metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.2Dic3, (C4×C12).2C4, (C6×D4).2C4, (C2×C12).5D4, (C2×Q8).27D6, (C2×D4).2Dic3, C4.4D4.3S3, (C6×Q8).3C22, C12.10D4⋊2C2, C6.24(C23⋊C4), C3⋊2(C42.C4), C2.9(C23.7D6), C22.15(C6.D4), (C2×C12).9(C2×C4), (C2×C4).7(C3⋊D4), (C2×C4).2(C2×Dic3), (C3×C4.4D4).1C2, (C2×C6).100(C22⋊C4), SmallGroup(192,101)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.Dic3
G = < a,b,c,d | a4=b4=1, c6=b2, d2=b2c3, ab=ba, cac-1=a-1b2, dad-1=a-1b-1, cbc-1=b-1, dbd-1=a2b-1, dcd-1=c5 >
Subgroups: 176 in 64 conjugacy classes, 23 normal (17 characteristic)
C1, C2, C2, C3, C4, C22, C22, C6, C6, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C12, C2×C6, C2×C6, C42, C22⋊C4, M4(2), C2×D4, C2×Q8, C3⋊C8, C2×C12, C2×C12, C2×C12, C3×D4, C3×Q8, C22×C6, C4.10D4, C4.4D4, C4.Dic3, C4×C12, C3×C22⋊C4, C6×D4, C6×Q8, C42.C4, C12.10D4, C3×C4.4D4, C42.Dic3
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Dic3, D6, C22⋊C4, C2×Dic3, C3⋊D4, C23⋊C4, C6.D4, C42.C4, C23.7D6, C42.Dic3
Character table of C42.Dic3
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 6A | 6B | 6C | 6D | 6E | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | |
| size | 1 | 1 | 2 | 8 | 2 | 4 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 8 | 8 | 24 | 24 | 24 | 24 | 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 | 1 | 1 | 1 | -1 | -1 | -i | i | i | -i | 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 | 1 | 1 | i | i | -i | -i | -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 | -1 | -1 | i | -i | -i | i | 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 | 1 | 1 | -i | -i | i | i | -1 | 1 | -1 | -1 | 1 | -1 | -1 | -1 | linear of order 4 |
| ρ9 | 2 | 2 | 2 | 0 | 2 | 0 | 2 | -2 | 0 | -2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | -2 | 0 | 2 | -2 | orthogonal lifted from D4 |
| ρ10 | 2 | 2 | 2 | 0 | 2 | 0 | -2 | -2 | 0 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | -2 | 0 | -2 | 2 | orthogonal lifted from D4 |
| ρ11 | 2 | 2 | 2 | -2 | -1 | -2 | 2 | 2 | -2 | 2 | -1 | -1 | -1 | 1 | 1 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | 1 | -1 | 1 | -1 | -1 | orthogonal lifted from D6 |
| ρ12 | 2 | 2 | 2 | 2 | -1 | 2 | 2 | 2 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ13 | 2 | 2 | 2 | 2 | -1 | -2 | -2 | 2 | -2 | -2 | -1 | -1 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | 1 | -1 | 1 | 1 | 1 | symplectic lifted from Dic3, Schur index 2 |
| ρ14 | 2 | 2 | 2 | -2 | -1 | 2 | -2 | 2 | 2 | -2 | -1 | -1 | -1 | 1 | 1 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | symplectic lifted from Dic3, Schur index 2 |
