Aliases: C12.14S4, SL2(𝔽3).10D6, C4.A4⋊2S3, C4.6(C3⋊S4), C6.35(C2×S4), C6.6S4⋊7C2, C6.5S4⋊7C2, (C3×Q8).17D6, C3⋊3(C4.6S4), (C3×SL2(𝔽3)).10C22, C2.9(C2×C3⋊S4), (C3×C4.A4)⋊1C2, (C3×C4○D4)⋊1S3, Q8.4(C2×C3⋊S3), C4○D4⋊1(C3⋊S3), SmallGroup(288,914)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C2 — Q8 — C3×SL2(𝔽3) — C12.14S4 |
| C3×SL2(𝔽3) — C12.14S4 |
Generators and relations for C12.14S4
G = < a,b,c,d,e | a12=d3=e2=1, b2=c2=a6, ab=ba, ac=ca, ad=da, eae=a5, cbc-1=a6b, dbd-1=a6bc, ebe=bc, dcd-1=b, ece=a6c, ede=d-1 >
Subgroups: 608 in 104 conjugacy classes, 21 normal (15 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C8, C2×C4, D4, Q8, Q8, C32, Dic3, C12, C12, D6, C2×C6, C2×C8, D8, SD16, Q16, C4○D4, C4○D4, C3⋊S3, C3×C6, C3⋊C8, SL2(𝔽3), Dic6, C4×S3, D12, C3⋊D4, C2×C12, C3×D4, C3×Q8, C4○D8, C3⋊Dic3, C3×C12, C2×C3⋊S3, C2×C3⋊C8, D4⋊S3, D4.S3, Q8⋊2S3, C3⋊Q16, CSU2(𝔽3), GL2(𝔽3), C4.A4, C4○D12, C3×C4○D4, C3×SL2(𝔽3), C4×C3⋊S3, Q8.13D6, C4.6S4, C6.5S4, C6.6S4, C3×C4.A4, C12.14S4
Quotients: C1, C2, C22, S3, D6, C3⋊S3, S4, C2×C3⋊S3, C2×S4, C3⋊S4, C4.6S4, C2×C3⋊S4, C12.14S4
Character table of C12.14S4
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 4A | 4B | 4C | 4D | 6A | 6B | 6C | 6D | 6E | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 12I | |
| size | 1 | 1 | 6 | 36 | 2 | 8 | 8 | 8 | 1 | 1 | 6 | 36 | 2 | 8 | 8 | 8 | 12 | 18 | 18 | 18 | 18 | 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 | 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 | 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 | -1 | -1 | 1 | linear of order 2 |
| ρ5 | 2 | 2 | 2 | 0 | -1 | -1 | 2 | -1 | 2 | 2 | 2 | 0 | -1 | -1 | 2 | -1 | -1 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 2 | 2 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ6 | 2 | 2 | -2 | 0 | 2 | -1 | -1 | -1 | -2 | -2 | 2 | 0 | 2 | -1 | -1 | -1 | -2 | 0 | 0 | 0 | 0 | -2 | -2 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | orthogonal lifted from D6 |
| ρ7 | 2 | 2 | -2 | 0 | -1 | 2 | -1 | -1 | -2 | -2 | 2 | 0 | -1 | 2 | -1 | -1 | 1 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | -2 | -2 | 1 | -1 | orthogonal lifted from D6 |
| ρ8 | 2 | 2 | 2 | 0 | 2 | -1 | -1 | -1 | 2 | 2 | 2 | 0 | 2 | -1 | -1 | -1 | 2 | 0 | 0 | 0 | 0 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | -1 | 2 | orthogonal lifted from S3 |
| ρ9 | 2 | 2 | -2 | 0 | -1 | -1 | -1 | 2 | -2 | -2 | 2 | 0 | -1 | -1 | -1 | 2 | 1 | 0 | 0 | 0 | 0 | 1 | 1 | -2 | 1 | 1 | 1 | 1 | -2 | -1 | orthogonal lifted from D6 |
