ94 | 2i | -2i | 0 | ζ97+ζ92 | ζ98+ζ9 | ζ95+ζ94 | ζ4ζ95+ζ4ζ94 | ζ43ζ95+ζ43ζ94 | ζ4ζ98+ζ4ζ9 | ζ43ζ98+ζ43ζ9 | ζ4ζ97+ζ4ζ92 | ζ43ζ97+ζ43ζ92 | complex faithful |
| ρ24 | 4 | -4 | 0 | -2 | 4i | -4i | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | -ζ95-ζ94 | -ζ97-ζ92 | -ζ98-ζ9 | -2i | 2i | 0 | ζ95+ζ94 | ζ97+ζ92 | ζ98+ζ9 | ζ43ζ98+ζ43ζ9 | ζ4ζ98+ζ4ζ9 | ζ43ζ97+ζ43ζ92 | ζ4ζ97+ζ4ζ92 | ζ43ζ95+ζ43ζ94 | ζ4ζ95+ζ4ζ94 | complex faithful |
| ρ25 | 4 | -4 | 0 | -2 | -4i | 4i | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | -ζ95-ζ94 | -ζ97-ζ92 | -ζ98-ζ9 | 2i | -2i | 0 | ζ95+ζ94 | ζ97+ζ92 | ζ98+ζ9 | ζ4ζ98+ζ4ζ9 | ζ43ζ98+ζ43ζ9 | ζ4ζ97+ζ4ζ92 | ζ43ζ97+ζ43ζ92 | ζ4ζ95+ζ4ζ94 | ζ43ζ95+ζ43ζ94 | complex faithful |
| ρ26 | 4 | -4 | 0 | -2 | -4i | 4i | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | -ζ98-ζ9 | -ζ95-ζ94 | -ζ97-ζ92 | 2i | -2i | 0 | ζ98+ζ9 | ζ95+ζ94 | ζ97+ζ92 | ζ4ζ97+ζ4ζ92 | ζ43ζ97+ζ43ζ92 | ζ4ζ95+ζ4ζ94 | ζ43ζ95+ζ43ζ94 | ζ4ζ98+ζ4ζ9 | ζ43ζ98+ζ43ζ9 | complex faithful |
| ρ27 | 4 | -4 | 0 | -2 | 4i | -4i | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | -ζ97-ζ92 | -ζ98-ζ9 | -ζ95-ζ94 | -2i | 2i | 0 | ζ97+ζ92 | ζ98+ζ9 | ζ95+ζ94 | ζ43ζ95+ζ43ζ94 | ζ4ζ95+ζ4ζ94 | ζ43ζ98+ζ43ζ9 | ζ4ζ98+ζ4ζ9 | ζ43ζ97+ζ43ζ92 | ζ4ζ97+ζ4ζ92 | complex faithful |
| ρ28 | 4 | -4 | 0 | -2 | 4i | -4i | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | -ζ98-ζ9 | -ζ95-ζ94 | -ζ97-ζ92 | -2i | 2i | 0 | ζ98+ζ9 | ζ95+ζ94 | ζ97+ζ92 | ζ43ζ97+ζ43ζ92 | ζ4ζ97+ζ4ζ92 | ζ43ζ95+ζ43ζ94 | ζ4ζ95+ζ4ζ94 | ζ43ζ98+ζ43ζ9 | ζ4ζ98+ζ4ζ9 | complex faithful |
| ρ29 | 6 | 6 | -2 | -3 | 6 | 6 | -2 | 0 | 0 | 0 | 0 | -3 | 1 | 0 | 0 | 0 | 0 | 0 | -3 | -3 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C3.S4 |
| ρ30 | 6 | 6 | 2 | -3 | -6 | -6 | -2 | 0 | 0 | 0 | 0 | -3 | -1 | 0 | 0 | 0 | 0 | 0 | 3 | 3 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from C6.S4, Schur index 2 |
►On 72 points
Generators in S72
(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)
(1 19 7 13)(2 20 8 14)(3 21 9 15)(4 22 10 16)(5 23 11 17)(6 24 12 18)(25 28 31 34)(26 29 32 35)(27 30 33 36)(37 72 43 66)(38 61 44 67)(39 62 45 68)(40 63 46 69)(41 64 47 70)(42 65 48 71)(49 58 55 52)(50 59 56 53)(51 60 57 54)
(1 4 7 10)(2 5 8 11)(3 6 9 12)(13 22 19 16)(14 23 20 17)(15 24 21 18)(25 50 31 56)(26 51 32 57)(27 52 33 58)(28 53 34 59)(29 54 35 60)(30 55 36 49)(37 63 43 69)(38 64 44 70)(39 65 45 71)(40 66 46 72)(41 67 47 61)(42 68 48 62)
