metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.3Dic3, C4⋊Q8.3S3, (C4×C12).3C4, (C2×C12).6D4, (C6×Q8).3C4, (C2×Q8).28D6, (C2×Q8).6Dic3, (C6×Q8).4C22, C6.26(C23⋊C4), C3⋊2(C42.3C4), C12.10D4.2C2, C2.11(C23.7D6), C22.17(C6.D4), (C3×C4⋊Q8).3C2, (C2×C12).11(C2×C4), (C2×C4).8(C3⋊D4), (C2×C4).4(C2×Dic3), (C2×C6).106(C22⋊C4), SmallGroup(192,107)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.3Dic3
G = < a,b,c,d | a4=b4=1, c6=b2, d2=b2c3, ab=ba, cac-1=a-1, dad-1=a-1b, cbc-1=b-1, dbd-1=a2b, dcd-1=c5 >
Subgroups: 144 in 60 conjugacy classes, 23 normal (17 characteristic)
C1, C2, C2, C3, C4, C22, C6, C6, C8, C2×C4, C2×C4, C2×C4, Q8, C12, C2×C6, C42, C4⋊C4, M4(2), C2×Q8, C3⋊C8, C2×C12, C2×C12, C2×C12, C3×Q8, C4.10D4, C4⋊Q8, C4.Dic3, C4×C12, C3×C4⋊C4, C6×Q8, C42.3C4, C12.10D4, C3×C4⋊Q8, C42.3Dic3
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, C42.3Dic3
Character table of C42.3Dic3
| class | 1 | 2A | 2B | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 12I | 12J | |
| size | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 8 | 2 | 2 | 2 | 24 | 24 | 24 | 24 | 4 | 4 | 4 | 4 | 4 | 4 | 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 | 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 | 1 | 1 | 1 | -i | i | i | -i | 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 | 1 | 1 | 1 | i | -i | -i | i | 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 | 1 | 1 | 1 | i | i | -i | -i | 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 | 1 | 1 | 1 | -i | -i | i | i | 1 | -1 | -1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | linear of order 4 |
| ρ9 | 2 | 2 | 2 | 2 | 0 | 2 | -2 | 0 | -2 | 0 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | -2 | 0 | 0 | 2 | -2 | 0 | 0 | orthogonal lifted from D4 |
| ρ10 | 2 | 2 | 2 | -1 | -2 | 2 | 2 | -2 | 2 | -2 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | -1 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | orthogonal lifted from D6 |
| ρ11 | 2 | 2 | 2 | 2 | 0 | -2 | 2 | 0 | -2 | 0 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | -2 | 0 | 0 | -2 | 2 | 0 | 0 | orthogonal lifted from D4 |
| ρ12 | 2 | 2 | 2 | -1 | 2 | 2 | 2 | 2 | 2 | 2 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ13 | 2 | 2 | 2 | -1 | -2 | -2 | -2 | -2 | 2 | 2 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | -1 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | symplectic lifted from Dic3, Schur index 2 |
| ρ14 | 2 | 2 | 2 | -1 | 2 | -2 | -2 | 2 | 2 | -2 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | symplectic lifted from Dic3, Schur index 2 |
| ρ15 | 2 | 2 | 2 | -1 | 0 | 2 | -2 | 0 | -2 | 0 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 1 | -√-3 | √-3 | 1 | -√-3 | √-3 | -1 | 1 | -√-3 | √-3 | complex lifted from C3⋊D4 |
| ρ16 | 2 | 2 | 2 | -1 | 0 | 2 | -2 | 0 | -2 | 0 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 1 | √-3 | -√-3 | 1 | √-3 | -√-3 | -1 | 1 | √-3 | -√-3 | complex lifted from C3⋊D4 |
| ρ17 | 2 | 2 | 2 | -1 | 0 | -2 | 2 | 0 | -2 | 0 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 1 | -√-3 | √-3 | 1 | -√-3 | √-3 | 1 | -1 | √-3 | -√-3 | complex lifted from C3⋊D4 |
| ρ18 | 2 | 2 | 2 | -1 | 0 | -2 | 2 | 0 | -2 | 0 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 1 | √-3 | -√-3 | 1 | √-3 | -√-3 | 1 | -1 | -√-3 | √-3 | complex lifted from C3⋊D4 |
| ρ19 | 4 | 4 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | -4 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C23⋊C4 |
| ρ20 | 4 | -4 | 0 | 4 | -2 | 0 | 0 | 2 | 0 | 0 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | symplectic lifted from C42.3C4, Schur index 2 |
| ρ21 | 4 | -4 | 0 | 4 | 2 | 0 | 0 | -2 | 0 | 0 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 0 | 2 | 2 | 0 | 0 | 0 | 0 | symplectic lifted from C42.3C4, Schur index 2 |
