Aliases: C12⋊S3.C4, Q8.(C32⋊C4), (Q8×C32).C4, C32⋊8(C8○D4), C12.26D6.4C2, C32⋊M4(2)⋊7C2, C3⋊Dic3.33C23, C32⋊2C8.12C22, C3⋊S3⋊3C8⋊5C2, C4.6(C2×C32⋊C4), (C3×C12).6(C2×C4), (C4×C3⋊S3).39C22, (C3×C6).32(C22×C4), C2.10(C22×C32⋊C4), (C2×C3⋊S3).21(C2×C4), SmallGroup(288,937)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C12⋊S3.C4
G = < a,b,c,d | a12=b3=c2=1, d4=a6, ab=ba, cac=a-1, dad-1=ab-1, cbc=b-1, dbd-1=a8b-1, cd=dc >
Subgroups: 520 in 102 conjugacy classes, 34 normal (10 characteristic)
C1, C2, C2, C3, C4, C4, C22, S3, C6, C8, C2×C4, D4, Q8, C32, Dic3, C12, D6, C2×C8, M4(2), C4○D4, C3⋊S3, C3×C6, C4×S3, D12, C3×Q8, C8○D4, C3⋊Dic3, C3×C12, C2×C3⋊S3, Q8⋊3S3, C32⋊2C8, C32⋊2C8, C4×C3⋊S3, C12⋊S3, Q8×C32, C3⋊S3⋊3C8, C32⋊M4(2), C12.26D6, C12⋊S3.C4
Quotients: C1, C2, C4, C22, C2×C4, C23, C22×C4, C8○D4, C32⋊C4, C2×C32⋊C4, C22×C32⋊C4, C12⋊S3.C4
Character table of C12⋊S3.C4
| class | 1 | 2A | 2B | 2C | 2D | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 6A | 6B | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 8I | 8J | 12A | 12B | 12C | 12D | 12E | 12F | |
| size | 1 | 1 | 18 | 18 | 18 | 4 | 4 | 2 | 2 | 2 | 9 | 9 | 4 | 4 | 9 | 9 | 9 | 9 | 18 | 18 | 18 | 18 | 18 | 18 | 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 | 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 | 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 |
| ρ6 | 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 |
| ρ7 | 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 |
| ρ8 | 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 |
| ρ9 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | -i | -i | i | i | i | i | -i | i | -i | -i | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ10 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | i | i | -i | -i | -i | -i | i | -i | i | i | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ11 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | -1 | 1 | 1 | i | i | -i | -i | i | i | i | -i | -i | -i | -1 | 1 | 1 | -1 | -1 | -1 | linear of order 4 |
| ρ12 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | -1 | 1 | 1 | -i | -i | i | i | -i | -i | -i | i | i | i | -1 | 1 | 1 | -1 | -1 | -1 | linear of order 4 |
| ρ13 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | 1 | i | i | -i | -i | -i | i | -i | i | i | -i | 1 | -1 | -1 | -1 | -1 | 1 | linear of order 4 |
| ρ14 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | 1 | -i | -i | i | i | i | -i | i | -i | -i | i | 1 | -1 | -1 | -1 | -1 | 1 | linear of order 4 |
| ρ15 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | -i | -i | i | i | -i | i | i | -i | i | -i | -1 | -1 | -1 | 1 | 1 | -1 | linear of order 4 |
| ρ16 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | i | i | -i | -i | i | -i | -i | i | -i | i | -1 | -1 | -1 | 1 | 1 | -1 | linear of order 4 |
