metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42⋊4Dic3, (C4×C12)⋊2C4, (C6×Q8)⋊2C4, (C2×D4).9D6, (C2×Q8)⋊4Dic3, C3⋊2(C42⋊3C4), C4.4D4.2S3, (C22×C6).16D4, C6.23(C23⋊C4), C23.7(C3⋊D4), (C6×D4).172C22, C23.7D6.4C2, C2.8(C23.7D6), C22.14(C6.D4), (C2×C12).8(C2×C4), (C2×C4).1(C2×Dic3), (C3×C4.4D4).9C2, (C2×C6).99(C22⋊C4), SmallGroup(192,100)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42⋊4Dic3
G = < a,b,c,d | a4=b4=c6=1, d2=c3, ab=ba, cac-1=a-1b2, dad-1=a-1b-1, cbc-1=b-1, dbd-1=a2b-1, dcd-1=c-1 >
Subgroups: 240 in 70 conjugacy classes, 23 normal (17 characteristic)
C1, C2, C2, C3, C4, C22, C22, C6, C6, C2×C4, C2×C4, D4, Q8, C23, Dic3, C12, C2×C6, C2×C6, C42, C22⋊C4, C2×D4, C2×Q8, C2×Dic3, C2×C12, C2×C12, C3×D4, C3×Q8, C22×C6, C23⋊C4, C4.4D4, C6.D4, C4×C12, C3×C22⋊C4, C6×D4, C6×Q8, C42⋊3C4, C23.7D6, C3×C4.4D4, C42⋊4Dic3
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Dic3, D6, C22⋊C4, C2×Dic3, C3⋊D4, C23⋊C4, C6.D4, C42⋊3C4, C23.7D6, C42⋊4Dic3
Character table of C42⋊4Dic3
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 6A | 6B | 6C | 6D | 6E | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | |
| size | 1 | 1 | 2 | 4 | 4 | 2 | 4 | 4 | 4 | 8 | 24 | 24 | 24 | 24 | 2 | 2 | 2 | 8 | 8 | 4 | 4 | 4 | 4 | 4 | 4 | 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 | 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 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | i | -i | -i | i | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | linear of order 4 |
| ρ6 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | -i | -i | i | i | 1 | 1 | 1 | -1 | -1 | -1 | 1 | -1 | -1 | 1 | -1 | 1 | 1 | linear of order 4 |
| ρ7 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -i | i | i | -i | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | linear of order 4 |
| ρ8 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | i | i | -i | -i | 1 | 1 | 1 | -1 | -1 | -1 | 1 | -1 | -1 | 1 | -1 | 1 | 1 | linear of order 4 |
| ρ9 | 2 | 2 | 2 | 2 | 2 | -1 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ10 | 2 | 2 | 2 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | 2 | -2 | 2 | 0 | -2 | 0 | 0 | -2 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ11 | 2 | 2 | 2 | 2 | 2 | -1 | 2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | 1 | -1 | 1 | 1 | -1 | 1 | 1 | 1 | orthogonal lifted from D6 |
| ρ12 | 2 | 2 | 2 | -2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | 2 | 2 | -2 | 0 | -2 | 0 | 0 | -2 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ13 | 2 | 2 | 2 | -2 | -2 | -1 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 1 | 1 | 1 | -1 | 1 | 1 | -1 | 1 | -1 | -1 | symplectic lifted from Dic3, Schur index 2 |