| ρ10 | 2 | 2 | 2 | 0 | -1 | -1 | -1 | 2 | 2 | 2 | 2 | 0 | -1 | -1 | -1 | 2 | -1 | 0 | 0 | 0 | 0 | -1 | -1 | 2 | -1 | -1 | -1 | -1 | 2 | -1 | orthogonal lifted from S3 |
| ρ11 | 2 | 2 | 2 | 0 | -1 | 2 | -1 | -1 | 2 | 2 | 2 | 0 | -1 | 2 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | 2 | 2 | -1 | -1 | orthogonal lifted from S3 |
| ρ12 | 2 | 2 | -2 | 0 | -1 | -1 | 2 | -1 | -2 | -2 | 2 | 0 | -1 | -1 | 2 | -1 | 1 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | -2 | -2 | 1 | 1 | 1 | -1 | orthogonal lifted from D6 |
| ρ13 | 2 | -2 | 0 | 0 | 2 | -1 | -1 | -1 | 2i | -2i | 0 | 0 | -2 | 1 | 1 | 1 | 0 | -√2 | √-2 | √2 | -√-2 | 2i | -2i | -i | i | -i | i | -i | i | 0 | complex lifted from C4.6S4 |
| ρ14 | 2 | -2 | 0 | 0 | 2 | -1 | -1 | -1 | -2i | 2i | 0 | 0 | -2 | 1 | 1 | 1 | 0 | √2 | √-2 | -√2 | -√-2 | -2i | 2i | i | -i | i | -i | i | -i | 0 | complex lifted from C4.6S4 |
| ρ15 | 2 | -2 | 0 | 0 | 2 | -1 | -1 | -1 | -2i | 2i | 0 | 0 | -2 | 1 | 1 | 1 | 0 | -√2 | -√-2 | √2 | √-2 | -2i | 2i | i | -i | i | -i | i | -i | 0 | complex lifted from C4.6S4 |
| ρ16 | 2 | -2 | 0 | 0 | 2 | -1 | -1 | -1 | 2i | -2i | 0 | 0 | -2 | 1 | 1 | 1 | 0 | √2 | -√-2 | -√2 | √-2 | 2i | -2i | -i | i | -i | i | -i | i | 0 | complex lifted from C4.6S4 |
| ρ17 | 3 | 3 | -1 | 1 | 3 | 0 | 0 | 0 | 3 | 3 | -1 | 1 | 3 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | 3 | 3 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | orthogonal lifted from S4 |
| ρ18 | 3 | 3 | 1 | -1 | 3 | 0 | 0 | 0 | -3 | -3 | -1 | 1 | 3 | 0 | 0 | 0 | 1 | -1 | 1 | -1 | 1 | -3 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | orthogonal lifted from C2×S4 |
| ρ19 | 3 | 3 | -1 | -1 | 3 | 0 | 0 | 0 | 3 | 3 | -1 | -1 | 3 | 0 | 0 | 0 | -1 | 1 | 1 | 1 | 1 | 3 | 3 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | orthogonal lifted from S4 |
| ρ20 | 3 | 3 | 1 | 1 | 3 | 0 | 0 | 0 | -3 | -3 | -1 | -1 | 3 | 0 | 0 | 0 | 1 | 1 | -1 | 1 | -1 | -3 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | orthogonal lifted from C2×S4 |
| ρ21 | 4 | -4 | 0 | 0 | -2 | 1 | -2 | 1 | 4i | -4i | 0 | 0 | 2 | -1 | 2 | -1 | 0 | 0 | 0 | 0 | 0 | -2i | 2i | i | 2i | -2i | -i | i | -i | 0 | complex faithful |
| ρ22 | 4 | -4 | 0 | 0 | 4 | 1 | 1 | 1 | -4i | 4i | 0 | 0 | -4 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | -4i | 4i | -i | i | -i | i | -i | i | 0 | complex lifted from C4.6S4 |
| ρ23 | 4 | -4 | 0 | 0 | 4 | 1 | 1 | 1 | 4i | -4i | 0 | 0 | -4 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 4i | -4i | i | -i | i | -i | i | -i | 0 | complex lifted from C4.6S4 |