| ρ15 | 2 | 2 | 2 | 0 | -1 | 0 | 2 | -2 | 0 | -2 | -1 | -1 | -1 | -√-3 | √-3 | 0 | 0 | 0 | 0 | -√-3 | 1 | √-3 | -√-3 | 1 | √-3 | -1 | 1 | complex lifted from C3⋊D4 |
| ρ16 | 2 | 2 | 2 | 0 | -1 | 0 | -2 | -2 | 0 | 2 | -1 | -1 | -1 | -√-3 | √-3 | 0 | 0 | 0 | 0 | √-3 | 1 | -√-3 | √-3 | 1 | -√-3 | 1 | -1 | complex lifted from C3⋊D4 |
| ρ17 | 2 | 2 | 2 | 0 | -1 | 0 | -2 | -2 | 0 | 2 | -1 | -1 | -1 | √-3 | -√-3 | 0 | 0 | 0 | 0 | -√-3 | 1 | √-3 | -√-3 | 1 | √-3 | 1 | -1 | complex lifted from C3⋊D4 |
| ρ18 | 2 | 2 | 2 | 0 | -1 | 0 | 2 | -2 | 0 | -2 | -1 | -1 | -1 | √-3 | -√-3 | 0 | 0 | 0 | 0 | √-3 | 1 | -√-3 | √-3 | 1 | -√-3 | -1 | 1 | complex lifted from C3⋊D4 |
| ρ19 | 4 | 4 | -4 | 0 | 4 | 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 | -4 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2√-3 | 0 | 0 | 2√-3 | 0 | 0 | 0 | complex lifted from C23.7D6 |
| ρ21 | 4 | 4 | -4 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2√-3 | 0 | 0 | -2√-3 | 0 | 0 | 0 | complex lifted from C23.7D6 |
| ρ22 | 4 | -4 | 0 | 0 | 4 | 2i | 0 | 0 | -2i | 0 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2i | 0 | 2i | 2i | 0 | -2i | 0 | 0 | complex lifted from C42.C4 |
| ρ23 | 4 | -4 | 0 | 0 | 4 | -2i | 0 | 0 | 2i | 0 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2i | 0 | -2i | -2i | 0 | 2i | 0 | 0 | complex lifted from C42.C4 |
| ρ24 | 4 | -4 | 0 | 0 | -2 | -2i | 0 | 0 | 2i | 0 | 2 | 2√-3 | -2√-3 | 0 | 0 | 0 | 0 | 0 | 0 | 2ζ4ζ32 | 0 | 2ζ43ζ3 | 2ζ43ζ32 | 0 | 2ζ4ζ3 | 0 | 0 | complex faithful |
| ρ25 | 4 | -4 | 0 | 0 | -2 | 2i | 0 | 0 | -2i | 0 | 2 | 2√-3 | -2√-3 | 0 | 0 | 0 | 0 | 0 | 0 | 2ζ43ζ32 | 0 | 2ζ4ζ3 | 2ζ4ζ32 | 0 | 2ζ43ζ3 | 0 | 0 | complex faithful |
| ρ26 | 4 | -4 | 0 | 0 | -2 | -2i | 0 | 0 | 2i | 0 | 2 | -2√-3 | 2√-3 | 0 | 0 | 0 | 0 | 0 | 0 | 2ζ4ζ3 | 0 | 2ζ43ζ32 | 2ζ43ζ3 | 0 | 2ζ4ζ32 | 0 | 0 | complex faithful |
| ρ27 | 4 | -4 | 0 | 0 | -2 | 2i | 0 | 0 | -2i | 0 | 2 | -2√-3 | 2√-3 | 0 | 0 | 0 | 0 | 0 | 0 | 2ζ43ζ3 | 0 | 2ζ4ζ32 | 2ζ4ζ3 | 0 | 2ζ43ζ32 | 0 | 0 | complex faithful |
►On 48 points
Generators in S48
(1 10 7 4)(2 11 8 5)(3 12 9 6)(13 43)(14 38)(15 45)(16 40)(17 47)(18 42)(19 37)(20 44)(21 39)(22 46)(23 41)(24 48)(25 28 31 34)(26 29 32 35)(27 30 33 36)
(1 33 7 27)(2 28 8 34)(3 35 9 29)(4 30 10 36)(5 25 11 31)(6 32 12 26)(13 46 19 40)(14 41 20 47)(15 48 21 42)(16 43 22 37)(17 38 23 44)(18 45 24 39)
(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 40 10 37 7 46 4 43)(2 45 11 42 8 39 5 48)(3 38 12 47 9 44 6 41)(13 30 22 27 19 36 16 33)(14 35 23 32 20 29 17 26)(15 28 24 25 21 34 18 31)