| ρ17 | 2 | -2 | 0 | 0 | 0 | 2 | 2 | 0 | 0 | 0 | -2i | 2i | -2 | -2 | 2ζ83 | 2ζ87 | 2ζ8 | 2ζ85 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C8○D4 |
| ρ18 | 2 | -2 | 0 | 0 | 0 | 2 | 2 | 0 | 0 | 0 | 2i | -2i | -2 | -2 | 2ζ8 | 2ζ85 | 2ζ83 | 2ζ87 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C8○D4 |
| ρ19 | 2 | -2 | 0 | 0 | 0 | 2 | 2 | 0 | 0 | 0 | -2i | 2i | -2 | -2 | 2ζ87 | 2ζ83 | 2ζ85 | 2ζ8 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C8○D4 |
| ρ20 | 2 | -2 | 0 | 0 | 0 | 2 | 2 | 0 | 0 | 0 | 2i | -2i | -2 | -2 | 2ζ85 | 2ζ8 | 2ζ87 | 2ζ83 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C8○D4 |
| ρ21 | 4 | 4 | 0 | 0 | 0 | 1 | -2 | 4 | -4 | -4 | 0 | 0 | -2 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -1 | 2 | 1 | -2 | -1 | orthogonal lifted from C2×C32⋊C4 |
| ρ22 | 4 | 4 | 0 | 0 | 0 | 1 | -2 | -4 | 4 | -4 | 0 | 0 | -2 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 1 | -2 | -1 | 2 | -1 | orthogonal lifted from C2×C32⋊C4 |
| ρ23 | 4 | 4 | 0 | 0 | 0 | -2 | 1 | 4 | 4 | 4 | 0 | 0 | 1 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -2 | 1 | -2 | 1 | -2 | orthogonal lifted from C32⋊C4 |
| ρ24 | 4 | 4 | 0 | 0 | 0 | -2 | 1 | -4 | -4 | 4 | 0 | 0 | 1 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 2 | -1 | 2 | -1 | -2 | orthogonal lifted from C2×C32⋊C4 |
| ρ25 | 4 | 4 | 0 | 0 | 0 | -2 | 1 | -4 | 4 | -4 | 0 | 0 | 1 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -2 | 1 | 2 | -1 | 2 | orthogonal lifted from C2×C32⋊C4 |
| ρ26 | 4 | 4 | 0 | 0 | 0 | -2 | 1 | 4 | -4 | -4 | 0 | 0 | 1 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 2 | -1 | -2 | 1 | 2 | orthogonal lifted from C2×C32⋊C4 |
| ρ27 | 4 | 4 | 0 | 0 | 0 | 1 | -2 | -4 | -4 | 4 | 0 | 0 | -2 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -1 | 2 | -1 | 2 | 1 | orthogonal lifted from C2×C32⋊C4 |
| ρ28 | 4 | 4 | 0 | 0 | 0 | 1 | -2 | 4 | 4 | 4 | 0 | 0 | -2 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 1 | -2 | 1 | -2 | 1 | orthogonal lifted from C32⋊C4 |
| ρ29 | 8 | -8 | 0 | 0 | 0 | -4 | 2 | 0 | 0 | 0 | 0 | 0 | -2 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal faithful, Schur index 2 |
| ρ30 | 8 | -8 | 0 | 0 | 0 | 2 | -4 | 0 | 0 | 0 | 0 | 0 | 4 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal faithful, Schur index 2 |
►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 9)(2 17 10)(3 18 11)(4 19 12)(5 16 22)(6 13 23)(7 14 24)(8 15 21)(25 29 33)(26 30 34)(27 31 35)(28 32 36)(37 41 45)(38 42 46)(39 43 47)(40 44 48)
(2 4)(5 14)(6 13)(7 16)(8 15)(9 20)(10 19)(11 18)(12 17)(22 24)(25 33)(26 32)(27 31)(28 30)(34 36)(37 43)(38 42)(39 41)(44 48)(45 47)
(1 35 23 46 3 29 21 40)(2 32 24 43 4 26 22 37)(5 45 10 36 7 39 12 30)(6 42 11 33 8 48 9 27)(13 38 18 25 15 44 20 31)(14 47 19 34 16 41 17 28)
