metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: (C2×D4).8D6, (C2×C12).4D4, (C22×C12)⋊2C4, C22⋊C4⋊2Dic3, (C6×D4).6C22, (C22×C4)⋊4Dic3, (C22×C6).15D4, C6.22(C23⋊C4), C12.D4.2C2, C3⋊3(C23.D4), C23.6(C3⋊D4), C23.7(C2×Dic3), C22.D4.1S3, C23.7D6.2C2, C2.7(C23.7D6), C22.13(C6.D4), (C3×C22⋊C4)⋊2C4, (C2×C4).6(C3⋊D4), (C22×C6).14(C2×C4), (C2×C6).97(C22⋊C4), (C3×C22.D4).1C2, SmallGroup(192,98)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for (C22×C12)⋊C4
G = < a,b,c,d | a2=b2=c12=d4=1, ab=ba, ac=ca, dad-1=abc6, bc=cb, dbd-1=bc6, dcd-1=abc-1 >
Subgroups: 208 in 68 conjugacy classes, 23 normal (all characteristic)
C1, C2, C2, C3, C4, C22, C22, C6, C6, C8, C2×C4, C2×C4, D4, C23, Dic3, C12, C2×C6, C2×C6, C22⋊C4, C22⋊C4, C4⋊C4, M4(2), C22×C4, C2×D4, C3⋊C8, C2×Dic3, C2×C12, C2×C12, C3×D4, C22×C6, C23⋊C4, C4.D4, C22.D4, C4.Dic3, C6.D4, C3×C22⋊C4, C3×C22⋊C4, C3×C4⋊C4, C22×C12, C6×D4, C23.D4, C12.D4, C23.7D6, C3×C22.D4, (C22×C12)⋊C4
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Dic3, D6, C22⋊C4, C2×Dic3, C3⋊D4, C23⋊C4, C6.D4, C23.D4, C23.7D6, (C22×C12)⋊C4
Character table of (C22×C12)⋊C4
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 6A | 6B | 6C | 6D | 6E | 6F | 8A | 8B | 12A | 12B | 12C | 12D | 12E | 12F | 12G | |
| size | 1 | 1 | 2 | 4 | 4 | 2 | 4 | 4 | 4 | 8 | 24 | 24 | 2 | 2 | 2 | 4 | 4 | 8 | 24 | 24 | 4 | 4 | 4 | 4 | 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 | -i | i | 1 | 1 | 1 | 1 | 1 | -1 | -i | i | 1 | 1 | 1 | 1 | -1 | -1 | -1 | linear of order 4 |
| ρ6 | 1 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | -1 | 1 | -i | i | 1 | 1 | 1 | 1 | 1 | -1 | i | -i | -1 | -1 | -1 | -1 | -1 | 1 | 1 | linear of order 4 |
| ρ7 | 1 | 1 | 1 | -1 | 1 | 1 | -1 | 1 | 1 | -1 | i | -i | 1 | 1 | 1 | 1 | 1 | -1 | i | -i | 1 | 1 | 1 | 1 | -1 | -1 | -1 | linear of order 4 |
| ρ8 | 1 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | -1 | 1 | i | -i | 1 | 1 | 1 | 1 | 1 | -1 | -i | i | -1 | -1 | -1 | -1 | -1 | 1 | 1 | linear of order 4 |
| ρ9 | 2 | 2 | 2 | 2 | 2 | -1 | 2 | 2 | 2 | 2 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ10 | 2 | 2 | 2 | -2 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | 2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | orthogonal lifted from D4 |
| ρ11 | 2 | 2 | 2 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | orthogonal lifted from D4 |
| ρ12 | 2 | 2 | 2 | 2 | 2 | -1 | 2 | -2 | -2 | -2 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | 0 | 0 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | orthogonal lifted from D6 |
| ρ13 | 2 | 2 | 2 | -2 | 2 | -1 | -2 | -2 | -2 | 2 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | 1 | 0 | 0 | 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 | -1 | -1 | -1 | -1 | -1 | 1 | 0 | 0 | -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 | -1 | -1 | -1 | 1 | 1 | 1 | 0 | 0 | √-3 | -√-3 | √-3 | -√-3 | -1 | √-3 | -√-3 | complex lifted from C3⋊D4 |