| ρ14 | 2 | 2 | 2 | -2 | -2 | -1 | 2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | symplectic lifted from Dic3, Schur index 2 |
| ρ15 | 2 | 2 | 2 | 2 | -2 | -1 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 1 | -1 | -√-3 | 1 | √-3 | -√-3 | 1 | √-3 | √-3 | -√-3 | complex lifted from C3⋊D4 |
| ρ16 | 2 | 2 | 2 | -2 | 2 | -1 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | 1 | √-3 | 1 | -√-3 | √-3 | 1 | -√-3 | √-3 | -√-3 | complex lifted from C3⋊D4 |
| ρ17 | 2 | 2 | 2 | -2 | 2 | -1 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | 1 | -√-3 | 1 | √-3 | -√-3 | 1 | √-3 | -√-3 | √-3 | complex lifted from C3⋊D4 |
| ρ18 | 2 | 2 | 2 | 2 | -2 | -1 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 1 | -1 | √-3 | 1 | -√-3 | √-3 | 1 | -√-3 | -√-3 | √-3 | complex lifted from C3⋊D4 |
| ρ19 | 4 | 4 | -4 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C23⋊C4 |
| ρ20 | 4 | 4 | -4 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 2 | 0 | 0 | 0 | -2√-3 | 0 | 0 | 2√-3 | 0 | 0 | 0 | complex lifted from C23.7D6 |
| ρ21 | 4 | 4 | -4 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 2 | 0 | 0 | 0 | 2√-3 | 0 | 0 | -2√-3 | 0 | 0 | 0 | complex lifted from C23.7D6 |
| ρ22 | 4 | -4 | 0 | 0 | 0 | 4 | 0 | -2i | 2i | 0 | 0 | 0 | 0 | 0 | -4 | 0 | 0 | 0 | 0 | -2i | 0 | 2i | 2i | 0 | -2i | 0 | 0 | complex lifted from C42⋊3C4 |
| ρ23 | 4 | -4 | 0 | 0 | 0 | 4 | 0 | 2i | -2i | 0 | 0 | 0 | 0 | 0 | -4 | 0 | 0 | 0 | 0 | 2i | 0 | -2i | -2i | 0 | 2i | 0 | 0 | complex lifted from C42⋊3C4 |
| ρ24 | 4 | -4 | 0 | 0 | 0 | -2 | 0 | -2i | 2i | 0 | 0 | 0 | 0 | 0 | 2 | 2√-3 | -2√-3 | 0 | 0 | 2ζ43ζ32 | 0 | 2ζ4ζ3 | 2ζ4ζ32 | 0 | 2ζ43ζ3 | 0 | 0 | complex faithful |
| ρ25 | 4 | -4 | 0 | 0 | 0 | -2 | 0 | -2i | 2i | 0 | 0 | 0 | 0 | 0 | 2 | -2√-3 | 2√-3 | 0 | 0 | 2ζ43ζ3 | 0 | 2ζ4ζ32 | 2ζ4ζ3 | 0 | 2ζ43ζ32 | 0 | 0 | complex faithful |
| ρ26 | 4 | -4 | 0 | 0 | 0 | -2 | 0 | 2i | -2i | 0 | 0 | 0 | 0 | 0 | 2 | -2√-3 | 2√-3 | 0 | 0 | 2ζ4ζ3 | 0 | 2ζ43ζ32 | 2ζ43ζ3 | 0 | 2ζ4ζ32 | 0 | 0 | complex faithful |
| ρ27 | 4 | -4 | 0 | 0 | 0 | -2 | 0 | 2i | -2i | 0 | 0 | 0 | 0 | 0 | 2 | 2√-3 | -2√-3 | 0 | 0 | 2ζ4ζ32 | 0 | 2ζ43ζ3 | 2ζ43ζ32 | 0 | 2ζ4ζ3 | 0 | 0 | complex faithful |
►On 48 points
Generators in S48
(1 32 11 29)(2 36 12 27)(3 34 10 25)(4 26 8 35)(5 30 9 33)(6 28 7 31)(13 48 20 45)(14 40 21 37)(15 44 22 47)(16 42 23 39)(17 46 24 43)(18 38 19 41)
(1 16 4 13)(2 14 5 17)(3 18 6 15)(7 22 10 19)(8 20 11 23)(9 24 12 21)(25 41 31 47)(26 48 32 42)(27 37 33 43)(28 44 34 38)(29 39 35 45)(30 46 36 40)
(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)
(2 3)(5 6)(7 12)(8 11)(9 10)(13 20 16 23)(14 19 17 22)(15 24 18 21)(25 37 28 40)(26 42 29 39)(27 41 30 38)(31 43 34 46)(32 48 35 45)(33 47 36 44)
