Aliases: C12.7S4, SL2(𝔽3)⋊4D6, C4.A4⋊1S3, C4.3(C3⋊S4), C6.36(C2×S4), C6.6S4⋊6C2, (C3×Q8).18D6, C3⋊2(C4.3S4), (C3×SL2(𝔽3))⋊4C22, C2.10(C2×C3⋊S4), (C3×C4.A4)⋊2C2, (C3×C4○D4)⋊2S3, Q8.5(C2×C3⋊S3), C4○D4⋊2(C3⋊S3), SmallGroup(288,915)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C2 — Q8 — C3×SL2(𝔽3) — C12.7S4 |
| C3×SL2(𝔽3) — C12.7S4 |
Generators and relations for C12.7S4
G = < a,b,c,d,e | a12=d3=e2=1, b2=c2=a6, ab=ba, ac=ca, ad=da, eae=a-1, cbc-1=a6b, dbd-1=a6bc, ebe=bc, dcd-1=b, ece=a6c, ede=d-1 >
Subgroups: 832 in 110 conjugacy classes, 21 normal (13 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C8, C2×C4, D4, Q8, C23, C32, C12, C12, D6, C2×C6, M4(2), D8, SD16, C2×D4, C4○D4, C3⋊S3, C3×C6, C3⋊C8, SL2(𝔽3), D12, C2×C12, C3×D4, C3×Q8, C22×S3, C8⋊C22, C3×C12, C2×C3⋊S3, C4.Dic3, D4⋊S3, Q8⋊2S3, GL2(𝔽3), C4.A4, C2×D12, C3×C4○D4, C3×SL2(𝔽3), C12⋊S3, D4⋊D6, C4.3S4, C6.6S4, C3×C4.A4, C12.7S4
Quotients: C1, C2, C22, S3, D6, C3⋊S3, S4, C2×C3⋊S3, C2×S4, C3⋊S4, C4.3S4, C2×C3⋊S4, C12.7S4
Character table of C12.7S4
| class | 1 | 2A | 2B | 2C | 2D | 3A | 3B | 3C | 3D | 4A | 4B | 6A | 6B | 6C | 6D | 6E | 8A | 8B | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 12I | |
| size | 1 | 1 | 6 | 36 | 36 | 2 | 8 | 8 | 8 | 2 | 6 | 2 | 8 | 8 | 8 | 12 | 36 | 36 | 2 | 2 | 8 | 8 | 8 | 8 | 8 | 8 | 12 | |
| 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 | 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 | 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 | 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 | -1 | -1 | 1 | linear of order 2 |
| ρ5 | 2 | 2 | -2 | 0 | 0 | -1 | -1 | -1 | 2 | -2 | 2 | -1 | 2 | -1 | -1 | 1 | 0 | 0 | 1 | 1 | 1 | -2 | -2 | 1 | 1 | 1 | -1 | orthogonal lifted from D6 |
| ρ6 | 2 | 2 | -2 | 0 | 0 | -1 | 2 | -1 | -1 | -2 | 2 | -1 | -1 | 2 | -1 | 1 | 0 | 0 | 1 | 1 | -2 | 1 | 1 | 1 | 1 | -2 | -1 | orthogonal lifted from D6 |
| ρ7 | 2 | 2 | -2 | 0 | 0 | -1 | -1 | 2 | -1 | -2 | 2 | -1 | -1 | -1 | 2 | 1 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | -2 | -2 | 1 | -1 | orthogonal lifted from D6 |
| ρ8 | 2 | 2 | 2 | 0 | 0 | -1 | -1 | -1 | 2 | 2 | 2 | -1 | 2 | -1 | -1 | -1 | 0 | 0 | -1 | -1 | -1 | 2 | 2 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ9 | 2 | 2 | -2 | 0 | 0 | 2 | -1 | -1 | -1 | -2 | 2 | 2 | -1 | -1 | -1 | -2 | 0 | 0 | -2 | -2 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | orthogonal lifted from D6 |