| ρ16 | 2 | 2 | 2 | 2 | -2 | -1 | -2 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 1 | 1 | -1 | 0 | 0 | √-3 | -√-3 | √-3 | -√-3 | 1 | -√-3 | √-3 | complex lifted from C3⋊D4 |
| ρ17 | 2 | 2 | 2 | 2 | -2 | -1 | -2 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 1 | 1 | -1 | 0 | 0 | -√-3 | √-3 | -√-3 | √-3 | 1 | √-3 | -√-3 | complex lifted from C3⋊D4 |
| ρ18 | 2 | 2 | 2 | -2 | -2 | -1 | 2 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 1 | 1 | 1 | 0 | 0 | -√-3 | √-3 | -√-3 | √-3 | -1 | -√-3 | √-3 | complex lifted from C3⋊D4 |
| ρ19 | 4 | 4 | -4 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | -4 | 4 | -4 | 0 | 0 | 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 | 2 | -2 | 2 | 2√-3 | -2√-3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C23.7D6 |
| ρ21 | 4 | 4 | -4 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 2 | -2√-3 | 2√-3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C23.7D6 |
| ρ22 | 4 | -4 | 0 | 0 | 0 | 4 | 0 | 2i | -2i | 0 | 0 | 0 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 2i | 2i | -2i | -2i | 0 | 0 | 0 | complex lifted from C23.D4 |
| ρ23 | 4 | -4 | 0 | 0 | 0 | 4 | 0 | -2i | 2i | 0 | 0 | 0 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | -2i | -2i | 2i | 2i | 0 | 0 | 0 | complex lifted from C23.D4 |
| ρ24 | 4 | -4 | 0 | 0 | 0 | -2 | 0 | 2i | -2i | 0 | 0 | 0 | 2√-3 | 2 | -2√-3 | 0 | 0 | 0 | 0 | 0 | 2ζ4ζ3 | 2ζ4ζ32 | 2ζ43ζ3 | 2ζ43ζ32 | 0 | 0 | 0 | complex faithful |
| ρ25 | 4 | -4 | 0 | 0 | 0 | -2 | 0 | 2i | -2i | 0 | 0 | 0 | -2√-3 | 2 | 2√-3 | 0 | 0 | 0 | 0 | 0 | 2ζ4ζ32 | 2ζ4ζ3 | 2ζ43ζ32 | 2ζ43ζ3 | 0 | 0 | 0 | complex faithful |
| ρ26 | 4 | -4 | 0 | 0 | 0 | -2 | 0 | -2i | 2i | 0 | 0 | 0 | -2√-3 | 2 | 2√-3 | 0 | 0 | 0 | 0 | 0 | 2ζ43ζ32 | 2ζ43ζ3 | 2ζ4ζ32 | 2ζ4ζ3 | 0 | 0 | 0 | complex faithful |
| ρ27 | 4 | -4 | 0 | 0 | 0 | -2 | 0 | -2i | 2i | 0 | 0 | 0 | 2√-3 | 2 | -2√-3 | 0 | 0 | 0 | 0 | 0 | 2ζ43ζ3 | 2ζ43ζ32 | 2ζ4ζ3 | 2ζ4ζ32 | 0 | 0 | 0 | complex faithful |
►On 48 points
Generators in S48
(1 27)(2 28)(3 29)(4 30)(5 31)(6 32)(7 33)(8 34)(9 35)(10 36)(11 25)(12 26)(13 47)(14 48)(15 37)(16 38)(17 39)(18 40)(19 41)(20 42)(21 43)(22 44)(23 45)(24 46)
(1 48)(2 37)(3 38)(4 39)(5 40)(6 41)(7 42)(8 43)(9 44)(10 45)(11 46)(12 47)(13 26)(14 27)(15 28)(16 29)(17 30)(18 31)(19 32)(20 33)(21 34)(22 35)(23 36)(24 25)
(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 32 37 13)(3 11)(4 30 39 23)(5 9)(6 28 41 21)(8 26 43 19)(10 36 45 17)(12 34 47 15)(14 33 20 27)(16 31 22 25)(18 29 24 35)(38 40)(42 48)(44 46)