G:=sub<Sym(48)| (1,32,11,29)(2,36,12,27)(3,34,10,25)(4,26,8,35)(5,30,9,33)(6,28,7,31)(13,48,20,45)(14,40,21,37)(15,44,22,47)(16,42,23,39)(17,46,24,43)(18,38,19,41), (1,16,4,13)(2,14,5,17)(3,18,6,15)(7,22,10,19)(8,20,11,23)(9,24,12,21)(25,41,31,47)(26,48,32,42)(27,37,33,43)(28,44,34,38)(29,39,35,45)(30,46,36,40), (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), (2,3)(5,6)(7,12)(8,11)(9,10)(13,20,16,23)(14,19,17,22)(15,24,18,21)(25,37,28,40)(26,42,29,39)(27,41,30,38)(31,43,34,46)(32,48,35,45)(33,47,36,44)>;
G:=Group( (1,32,11,29)(2,36,12,27)(3,34,10,25)(4,26,8,35)(5,30,9,33)(6,28,7,31)(13,48,20,45)(14,40,21,37)(15,44,22,47)(16,42,23,39)(17,46,24,43)(18,38,19,41), (1,16,4,13)(2,14,5,17)(3,18,6,15)(7,22,10,19)(8,20,11,23)(9,24,12,21)(25,41,31,47)(26,48,32,42)(27,37,33,43)(28,44,34,38)(29,39,35,45)(30,46,36,40), (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), (2,3)(5,6)(7,12)(8,11)(9,10)(13,20,16,23)(14,19,17,22)(15,24,18,21)(25,37,28,40)(26,42,29,39)(27,41,30,38)(31,43,34,46)(32,48,35,45)(33,47,36,44) );
G=PermutationGroup([[(1,32,11,29),(2,36,12,27),(3,34,10,25),(4,26,8,35),(5,30,9,33),(6,28,7,31),(13,48,20,45),(14,40,21,37),(15,44,22,47),(16,42,23,39),(17,46,24,43),(18,38,19,41)], [(1,16,4,13),(2,14,5,17),(3,18,6,15),(7,22,10,19),(8,20,11,23),(9,24,12,21),(25,41,31,47),(26,48,32,42),(27,37,33,43),(28,44,34,38),(29,39,35,45),(30,46,36,40)], [(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)], [(2,3),(5,6),(7,12),(8,11),(9,10),(13,20,16,23),(14,19,17,22),(15,24,18,21),(25,37,28,40),(26,42,29,39),(27,41,30,38),(31,43,34,46),(32,48,35,45),(33,47,36,44)]])
Matrix representation of C42⋊4Dic3 ►in GL4(𝔽13) generated by
| 12 | 6 | 7 | 2 |
| 1 | 3 | 5 | 7 |
| 10 | 4 | 8 | 0 |
| 8 | 4 | 6 | 6 |
| 0 | 0 | 12 | 1 |
| 12 | 12 | 11 | 12 |
| 5 | 5 | 1 | 0 |
| 4 | 5 | 1 | 0 |
| 12 | 1 | 0 | 0 |
| 12 | 0 | 0 | 0 |
| 9 | 8 | 0 | 12 |
| 10 | 9 | 1 | 1 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 6 | 6 | 2 | 4 |
| 4 | 4 | 2 | 11 |
G:=sub<GL(4,GF(13))| [12,1,10,8,6,3,4,4,7,5,8,6,2,7,0,6],[0,12,5,4,0,12,5,5,12,11,1,1,1,12,0,0],[12,12,9,10,1,0,8,9,0,0,0,1,0,0,12,1],[0,1,6,4,1,0,6,4,0,0,2,2,0,0,4,11] >;
C42⋊4Dic3 in GAP, Magma, Sage, TeX
C_4^2\rtimes_4{\rm Dic}_3 % in TeX
G:=Group("C4^2:4Dic3"); // GroupNames label
G:=SmallGroup(192,100);
// by ID
G=gap.SmallGroup(192,100);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,28,141,232,219,1571,570,297,136,1684,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^6=1,d^2=c^3,a*b=b*a,c*a*c^-1=a^-1*b^2,d*a*d^-1=a^-1*b^-1,c*b*c^-1=b^-1,d*b*d^-1=a^2*b^-1,d*c*d^-1=c^-1>;
// generators/relations
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