| ρ22 | 4 | -4 | 0 | -2 | 2 | 0 | 0 | -2 | 0 | 0 | -2√-3 | 2 | 2√-3 | 0 | 0 | 0 | 0 | 0 | 1-√-3 | 1+√-3 | 0 | -1+√-3 | -1-√-3 | 0 | 0 | 0 | 0 | complex faithful |
| ρ23 | 4 | -4 | 0 | -2 | 2 | 0 | 0 | -2 | 0 | 0 | 2√-3 | 2 | -2√-3 | 0 | 0 | 0 | 0 | 0 | 1+√-3 | 1-√-3 | 0 | -1-√-3 | -1+√-3 | 0 | 0 | 0 | 0 | complex faithful |
| ρ24 | 4 | 4 | -4 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | -2√-3 | 0 | 0 | 2√-3 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C23.7D6 |
| ρ25 | 4 | -4 | 0 | -2 | -2 | 0 | 0 | 2 | 0 | 0 | 2√-3 | 2 | -2√-3 | 0 | 0 | 0 | 0 | 0 | -1-√-3 | -1+√-3 | 0 | 1+√-3 | 1-√-3 | 0 | 0 | 0 | 0 | complex faithful |
| ρ26 | 4 | -4 | 0 | -2 | -2 | 0 | 0 | 2 | 0 | 0 | -2√-3 | 2 | 2√-3 | 0 | 0 | 0 | 0 | 0 | -1+√-3 | -1-√-3 | 0 | 1-√-3 | 1+√-3 | 0 | 0 | 0 | 0 | complex faithful |
| ρ27 | 4 | 4 | -4 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 2√-3 | 0 | 0 | -2√-3 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C23.7D6 |
►On 48 points
Generators in S48
(13 28 19 34)(14 35 20 29)(15 30 21 36)(16 25 22 31)(17 32 23 26)(18 27 24 33)
(1 39 7 45)(2 46 8 40)(3 41 9 47)(4 48 10 42)(5 43 11 37)(6 38 12 44)(13 28 19 34)(14 35 20 29)(15 30 21 36)(16 25 22 31)(17 32 23 26)(18 27 24 33)
(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 28 10 25 7 34 4 31)(2 33 11 30 8 27 5 36)(3 26 12 35 9 32 6 29)(13 42 22 39 19 48 16 45)(14 47 23 44 20 41 17 38)(15 40 24 37 21 46 18 43)
G:=sub<Sym(48)| (13,28,19,34)(14,35,20,29)(15,30,21,36)(16,25,22,31)(17,32,23,26)(18,27,24,33), (1,39,7,45)(2,46,8,40)(3,41,9,47)(4,48,10,42)(5,43,11,37)(6,38,12,44)(13,28,19,34)(14,35,20,29)(15,30,21,36)(16,25,22,31)(17,32,23,26)(18,27,24,33), (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), (1,28,10,25,7,34,4,31)(2,33,11,30,8,27,5,36)(3,26,12,35,9,32,6,29)(13,42,22,39,19,48,16,45)(14,47,23,44,20,41,17,38)(15,40,24,37,21,46,18,43)>;
G:=Group( (13,28,19,34)(14,35,20,29)(15,30,21,36)(16,25,22,31)(17,32,23,26)(18,27,24,33), (1,39,7,45)(2,46,8,40)(3,41,9,47)(4,48,10,42)(5,43,11,37)(6,38,12,44)(13,28,19,34)(14,35,20,29)(15,30,21,36)(16,25,22,31)(17,32,23,26)(18,27,24,33), (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), (1,28,10,25,7,34,4,31)(2,33,11,30,8,27,5,36)(3,26,12,35,9,32,6,29)(13,42,22,39,19,48,16,45)(14,47,23,44,20,41,17,38)(15,40,24,37,21,46,18,43) );
G=PermutationGroup([[(13,28,19,34),(14,35,20,29),(15,30,21,36),(16,25,22,31),(17,32,23,26),(18,27,24,33)], [(1,39,7,45),(2,46,8,40),(3,41,9,47),(4,48,10,42),(5,43,11,37),(6,38,12,44),(13,28,19,34),(14,35,20,29),(15,30,21,36),(16,25,22,31),(17,32,23,26),(18,27,24,33)], [(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)], [(1,28,10,25,7,34,4,31),(2,33,11,30,8,27,5,36),(3,26,12,35,9,32,6,29),(13,42,22,39,19,48,16,45),(14,47,23,44,20,41,17,38),(15,40,24,37,21,46,18,43)]])
Matrix representation of C42.3Dic3 ►in GL4(𝔽7) generated by
| 6 | 0 | 6 | 0 |
| 3 | 2 | 3 | 1 |
| 1 | 1 | 5 | 5 |
| 0 | 0 | 0 | 6 |
| 3 | 5 | 0 | 0 |
| 5 | 4 | 0 | 0 |
| 6 | 6 | 2 | 2 |
| 5 | 2 | 1 | 5 |
| 1 | 3 | 6 | 2 |
| 2 | 5 | 2 | 0 |
| 4 | 4 | 5 | 6 |
| 4 | 3 | 5 | 3 |
| 0 | 3 | 6 | 2 |
| 5 | 0 | 6 | 5 |
| 6 | 1 | 4 | 4 |
| 2 | 2 | 1 | 3 |
G:=sub<GL(4,GF(7))| [6,3,1,0,0,2,1,0,6,3,5,0,0,1,5,6],[3,5,6,5,5,4,6,2,0,0,2,1,0,0,2,5],[1,2,4,4,3,5,4,3,6,2,5,5,2,0,6,3],[0,5,6,2,3,0,1,2,6,6,4,1,2,5,4,3] >;
C42.3Dic3 in GAP, Magma, Sage, TeX
C_4^2._3{\rm Dic}_3 % in TeX
G:=Group("C4^2.3Dic3"); // GroupNames label
G:=SmallGroup(192,107);
// by ID
G=gap.SmallGroup(192,107);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,28,141,232,219,184,1571,570,297,136,1684,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^6=b^2,d^2=b^2*c^3,a*b=b*a,c*a*c^-1=a^-1,d*a*d^-1=a^-1*b,c*b*c^-1=b^-1,d*b*d^-1=a^2*b,d*c*d^-1=c^5>;
// generators/relations
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