(1 67 33 5 71 25 9 63 29)(2 68 34 6 72 26 10 64 30)(3 69 35 7 61 27 11 65 31)(4 70 36 8 62 28 12 66 32)(13 41 49 17 45 53 21 37 57)(14 42 50 18 46 54 22 38 58)(15 43 51 19 47 55 23 39 59)(16 44 52 20 48 56 24 40 60)
(1 19 10 16 7 13 4 22)(2 24 11 21 8 18 5 15)(3 17 12 14 9 23 6 20)(25 39 34 48 31 45 28 42)(26 44 35 41 32 38 29 47)(27 37 36 46 33 43 30 40)(49 66 58 63 55 72 52 69)(50 71 59 68 56 65 53 62)(51 64 60 61 57 70 54 67)
G:=sub<Sym(72)| (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), (1,19,7,13)(2,20,8,14)(3,21,9,15)(4,22,10,16)(5,23,11,17)(6,24,12,18)(25,28,31,34)(26,29,32,35)(27,30,33,36)(37,72,43,66)(38,61,44,67)(39,62,45,68)(40,63,46,69)(41,64,47,70)(42,65,48,71)(49,58,55,52)(50,59,56,53)(51,60,57,54), (1,4,7,10)(2,5,8,11)(3,6,9,12)(13,22,19,16)(14,23,20,17)(15,24,21,18)(25,50,31,56)(26,51,32,57)(27,52,33,58)(28,53,34,59)(29,54,35,60)(30,55,36,49)(37,63,43,69)(38,64,44,70)(39,65,45,71)(40,66,46,72)(41,67,47,61)(42,68,48,62), (1,67,33,5,71,25,9,63,29)(2,68,34,6,72,26,10,64,30)(3,69,35,7,61,27,11,65,31)(4,70,36,8,62,28,12,66,32)(13,41,49,17,45,53,21,37,57)(14,42,50,18,46,54,22,38,58)(15,43,51,19,47,55,23,39,59)(16,44,52,20,48,56,24,40,60), (1,19,10,16,7,13,4,22)(2,24,11,21,8,18,5,15)(3,17,12,14,9,23,6,20)(25,39,34,48,31,45,28,42)(26,44,35,41,32,38,29,47)(27,37,36,46,33,43,30,40)(49,66,58,63,55,72,52,69)(50,71,59,68,56,65,53,62)(51,64,60,61,57,70,54,67)>;
G:=Group( (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), (1,19,7,13)(2,20,8,14)(3,21,9,15)(4,22,10,16)(5,23,11,17)(6,24,12,18)(25,28,31,34)(26,29,32,35)(27,30,33,36)(37,72,43,66)(38,61,44,67)(39,62,45,68)(40,63,46,69)(41,64,47,70)(42,65,48,71)(49,58,55,52)(50,59,56,53)(51,60,57,54), (1,4,7,10)(2,5,8,11)(3,6,9,12)(13,22,19,16)(14,23,20,17)(15,24,21,18)(25,50,31,56)(26,51,32,57)(27,52,33,58)(28,53,34,59)(29,54,35,60)(30,55,36,49)(37,63,43,69)(38,64,44,70)(39,65,45,71)(40,66,46,72)(41,67,47,61)(42,68,48,62), (1,67,33,5,71,25,9,63,29)(2,68,34,6,72,26,10,64,30)(3,69,35,7,61,27,11,65,31)(4,70,36,8,62,28,12,66,32)(13,41,49,17,45,53,21,37,57)(14,42,50,18,46,54,22,38,58)(15,43,51,19,47,55,23,39,59)(16,44,52,20,48,56,24,40,60), (1,19,10,16,7,13,4,22)(2,24,11,21,8,18,5,15)(3,17,12,14,9,23,6,20)(25,39,34,48,31,45,28,42)(26,44,35,41,32,38,29,47)(27,37,36,46,33,43,30,40)(49,66,58,63,55,72,52,69)(50,71,59,68,56,65,53,62)(51,64,60,61,57,70,54,67) );