| ρ24 | 4 | -4 | 0 | 0 | -2 | 1 | -2 | 1 | -4i | 4i | 0 | 0 | 2 | -1 | 2 | -1 | 0 | 0 | 0 | 0 | 0 | 2i | -2i | -i | -2i | 2i | i | -i | i | 0 | complex faithful |
| ρ25 | 4 | -4 | 0 | 0 | -2 | -2 | 1 | 1 | 4i | -4i | 0 | 0 | 2 | 2 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | -2i | 2i | i | -i | i | 2i | -2i | -i | 0 | complex faithful |
| ρ26 | 4 | -4 | 0 | 0 | -2 | -2 | 1 | 1 | -4i | 4i | 0 | 0 | 2 | 2 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 2i | -2i | -i | i | -i | -2i | 2i | i | 0 | complex faithful |
| ρ27 | 4 | -4 | 0 | 0 | -2 | 1 | 1 | -2 | -4i | 4i | 0 | 0 | 2 | -1 | -1 | 2 | 0 | 0 | 0 | 0 | 0 | 2i | -2i | 2i | i | -i | i | -i | -2i | 0 | complex faithful |
| ρ28 | 4 | -4 | 0 | 0 | -2 | 1 | 1 | -2 | 4i | -4i | 0 | 0 | 2 | -1 | -1 | 2 | 0 | 0 | 0 | 0 | 0 | -2i | 2i | -2i | -i | i | -i | i | 2i | 0 | complex faithful |
| ρ29 | 6 | 6 | -2 | 0 | -3 | 0 | 0 | 0 | 6 | 6 | -2 | 0 | -3 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | -3 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | orthogonal lifted from C3⋊S4 |
| ρ30 | 6 | 6 | 2 | 0 | -3 | 0 | 0 | 0 | -6 | -6 | -2 | 0 | -3 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 3 | 3 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | orthogonal lifted from C2×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 20 7 14)(2 21 8 15)(3 22 9 16)(4 23 10 17)(5 24 11 18)(6 13 12 19)(25 45 31 39)(26 46 32 40)(27 47 33 41)(28 48 34 42)(29 37 35 43)(30 38 36 44)
(1 30 7 36)(2 31 8 25)(3 32 9 26)(4 33 10 27)(5 34 11 28)(6 35 12 29)(13 37 19 43)(14 38 20 44)(15 39 21 45)(16 40 22 46)(17 41 23 47)(18 42 24 48)
(1 9 5)(2 10 6)(3 11 7)(4 12 8)(13 31 47)(14 32 48)(15 33 37)(16 34 38)(17 35 39)(18 36 40)(19 25 41)(20 26 42)(21 27 43)(22 28 44)(23 29 45)(24 30 46)
(2 6)(3 11)(5 9)(8 12)(13 45)(14 38)(15 43)(16 48)(17 41)(18 46)(19 39)(20 44)(21 37)(22 42)(23 47)(24 40)(25 35)(26 28)(27 33)(29 31)(30 36)(32 34)
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), (1,20,7,14)(2,21,8,15)(3,22,9,16)(4,23,10,17)(5,24,11,18)(6,13,12,19)(25,45,31,39)(26,46,32,40)(27,47,33,41)(28,48,34,42)(29,37,35,43)(30,38,36,44), (1,30,7,36)(2,31,8,25)(3,32,9,26)(4,33,10,27)(5,34,11,28)(6,35,12,29)(13,37,19,43)(14,38,20,44)(15,39,21,45)(16,40,22,46)(17,41,23,47)(18,42,24,48), (1,9,5)(2,10,6)(3,11,7)(4,12,8)(13,31,47)(14,32,48)(15,33,37)(16,34,38)(17,35,39)(18,36,40)(19,25,41)(20,26,42)(21,27,43)(22,28,44)(23,29,45)(24,30,46), (2,6)(3,11)(5,9)(8,12)(13,45)(14,38)(15,43)(16,48)(17,41)(18,46)(19,39)(20,44)(21,37)(22,42)(23,47)(24,40)(25,35)(26,28)(27,33)(29,31)(30,36)(32,34)>;