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), (1,20,9)(2,17,10)(3,18,11)(4,19,12)(5,16,22)(6,13,23)(7,14,24)(8,15,21)(25,29,33)(26,30,34)(27,31,35)(28,32,36)(37,41,45)(38,42,46)(39,43,47)(40,44,48), (2,4)(5,14)(6,13)(7,16)(8,15)(9,20)(10,19)(11,18)(12,17)(22,24)(25,33)(26,32)(27,31)(28,30)(34,36)(37,43)(38,42)(39,41)(44,48)(45,47), (1,35,23,46,3,29,21,40)(2,32,24,43,4,26,22,37)(5,45,10,36,7,39,12,30)(6,42,11,33,8,48,9,27)(13,38,18,25,15,44,20,31)(14,47,19,34,16,41,17,28)>;
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), (1,20,9)(2,17,10)(3,18,11)(4,19,12)(5,16,22)(6,13,23)(7,14,24)(8,15,21)(25,29,33)(26,30,34)(27,31,35)(28,32,36)(37,41,45)(38,42,46)(39,43,47)(40,44,48), (2,4)(5,14)(6,13)(7,16)(8,15)(9,20)(10,19)(11,18)(12,17)(22,24)(25,33)(26,32)(27,31)(28,30)(34,36)(37,43)(38,42)(39,41)(44,48)(45,47), (1,35,23,46,3,29,21,40)(2,32,24,43,4,26,22,37)(5,45,10,36,7,39,12,30)(6,42,11,33,8,48,9,27)(13,38,18,25,15,44,20,31)(14,47,19,34,16,41,17,28) );
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)], [(1,20,9),(2,17,10),(3,18,11),(4,19,12),(5,16,22),(6,13,23),(7,14,24),(8,15,21),(25,29,33),(26,30,34),(27,31,35),(28,32,36),(37,41,45),(38,42,46),(39,43,47),(40,44,48)], [(2,4),(5,14),(6,13),(7,16),(8,15),(9,20),(10,19),(11,18),(12,17),(22,24),(25,33),(26,32),(27,31),(28,30),(34,36),(37,43),(38,42),(39,41),(44,48),(45,47)], [(1,35,23,46,3,29,21,40),(2,32,24,43,4,26,22,37),(5,45,10,36,7,39,12,30),(6,42,11,33,8,48,9,27),(13,38,18,25,15,44,20,31),(14,47,19,34,16,41,17,28)]])
Matrix representation of C12⋊S3.C4 ►in GL6(𝔽73)
| 51 | 71 | 0 | 0 | 0 | 0 |
| 60 | 22 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 46 | 46 | 1 | 1 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 1 | 72 | 0 | 0 |
| 0 | 0 | 27 | 0 | 72 | 72 |
| 0 | 0 | 0 | 46 | 1 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 51 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 27 | 27 | 72 | 72 |
| 22 | 0 | 0 | 0 | 0 | 0 |
| 0 | 22 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 1 |
| 0 | 0 | 27 | 27 | 71 | 72 |
| 0 | 0 | 48 | 48 | 46 | 0 |
| 0 | 0 | 48 | 49 | 46 | 0 |
G:=sub<GL(6,GF(73))| [51,60,0,0,0,0,71,22,0,0,0,0,0,0,72,0,46,0,0,0,0,72,46,0,0,0,0,0,1,72,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,27,0,0,0,72,72,0,46,0,0,0,0,72,1,0,0,0,0,72,0],[1,51,0,0,0,0,0,72,0,0,0,0,0,0,0,1,0,27,0,0,1,0,0,27,0,0,0,0,1,72,0,0,0,0,0,72],[22,0,0,0,0,0,0,22,0,0,0,0,0,0,0,27,48,48,0,0,0,27,48,49,0,0,72,71,46,46,0,0,1,72,0,0] >;
C12⋊S3.C4 in GAP, Magma, Sage, TeX
C_{12}\rtimes S_3.C_4 % in TeX
G:=Group("C12:S3.C4"); // GroupNames label
G:=SmallGroup(288,937);
// by ID
G=gap.SmallGroup(288,937);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,56,120,219,100,80,9413,362,12550,1203]);
// Polycyclic
G:=Group<a,b,c,d|a^12=b^3=c^2=1,d^4=a^6,a*b=b*a,c*a*c=a^-1,d*a*d^-1=a*b^-1,c*b*c=b^-1,d*b*d^-1=a^8*b^-1,c*d=d*c>;
// generators/relations
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