Aliases: C62.(C2×C4), D4.(C32⋊C4), (D4×C32).C4, C32⋊7D4.C4, C32⋊4Q8.C4, C32⋊7(C8○D4), C62.C4⋊5C2, C12.D6.4C2, C32⋊M4(2)⋊6C2, C3⋊Dic3.32C23, C32⋊2C8.11C22, C3⋊S3⋊3C8⋊4C2, C4.5(C2×C32⋊C4), (C3×C12).5(C2×C4), (C2×C32⋊2C8)⋊8C2, C22.1(C2×C32⋊C4), C2.8(C22×C32⋊C4), (C4×C3⋊S3).37C22, C3⋊Dic3.23(C2×C4), (C3×C6).30(C22×C4), (C2×C3⋊Dic3).100C22, (C2×C3⋊S3).20(C2×C4), SmallGroup(288,935)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C62.(C2×C4)
G = < a,b,c,d | a6=b6=c2=1, d4=b3, ab=ba, cac=ab3, dad-1=a-1b4, bc=cb, dbd-1=a4b, cd=dc >
Subgroups: 456 in 102 conjugacy classes, 34 normal (18 characteristic)
C1, C2, C2 [×3], C3 [×2], C4, C4 [×3], C22 [×2], C22, S3 [×2], C6 [×6], C8 [×4], C2×C4 [×3], D4, D4 [×2], Q8, C32, Dic3 [×6], C12 [×2], D6 [×2], C2×C6 [×4], C2×C8 [×3], M4(2) [×3], C4○D4, C3⋊S3, C3×C6, C3×C6 [×2], Dic6 [×2], C4×S3 [×2], C2×Dic3 [×4], C3⋊D4 [×4], C3×D4 [×2], C8○D4, C3⋊Dic3, C3⋊Dic3 [×2], C3×C12, C2×C3⋊S3, C62 [×2], D4⋊2S3 [×2], C32⋊2C8 [×2], C32⋊2C8 [×2], C32⋊4Q8, C4×C3⋊S3, C2×C3⋊Dic3 [×2], C32⋊7D4 [×2], D4×C32, C3⋊S3⋊3C8, C32⋊M4(2), C2×C32⋊2C8 [×2], C62.C4 [×2], C12.D6, C62.(C2×C4)
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], C23, C22×C4, C8○D4, C32⋊C4, C2×C32⋊C4 [×3], C22×C32⋊C4, C62.(C2×C4)
Character table of C62.(C2×C4)
| class | 1 | 2A | 2B | 2C | 2D | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 6A | 6B | 6C | 6D | 6E | 6F | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 8I | 8J | 12A | 12B | |
| size | 1 | 1 | 2 | 2 | 18 | 4 | 4 | 2 | 9 | 9 | 18 | 18 | 4 | 4 | 8 | 8 | 8 | 8 | 9 | 9 | 9 | 9 | 18 | 18 | 18 | 18 | 18 | 18 | 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 | -1 | -1 | -1 | -1 | i | i | -i | -i | -i | -i | i | -i | i | i | 1 | 1 | linear of order 4 |
| ρ10 | 1 | 1 | 1 | 1 | -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 | linear of order 4 |
| ρ11 | 1 | 1 | 1 | -1 | 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 | linear of order 4 |
| ρ12 | 1 | 1 | -1 | 1 | 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 | linear of order 4 |
| ρ13 | 1 | 1 | -1 | -1 | -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 | linear of order 4 |
| ρ14 | 1 | 1 | 1 | 1 | -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 | linear of order 4 |
| ρ15 | 1 | 1 | 1 | -1 | 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 | linear of order 4 |
| ρ16 | 1 | 1 | -1 | 1 | 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 | linear of order 4 |
| ρ17 | 2 | -2 | 0 | 0 | 0 | 2 | 2 | 0 | -2i | 2i | 0 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | 2ζ87 | 2ζ83 | 2ζ8 | 2ζ85 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C8○D4 |
| ρ18 | 2 | -2 | 0 | 0 | 0 | 2 | 2 | 0 | 2i | -2i | 0 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | 2ζ85 | 2ζ8 | 2ζ83 | 2ζ87 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C8○D4 |
| ρ19 | 2 | -2 | 0 | 0 | 0 | 2 | 2 | 0 | -2i | 2i | 0 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | 2ζ83 | 2ζ87 | 2ζ85 | 2ζ8 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C8○D4 |
| ρ20 | 2 | -2 | 0 | 0 | 0 | 2 | 2 | 0 | 2i | -2i | 0 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | 2ζ8 | 2ζ85 | 2ζ87 | 2ζ83 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C8○D4 |
| ρ21 | 4 | 4 | 4 | 4 | 0 | -2 | 1 | 4 | 0 | 0 | 0 | 0 | 1 | -2 | -2 | 1 | -2 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -2 | orthogonal lifted from C32⋊C4 |
| ρ22 | 4 | 4 | -4 | 4 | 0 | -2 | 1 | -4 | 0 | 0 | 0 | 0 | 1 | -2 | -2 | -1 | 2 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 2 | orthogonal lifted from C2×C32⋊C4 |
| ρ23 | 4 | 4 | 4 | 4 | 0 | 1 | -2 | 4 | 0 | 0 | 0 | 0 | -2 | 1 | 1 | -2 | 1 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 1 | orthogonal lifted from C32⋊C4 |
| ρ24 | 4 | 4 | -4 | 4 | 0 | 1 | -2 | -4 | 0 | 0 | 0 | 0 | -2 | 1 | 1 | 2 | -1 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -1 | orthogonal lifted from C2×C32⋊C4 |
| ρ25 | 4 | 4 | 4 | -4 | 0 | 1 | -2 | -4 | 0 | 0 | 0 | 0 | -2 | 1 | -1 | -2 | 1 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -1 | orthogonal lifted from C2×C32⋊C4 |