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), (1,20,7,14)(2,21,8,15)(3,22,9,16)(4,23,10,17)(5,24,11,18)(6,13,12,19)(25,45,31,39)(26,46,32,40)(27,47,33,41)(28,48,34,42)(29,37,35,43)(30,38,36,44), (1,30,7,36)(2,31,8,25)(3,32,9,26)(4,33,10,27)(5,34,11,28)(6,35,12,29)(13,37,19,43)(14,38,20,44)(15,39,21,45)(16,40,22,46)(17,41,23,47)(18,42,24,48), (1,9,5)(2,10,6)(3,11,7)(4,12,8)(13,31,47)(14,32,48)(15,33,37)(16,34,38)(17,35,39)(18,36,40)(19,25,41)(20,26,42)(21,27,43)(22,28,44)(23,29,45)(24,30,46), (2,6)(3,11)(5,9)(8,12)(13,45)(14,38)(15,43)(16,48)(17,41)(18,46)(19,39)(20,44)(21,37)(22,42)(23,47)(24,40)(25,35)(26,28)(27,33)(29,31)(30,36)(32,34) );
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)], [(1,20,7,14),(2,21,8,15),(3,22,9,16),(4,23,10,17),(5,24,11,18),(6,13,12,19),(25,45,31,39),(26,46,32,40),(27,47,33,41),(28,48,34,42),(29,37,35,43),(30,38,36,44)], [(1,30,7,36),(2,31,8,25),(3,32,9,26),(4,33,10,27),(5,34,11,28),(6,35,12,29),(13,37,19,43),(14,38,20,44),(15,39,21,45),(16,40,22,46),(17,41,23,47),(18,42,24,48)], [(1,9,5),(2,10,6),(3,11,7),(4,12,8),(13,31,47),(14,32,48),(15,33,37),(16,34,38),(17,35,39),(18,36,40),(19,25,41),(20,26,42),(21,27,43),(22,28,44),(23,29,45),(24,30,46)], [(2,6),(3,11),(5,9),(8,12),(13,45),(14,38),(15,43),(16,48),(17,41),(18,46),(19,39),(20,44),(21,37),(22,42),(23,47),(24,40),(25,35),(26,28),(27,33),(29,31),(30,36),(32,34)]])
Matrix representation of C12.14S4 ►in GL4(𝔽5) generated by
| 4 | 0 | 2 | 0 |
| 0 | 1 | 1 | 2 |
| 4 | 0 | 3 | 0 |
| 3 | 1 | 4 | 1 |
| 1 | 2 | 0 | 0 |
| 4 | 4 | 0 | 0 |
| 0 | 2 | 2 | 2 |
| 1 | 4 | 0 | 3 |
| 3 | 0 | 0 | 0 |
| 2 | 2 | 0 | 0 |
| 0 | 0 | 3 | 0 |
| 3 | 0 | 4 | 2 |
| 4 | 2 | 0 | 0 |
| 2 | 0 | 0 | 0 |
| 0 | 2 | 0 | 2 |
| 3 | 4 | 2 | 4 |
| 0 | 1 | 1 | 1 |
| 0 | 2 | 4 | 2 |
| 4 | 3 | 0 | 0 |
| 2 | 0 | 1 | 3 |
G:=sub<GL(4,GF(5))| [4,0,4,3,0,1,0,1,2,1,3,4,0,2,0,1],[1,4,0,1,2,4,2,4,0,0,2,0,0,0,2,3],[3,2,0,3,0,2,0,0,0,0,3,4,0,0,0,2],[4,2,0,3,2,0,2,4,0,0,0,2,0,0,2,4],[0,0,4,2,1,2,3,0,1,4,0,1,1,2,0,3] >;
C12.14S4 in GAP, Magma, Sage, TeX
C_{12}._{14}S_4 % in TeX
G:=Group("C12.14S4"); // GroupNames label
G:=SmallGroup(288,914);
// by ID
G=gap.SmallGroup(288,914);
# by ID
G:=PCGroup([7,-2,-2,-3,-3,-2,2,-2,1016,170,675,2524,1908,172,1517,1153,285,124]);
// 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^5,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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