| ρ10 | 2 | 2 | 2 | 0 | 0 | -1 | -1 | 2 | -1 | 2 | 2 | -1 | -1 | -1 | 2 | -1 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | 2 | 2 | -1 | -1 | orthogonal lifted from S3 |
| ρ11 | 2 | 2 | 2 | 0 | 0 | -1 | 2 | -1 | -1 | 2 | 2 | -1 | -1 | 2 | -1 | -1 | 0 | 0 | -1 | -1 | 2 | -1 | -1 | -1 | -1 | 2 | -1 | orthogonal lifted from S3 |
| ρ12 | 2 | 2 | 2 | 0 | 0 | 2 | -1 | -1 | -1 | 2 | 2 | 2 | -1 | -1 | -1 | 2 | 0 | 0 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | -1 | 2 | orthogonal lifted from S3 |
| ρ13 | 3 | 3 | -1 | -1 | -1 | 3 | 0 | 0 | 0 | 3 | -1 | 3 | 0 | 0 | 0 | -1 | 1 | 1 | 3 | 3 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | orthogonal lifted from S4 |
| ρ14 | 3 | 3 | 1 | 1 | -1 | 3 | 0 | 0 | 0 | -3 | -1 | 3 | 0 | 0 | 0 | 1 | -1 | 1 | -3 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | orthogonal lifted from C2×S4 |
| ρ15 | 3 | 3 | -1 | 1 | 1 | 3 | 0 | 0 | 0 | 3 | -1 | 3 | 0 | 0 | 0 | -1 | -1 | -1 | 3 | 3 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | orthogonal lifted from S4 |
| ρ16 | 3 | 3 | 1 | -1 | 1 | 3 | 0 | 0 | 0 | -3 | -1 | 3 | 0 | 0 | 0 | 1 | 1 | -1 | -3 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | orthogonal lifted from C2×S4 |
| ρ17 | 4 | -4 | 0 | 0 | 0 | 4 | -2 | -2 | -2 | 0 | 0 | -4 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C4.3S4 |
| ρ18 | 4 | -4 | 0 | 0 | 0 | -2 | -2 | 1 | 1 | 0 | 0 | 2 | -1 | 2 | -1 | 0 | 0 | 0 | 2√3 | -2√3 | 0 | -√3 | √3 | √3 | -√3 | 0 | 0 | orthogonal faithful |
| ρ19 | 4 | -4 | 0 | 0 | 0 | 4 | 1 | 1 | 1 | 0 | 0 | -4 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | -√3 | √3 | -√3 | √3 | -√3 | √3 | 0 | orthogonal lifted from C4.3S4 |
| ρ20 | 4 | -4 | 0 | 0 | 0 | 4 | 1 | 1 | 1 | 0 | 0 | -4 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | √3 | -√3 | √3 | -√3 | √3 | -√3 | 0 | orthogonal lifted from C4.3S4 |
| ρ21 | 4 | -4 | 0 | 0 | 0 | -2 | 1 | 1 | -2 | 0 | 0 | 2 | 2 | -1 | -1 | 0 | 0 | 0 | -2√3 | 2√3 | √3 | 0 | 0 | √3 | -√3 | -√3 | 0 | orthogonal faithful |
| ρ22 | 4 | -4 | 0 | 0 | 0 | -2 | -2 | 1 | 1 | 0 | 0 | 2 | -1 | 2 | -1 | 0 | 0 | 0 | -2√3 | 2√3 | 0 | √3 | -√3 | -√3 | √3 | 0 | 0 | orthogonal faithful |