| ρ26 | 4 | 4 | -4 | -4 | 0 | 1 | -2 | 4 | 0 | 0 | 0 | 0 | -2 | 1 | -1 | 2 | -1 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 1 | orthogonal lifted from C2×C32⋊C4 |
| ρ27 | 4 | 4 | 4 | -4 | 0 | -2 | 1 | -4 | 0 | 0 | 0 | 0 | 1 | -2 | 2 | 1 | -2 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 2 | orthogonal lifted from C2×C32⋊C4 |
| ρ28 | 4 | 4 | -4 | -4 | 0 | -2 | 1 | 4 | 0 | 0 | 0 | 0 | 1 | -2 | 2 | -1 | 2 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -2 | orthogonal lifted from C2×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 | symplectic 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 | symplectic faithful, Schur index 2 |
►On 48 points
Generators in S48
(1 39 47 28 13 24)(2 17 14 29 48 40)(3 18 15 30 41 33)(4 34 42 31 16 19)(5 35 43 32 9 20)(6 21 10 25 44 36)(7 22 11 26 45 37)(8 38 46 27 12 23)
(1 5)(2 10 48 6 14 44)(3 7)(4 46 16 8 42 12)(9 13)(11 15)(17 25 40 21 29 36)(18 22)(19 38 31 23 34 27)(20 24)(26 30)(28 32)(33 37)(35 39)(41 45)(43 47)
(1 30)(2 31)(3 32)(4 25)(5 26)(6 27)(7 28)(8 29)(9 22)(10 23)(11 24)(12 17)(13 18)(14 19)(15 20)(16 21)(33 47)(34 48)(35 41)(36 42)(37 43)(38 44)(39 45)(40 46)
(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)
G:=sub<Sym(48)| (1,39,47,28,13,24)(2,17,14,29,48,40)(3,18,15,30,41,33)(4,34,42,31,16,19)(5,35,43,32,9,20)(6,21,10,25,44,36)(7,22,11,26,45,37)(8,38,46,27,12,23), (1,5)(2,10,48,6,14,44)(3,7)(4,46,16,8,42,12)(9,13)(11,15)(17,25,40,21,29,36)(18,22)(19,38,31,23,34,27)(20,24)(26,30)(28,32)(33,37)(35,39)(41,45)(43,47), (1,30)(2,31)(3,32)(4,25)(5,26)(6,27)(7,28)(8,29)(9,22)(10,23)(11,24)(12,17)(13,18)(14,19)(15,20)(16,21)(33,47)(34,48)(35,41)(36,42)(37,43)(38,44)(39,45)(40,46), (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)>;
G:=Group( (1,39,47,28,13,24)(2,17,14,29,48,40)(3,18,15,30,41,33)(4,34,42,31,16,19)(5,35,43,32,9,20)(6,21,10,25,44,36)(7,22,11,26,45,37)(8,38,46,27,12,23), (1,5)(2,10,48,6,14,44)(3,7)(4,46,16,8,42,12)(9,13)(11,15)(17,25,40,21,29,36)(18,22)(19,38,31,23,34,27)(20,24)(26,30)(28,32)(33,37)(35,39)(41,45)(43,47), (1,30)(2,31)(3,32)(4,25)(5,26)(6,27)(7,28)(8,29)(9,22)(10,23)(11,24)(12,17)(13,18)(14,19)(15,20)(16,21)(33,47)(34,48)(35,41)(36,42)(37,43)(38,44)(39,45)(40,46), (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) );
G=PermutationGroup([(1,39,47,28,13,24),(2,17,14,29,48,40),(3,18,15,30,41,33),(4,34,42,31,16,19),(5,35,43,32,9,20),(6,21,10,25,44,36),(7,22,11,26,45,37),(8,38,46,27,12,23)], [(1,5),(2,10,48,6,14,44),(3,7),(4,46,16,8,42,12),(9,13),(11,15),(17,25,40,21,29,36),(18,22),(19,38,31,23,34,27),(20,24),(26,30),(28,32),(33,37),(35,39),(41,45),(43,47)], [(1,30),(2,31),(3,32),(4,25),(5,26),(6,27),(7,28),(8,29),(9,22),(10,23),(11,24),(12,17),(13,18),(14,19),(15,20),(16,21),(33,47),(34,48),(35,41),(36,42),(37,43),(38,44),(39,45),(40,46)], [(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)])
Matrix representation of C62.(C2×C4) ►in GL6(𝔽73)
| 0 | 46 | 0 | 0 | 0 | 0 |
| 27 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 1 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 |
| 0 | 0 | 0 | 0 | 1 | 72 |
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 1 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 22 | 0 | 0 | 0 | 0 | 0 |
| 0 | 22 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
G:=sub<GL(6,GF(73))| [0,27,0,0,0,0,46,0,0,0,0,0,0,0,0,1,0,0,0,0,72,72,0,0,0,0,0,0,0,1,0,0,0,0,72,72],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,72,0,0,0,0,1,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[22,0,0,0,0,0,0,22,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1,0,0] >;
C62.(C2×C4) in GAP, Magma, Sage, TeX
C_6^2.(C_2\times C_4)
% in TeX
G:=Group("C6^2.(C2xC4)"); // GroupNames label
G:=SmallGroup(288,935);
// by ID
G=gap.SmallGroup(288,935);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,56,219,80,9413,362,12550,1203]);
// Polycyclic
G:=Group<a,b,c,d|a^6=b^6=c^2=1,d^4=b^3,a*b=b*a,c*a*c=a*b^3,d*a*d^-1=a^-1*b^4,b*c=c*b,d*b*d^-1=a^4*b,c*d=d*c>;
// generators/relations
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