G=PermutationGroup([[(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)], [(1,19,7,13),(2,20,8,14),(3,21,9,15),(4,22,10,16),(5,23,11,17),(6,24,12,18),(25,28,31,34),(26,29,32,35),(27,30,33,36),(37,72,43,66),(38,61,44,67),(39,62,45,68),(40,63,46,69),(41,64,47,70),(42,65,48,71),(49,58,55,52),(50,59,56,53),(51,60,57,54)], [(1,4,7,10),(2,5,8,11),(3,6,9,12),(13,22,19,16),(14,23,20,17),(15,24,21,18),(25,50,31,56),(26,51,32,57),(27,52,33,58),(28,53,34,59),(29,54,35,60),(30,55,36,49),(37,63,43,69),(38,64,44,70),(39,65,45,71),(40,66,46,72),(41,67,47,61),(42,68,48,62)], [(1,67,33,5,71,25,9,63,29),(2,68,34,6,72,26,10,64,30),(3,69,35,7,61,27,11,65,31),(4,70,36,8,62,28,12,66,32),(13,41,49,17,45,53,21,37,57),(14,42,50,18,46,54,22,38,58),(15,43,51,19,47,55,23,39,59),(16,44,52,20,48,56,24,40,60)], [(1,19,10,16,7,13,4,22),(2,24,11,21,8,18,5,15),(3,17,12,14,9,23,6,20),(25,39,34,48,31,45,28,42),(26,44,35,41,32,38,29,47),(27,37,36,46,33,43,30,40),(49,66,58,63,55,72,52,69),(50,71,59,68,56,65,53,62),(51,64,60,61,57,70,54,67)]])
Matrix representation of C12.9S4 ►in GL4(𝔽73) generated by
| 27 | 0 | 0 | 0 |
| 0 | 27 | 0 | 0 |
| 0 | 0 | 1 | 2 |
| 0 | 0 | 35 | 71 |
| 0 | 27 | 0 | 0 |
| 27 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 27 | 0 | 0 | 0 |
| 0 | 46 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 59 | 59 | 0 | 0 |
| 60 | 13 | 0 | 0 |
| 0 | 0 | 34 | 6 |
| 0 | 0 | 32 | 25 |
| 0 | 27 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 35 | 72 |
G:=sub<GL(4,GF(73))| [27,0,0,0,0,27,0,0,0,0,1,35,0,0,2,71],[0,27,0,0,27,0,0,0,0,0,1,0,0,0,0,1],[27,0,0,0,0,46,0,0,0,0,1,0,0,0,0,1],[59,60,0,0,59,13,0,0,0,0,34,32,0,0,6,25],[0,1,0,0,27,0,0,0,0,0,1,35,0,0,0,72] >;
C12.9S4 in GAP, Magma, Sage, TeX
C_{12}._9S_4 % in TeX
G:=Group("C12.9S4"); // GroupNames label
G:=SmallGroup(288,70);
// by ID
G=gap.SmallGroup(288,70);
# by ID
G:=PCGroup([7,-2,-2,-3,-3,-2,2,-2,14,1016,422,142,675,2524,1908,172,1517,1153,285,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^12=1,b^2=c^2=a^6,d^3=a^4,e^2=a^9,a*b=b*a,a*c=c*a,a*d=d*a,e*a*e^-1=a^5,c*b*c^-1=a^6*b,d*b*d^-1=a^6*b*c,e*b*e^-1=b*c,d*c*d^-1=b,e*c*e^-1=a^6*c,e*d*e^-1=a^8*d^2>;
// generators/relations
Export
Subgroup lattice of C12.9S4 in TeX
Character table of C12.9S4 in TeX