| ρ23 | 4 | -4 | 0 | 0 | 0 | -2 | 1 | -2 | 1 | 0 | 0 | 2 | -1 | -1 | 2 | 0 | 0 | 0 | -2√3 | 2√3 | -√3 | -√3 | √3 | 0 | 0 | √3 | 0 | orthogonal faithful |
| ρ24 | 4 | -4 | 0 | 0 | 0 | -2 | 1 | 1 | -2 | 0 | 0 | 2 | 2 | -1 | -1 | 0 | 0 | 0 | 2√3 | -2√3 | -√3 | 0 | 0 | -√3 | √3 | √3 | 0 | orthogonal faithful |
| ρ25 | 4 | -4 | 0 | 0 | 0 | -2 | 1 | -2 | 1 | 0 | 0 | 2 | -1 | -1 | 2 | 0 | 0 | 0 | 2√3 | -2√3 | √3 | √3 | -√3 | 0 | 0 | -√3 | 0 | orthogonal faithful |
| ρ26 | 6 | 6 | 2 | 0 | 0 | -3 | 0 | 0 | 0 | -6 | -2 | -3 | 0 | 0 | 0 | -1 | 0 | 0 | 3 | 3 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | orthogonal lifted from C2×C3⋊S4 |
| ρ27 | 6 | 6 | -2 | 0 | 0 | -3 | 0 | 0 | 0 | 6 | -2 | -3 | 0 | 0 | 0 | 1 | 0 | 0 | -3 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | orthogonal lifted from C3⋊S4 |
►On 48 points
Generators in S48
(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 45 7 39)(2 46 8 40)(3 47 9 41)(4 48 10 42)(5 37 11 43)(6 38 12 44)(13 36 19 30)(14 25 20 31)(15 26 21 32)(16 27 22 33)(17 28 23 34)(18 29 24 35)
(1 23 7 17)(2 24 8 18)(3 13 9 19)(4 14 10 20)(5 15 11 21)(6 16 12 22)(25 48 31 42)(26 37 32 43)(27 38 33 44)(28 39 34 45)(29 40 35 46)(30 41 36 47)
(13 36 47)(14 25 48)(15 26 37)(16 27 38)(17 28 39)(18 29 40)(19 30 41)(20 31 42)(21 32 43)(22 33 44)(23 34 45)(24 35 46)
(2 12)(3 11)(4 10)(5 9)(6 8)(13 15)(16 24)(17 23)(18 22)(19 21)(25 48)(26 47)(27 46)(28 45)(29 44)(30 43)(31 42)(32 41)(33 40)(34 39)(35 38)(36 37)
G:=sub<Sym(48)| (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), (1,45,7,39)(2,46,8,40)(3,47,9,41)(4,48,10,42)(5,37,11,43)(6,38,12,44)(13,36,19,30)(14,25,20,31)(15,26,21,32)(16,27,22,33)(17,28,23,34)(18,29,24,35), (1,23,7,17)(2,24,8,18)(3,13,9,19)(4,14,10,20)(5,15,11,21)(6,16,12,22)(25,48,31,42)(26,37,32,43)(27,38,33,44)(28,39,34,45)(29,40,35,46)(30,41,36,47), (13,36,47)(14,25,48)(15,26,37)(16,27,38)(17,28,39)(18,29,40)(19,30,41)(20,31,42)(21,32,43)(22,33,44)(23,34,45)(24,35,46), (2,12)(3,11)(4,10)(5,9)(6,8)(13,15)(16,24)(17,23)(18,22)(19,21)(25,48)(26,47)(27,46)(28,45)(29,44)(30,43)(31,42)(32,41)(33,40)(34,39)(35,38)(36,37)>;