G:=sub<Sym(48)| (1,27)(2,28)(3,29)(4,30)(5,31)(6,32)(7,33)(8,34)(9,35)(10,36)(11,25)(12,26)(13,47)(14,48)(15,37)(16,38)(17,39)(18,40)(19,41)(20,42)(21,43)(22,44)(23,45)(24,46), (1,48)(2,37)(3,38)(4,39)(5,40)(6,41)(7,42)(8,43)(9,44)(10,45)(11,46)(12,47)(13,26)(14,27)(15,28)(16,29)(17,30)(18,31)(19,32)(20,33)(21,34)(22,35)(23,36)(24,25), (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), (2,32,37,13)(3,11)(4,30,39,23)(5,9)(6,28,41,21)(8,26,43,19)(10,36,45,17)(12,34,47,15)(14,33,20,27)(16,31,22,25)(18,29,24,35)(38,40)(42,48)(44,46)>;
G:=Group( (1,27)(2,28)(3,29)(4,30)(5,31)(6,32)(7,33)(8,34)(9,35)(10,36)(11,25)(12,26)(13,47)(14,48)(15,37)(16,38)(17,39)(18,40)(19,41)(20,42)(21,43)(22,44)(23,45)(24,46), (1,48)(2,37)(3,38)(4,39)(5,40)(6,41)(7,42)(8,43)(9,44)(10,45)(11,46)(12,47)(13,26)(14,27)(15,28)(16,29)(17,30)(18,31)(19,32)(20,33)(21,34)(22,35)(23,36)(24,25), (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), (2,32,37,13)(3,11)(4,30,39,23)(5,9)(6,28,41,21)(8,26,43,19)(10,36,45,17)(12,34,47,15)(14,33,20,27)(16,31,22,25)(18,29,24,35)(38,40)(42,48)(44,46) );
G=PermutationGroup([[(1,27),(2,28),(3,29),(4,30),(5,31),(6,32),(7,33),(8,34),(9,35),(10,36),(11,25),(12,26),(13,47),(14,48),(15,37),(16,38),(17,39),(18,40),(19,41),(20,42),(21,43),(22,44),(23,45),(24,46)], [(1,48),(2,37),(3,38),(4,39),(5,40),(6,41),(7,42),(8,43),(9,44),(10,45),(11,46),(12,47),(13,26),(14,27),(15,28),(16,29),(17,30),(18,31),(19,32),(20,33),(21,34),(22,35),(23,36),(24,25)], [(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)], [(2,32,37,13),(3,11),(4,30,39,23),(5,9),(6,28,41,21),(8,26,43,19),(10,36,45,17),(12,34,47,15),(14,33,20,27),(16,31,22,25),(18,29,24,35),(38,40),(42,48),(44,46)]])
Matrix representation of (C22×C12)⋊C4 ►in GL4(𝔽73) generated by
| 72 | 0 | 71 | 0 |
| 0 | 72 | 0 | 71 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 30 | 60 | 0 | 0 |
| 13 | 43 | 0 | 0 |
| 0 | 0 | 30 | 60 |
| 0 | 0 | 13 | 43 |
| 59 | 66 | 59 | 39 |
| 7 | 66 | 34 | 20 |
| 0 | 0 | 0 | 27 |
| 0 | 0 | 46 | 46 |
| 1 | 0 | 0 | 0 |
| 72 | 72 | 0 | 0 |
| 21 | 43 | 43 | 13 |
| 22 | 52 | 43 | 30 |
G:=sub<GL(4,GF(73))| [72,0,0,0,0,72,0,0,71,0,1,0,0,71,0,1],[30,13,0,0,60,43,0,0,0,0,30,13,0,0,60,43],[59,7,0,0,66,66,0,0,59,34,0,46,39,20,27,46],[1,72,21,22,0,72,43,52,0,0,43,43,0,0,13,30] >;
(C22×C12)⋊C4 in GAP, Magma, Sage, TeX
(C_2^2\times C_{12})\rtimes C_4 % in TeX
G:=Group("(C2^2xC12):C4"); // GroupNames label
G:=SmallGroup(192,98);
// by ID
G=gap.SmallGroup(192,98);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,28,141,232,219,675,297,1684,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^2=c^12=d^4=1,a*b=b*a,a*c=c*a,d*a*d^-1=a*b*c^6,b*c=c*b,d*b*d^-1=b*c^6,d*c*d^-1=a*b*c^-1>;
// generators/relations
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