G:=sub<Sym(48)| (1,10,7,4)(2,11,8,5)(3,12,9,6)(13,43)(14,38)(15,45)(16,40)(17,47)(18,42)(19,37)(20,44)(21,39)(22,46)(23,41)(24,48)(25,28,31,34)(26,29,32,35)(27,30,33,36), (1,33,7,27)(2,28,8,34)(3,35,9,29)(4,30,10,36)(5,25,11,31)(6,32,12,26)(13,46,19,40)(14,41,20,47)(15,48,21,42)(16,43,22,37)(17,38,23,44)(18,45,24,39), (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), (1,40,10,37,7,46,4,43)(2,45,11,42,8,39,5,48)(3,38,12,47,9,44,6,41)(13,30,22,27,19,36,16,33)(14,35,23,32,20,29,17,26)(15,28,24,25,21,34,18,31)>;
G:=Group( (1,10,7,4)(2,11,8,5)(3,12,9,6)(13,43)(14,38)(15,45)(16,40)(17,47)(18,42)(19,37)(20,44)(21,39)(22,46)(23,41)(24,48)(25,28,31,34)(26,29,32,35)(27,30,33,36), (1,33,7,27)(2,28,8,34)(3,35,9,29)(4,30,10,36)(5,25,11,31)(6,32,12,26)(13,46,19,40)(14,41,20,47)(15,48,21,42)(16,43,22,37)(17,38,23,44)(18,45,24,39), (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), (1,40,10,37,7,46,4,43)(2,45,11,42,8,39,5,48)(3,38,12,47,9,44,6,41)(13,30,22,27,19,36,16,33)(14,35,23,32,20,29,17,26)(15,28,24,25,21,34,18,31) );
G=PermutationGroup([[(1,10,7,4),(2,11,8,5),(3,12,9,6),(13,43),(14,38),(15,45),(16,40),(17,47),(18,42),(19,37),(20,44),(21,39),(22,46),(23,41),(24,48),(25,28,31,34),(26,29,32,35),(27,30,33,36)], [(1,33,7,27),(2,28,8,34),(3,35,9,29),(4,30,10,36),(5,25,11,31),(6,32,12,26),(13,46,19,40),(14,41,20,47),(15,48,21,42),(16,43,22,37),(17,38,23,44),(18,45,24,39)], [(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)], [(1,40,10,37,7,46,4,43),(2,45,11,42,8,39,5,48),(3,38,12,47,9,44,6,41),(13,30,22,27,19,36,16,33),(14,35,23,32,20,29,17,26),(15,28,24,25,21,34,18,31)]])
Matrix representation of C42.Dic3 ►in GL4(𝔽73) generated by
| 46 | 0 | 0 | 0 |
| 0 | 46 | 0 | 0 |
| 0 | 0 | 46 | 19 |
| 0 | 0 | 27 | 27 |
| 72 | 71 | 0 | 0 |
| 1 | 1 | 0 | 0 |
| 0 | 0 | 1 | 2 |
| 0 | 0 | 72 | 72 |
| 3 | 0 | 0 | 0 |
| 70 | 70 | 0 | 0 |
| 0 | 0 | 24 | 0 |
| 0 | 0 | 49 | 49 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 46 | 0 | 0 | 0 |
| 27 | 27 | 0 | 0 |
G:=sub<GL(4,GF(73))| [46,0,0,0,0,46,0,0,0,0,46,27,0,0,19,27],[72,1,0,0,71,1,0,0,0,0,1,72,0,0,2,72],[3,70,0,0,0,70,0,0,0,0,24,49,0,0,0,49],[0,0,46,27,0,0,0,27,1,0,0,0,0,1,0,0] >;
C42.Dic3 in GAP, Magma, Sage, TeX
C_4^2.{\rm Dic}_3 % in TeX
G:=Group("C4^2.Dic3"); // GroupNames label
G:=SmallGroup(192,101);
// by ID
G=gap.SmallGroup(192,101);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,28,141,219,184,1571,570,297,136,1684,6278]);
// 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*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^5>;
// generators/relations
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