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), (1,45,7,39)(2,46,8,40)(3,47,9,41)(4,48,10,42)(5,37,11,43)(6,38,12,44)(13,36,19,30)(14,25,20,31)(15,26,21,32)(16,27,22,33)(17,28,23,34)(18,29,24,35), (1,23,7,17)(2,24,8,18)(3,13,9,19)(4,14,10,20)(5,15,11,21)(6,16,12,22)(25,48,31,42)(26,37,32,43)(27,38,33,44)(28,39,34,45)(29,40,35,46)(30,41,36,47), (13,36,47)(14,25,48)(15,26,37)(16,27,38)(17,28,39)(18,29,40)(19,30,41)(20,31,42)(21,32,43)(22,33,44)(23,34,45)(24,35,46), (2,12)(3,11)(4,10)(5,9)(6,8)(13,15)(16,24)(17,23)(18,22)(19,21)(25,48)(26,47)(27,46)(28,45)(29,44)(30,43)(31,42)(32,41)(33,40)(34,39)(35,38)(36,37) );
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)], [(1,45,7,39),(2,46,8,40),(3,47,9,41),(4,48,10,42),(5,37,11,43),(6,38,12,44),(13,36,19,30),(14,25,20,31),(15,26,21,32),(16,27,22,33),(17,28,23,34),(18,29,24,35)], [(1,23,7,17),(2,24,8,18),(3,13,9,19),(4,14,10,20),(5,15,11,21),(6,16,12,22),(25,48,31,42),(26,37,32,43),(27,38,33,44),(28,39,34,45),(29,40,35,46),(30,41,36,47)], [(13,36,47),(14,25,48),(15,26,37),(16,27,38),(17,28,39),(18,29,40),(19,30,41),(20,31,42),(21,32,43),(22,33,44),(23,34,45),(24,35,46)], [(2,12),(3,11),(4,10),(5,9),(6,8),(13,15),(16,24),(17,23),(18,22),(19,21),(25,48),(26,47),(27,46),(28,45),(29,44),(30,43),(31,42),(32,41),(33,40),(34,39),(35,38),(36,37)]])
Matrix representation of C12.7S4 ►in GL4(𝔽73) generated by
| 59 | 0 | 66 | 66 |
| 66 | 59 | 0 | 66 |
| 0 | 66 | 59 | 66 |
| 7 | 7 | 7 | 0 |
| 0 | 0 | 1 | 0 |
| 72 | 72 | 72 | 71 |
| 72 | 0 | 0 | 0 |
| 1 | 1 | 0 | 1 |
| 1 | 1 | 1 | 2 |
| 0 | 0 | 1 | 0 |
| 0 | 72 | 0 | 0 |
| 72 | 0 | 72 | 72 |
| 1 | 0 | 0 | 0 |
| 72 | 72 | 72 | 71 |
| 0 | 72 | 0 | 0 |
| 0 | 1 | 1 | 1 |
| 72 | 0 | 72 | 72 |
| 0 | 72 | 72 | 72 |
| 72 | 72 | 0 | 72 |
| 1 | 1 | 1 | 2 |
G:=sub<GL(4,GF(73))| [59,66,0,7,0,59,66,7,66,0,59,7,66,66,66,0],[0,72,72,1,0,72,0,1,1,72,0,0,0,71,0,1],[1,0,0,72,1,0,72,0,1,1,0,72,2,0,0,72],[1,72,0,0,0,72,72,1,0,72,0,1,0,71,0,1],[72,0,72,1,0,72,72,1,72,72,0,1,72,72,72,2] >;
C12.7S4 in GAP, Magma, Sage, TeX
C_{12}._7S_4 % in TeX
G:=Group("C12.7S4"); // GroupNames label
G:=SmallGroup(288,915);
// by ID
G=gap.SmallGroup(288,915);
# by ID
G:=PCGroup([7,-2,-2,-3,-3,-2,2,-2,2045,1016,170,675,2524,1908,172,1517,1153,285,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^12=d^3=e^2=1,b^2=c^2=a^6,a*b=b*a,a*c=c*a,a*d=d*a,e*a*e=a^-1,c*b*c^-1=a^6*b,d*b*d^-1=a^6*b*c,e*b*e=b*c,d*c*d^-1=b,e*c*e=a^6*c,e*d*e=d^-1>;
// generators/relations
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