metabelian, soluble, monomial, A-group
Aliases: C42⋊2A4, C24.10A4, C22⋊(C42⋊C3), (C22×C42)⋊3C3, C22.1(C22⋊A4), SmallGroup(192,1020)
Series: Derived ►Chief ►Lower central ►Upper central
| C22×C42 — C42⋊2A4 |
Subgroups: 414 in 103 conjugacy classes, 13 normal (5 characteristic)
C1, C2 [×5], C3, C4 [×8], C22, C22 [×4], C22 [×10], C2×C4 [×28], C23 [×5], A4 [×5], C42 [×4], C42 [×4], C22×C4 [×14], C24, C2×C42 [×4], C23×C4, C42⋊C3 [×4], C22⋊A4, C22×C42, C42⋊2A4
Quotients:
C1, C3, A4 [×5], C42⋊C3 [×4], C22⋊A4, C42⋊2A4
Generators and relations
G = < a,b,c,d,e | a4=b4=c2=d2=e3=1, ab=ba, ac=ca, ad=da, eae-1=ab-1, bc=cb, bd=db, ebe-1=a-1b2, ece-1=cd=dc, ede-1=c >
►On 24 points - transitive group 24T388
Generators in S24
(1 2)(3 4)(5 6)(7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)
(1 5 2 6)(3 7 4 8)(13 16 15 14)(17 20 19 18)
(1 4)(2 3)(5 8)(6 7)(9 11)(10 12)(13 18)(14 19)(15 20)(16 17)(21 23)(22 24)
(1 3)(2 4)(5 7)(6 8)(9 21)(10 22)(11 23)(12 24)(13 15)(14 16)(17 19)(18 20)
(1 24 15)(2 22 13)(3 10 20)(4 12 18)(5 21 16)(6 23 14)(7 11 17)(8 9 19)
G:=sub<Sym(24)| (1,2)(3,4)(5,6)(7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,5,2,6)(3,7,4,8)(13,16,15,14)(17,20,19,18), (1,4)(2,3)(5,8)(6,7)(9,11)(10,12)(13,18)(14,19)(15,20)(16,17)(21,23)(22,24), (1,3)(2,4)(5,7)(6,8)(9,21)(10,22)(11,23)(12,24)(13,15)(14,16)(17,19)(18,20), (1,24,15)(2,22,13)(3,10,20)(4,12,18)(5,21,16)(6,23,14)(7,11,17)(8,9,19)>;
G:=Group( (1,2)(3,4)(5,6)(7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,5,2,6)(3,7,4,8)(13,16,15,14)(17,20,19,18), (1,4)(2,3)(5,8)(6,7)(9,11)(10,12)(13,18)(14,19)(15,20)(16,17)(21,23)(22,24), (1,3)(2,4)(5,7)(6,8)(9,21)(10,22)(11,23)(12,24)(13,15)(14,16)(17,19)(18,20), (1,24,15)(2,22,13)(3,10,20)(4,12,18)(5,21,16)(6,23,14)(7,11,17)(8,9,19) );
G=PermutationGroup([(1,2),(3,4),(5,6),(7,8),(9,10,11,12),(13,14,15,16),(17,18,19,20),(21,22,23,24)], [(1,5,2,6),(3,7,4,8),(13,16,15,14),(17,20,19,18)], [(1,4),(2,3),(5,8),(6,7),(9,11),(10,12),(13,18),(14,19),(15,20),(16,17),(21,23),(22,24)], [(1,3),(2,4),(5,7),(6,8),(9,21),(10,22),(11,23),(12,24),(13,15),(14,16),(17,19),(18,20)], [(1,24,15),(2,22,13),(3,10,20),(4,12,18),(5,21,16),(6,23,14),(7,11,17),(8,9,19)])
G:=TransitiveGroup(24,388);
Matrix representation ►G ⊆ GL6(𝔽13)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 5 | 0 |
| 0 | 0 | 0 | 0 | 0 | 8 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 5 | 0 | 0 |
| 0 | 0 | 0 | 0 | 5 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 1 | 0 | 0 |
G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,5,0,0,0,0,0,0,8],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,5,0,0,0,0,0,0,5,0,0,0,0,0,0,12],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,12],[12,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[0,0,1,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1,0] >;
Character table of C42⋊2A4
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 4P | |
| size | 1 | 3 | 3 | 3 | 3 | 3 | 64 | 64 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | |
| 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 | trivial |
| ρ2 | 1 | 1 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 3 |
| ρ3 | 1 | 1 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 3 |
| ρ4 | 3 | -1 | -1 | -1 | -1 | 3 | 0 | 0 | -1 | -1 | 3 | 3 | 3 | 3 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from A4 |
| ρ5 | 3 | -1 | -1 | -1 | -1 | 3 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 3 | 3 | 3 | 3 | -1 | -1 | orthogonal lifted from A4 |
| ρ6 | 3 | -1 | -1 | -1 | -1 | 3 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | 3 | 3 | 3 | 3 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from A4 |
| ρ7 | 3 | 3 | 3 | 3 | 3 | 3 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from A4 |
| ρ8 | 3 | -1 | -1 | -1 | -1 | 3 | 0 | 0 | 3 | 3 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 3 | 3 | orthogonal lifted from A4 |
| ρ9 | 3 | 3 | -1 | -1 | -1 | -1 | 0 | 0 | -1+2i | 1 | 1 | -1+2i | -1-2i | 1 | -1-2i | 1 | 1 | -1+2i | -1+2i | 1 | 1 | -1-2i | 1 | -1-2i | complex lifted from C42⋊C3 |
| ρ10 | 3 | -1 | 3 | -1 | -1 | -1 | 0 | 0 | 1 | -1-2i | -1+2i | 1 | 1 | -1-2i | -1+2i | 1 | 1 | -1-2i | -1+2i | 1 | 1 | -1-2i | -1+2i | 1 | complex lifted from C42⋊C3 |
| ρ11 | 3 | -1 | -1 | -1 | 3 | -1 | 0 | 0 | -1-2i | 1 | -1-2i | 1 | 1 | -1+2i | 1 | -1-2i | -1+2i | 1 | -1+2i | 1 | 1 | -1-2i | 1 | -1+2i | complex lifted from C42⋊C3 |
| ρ12 | 3 | 3 | -1 | -1 | -1 | -1 | 0 | 0 | 1 | -1-2i | -1-2i | 1 | 1 | -1+2i | 1 | -1+2i | -1-2i | 1 | 1 | -1-2i | -1+2i | 1 | -1+2i | 1 | complex lifted from C42⋊C3 |
| ρ13 | 3 | -1 | 3 | -1 | -1 | -1 | 0 | 0 | -1+2i | 1 | 1 | -1-2i | -1+2i | 1 | 1 | -1-2i | -1+2i | 1 | 1 | -1-2i | -1+2i | 1 | 1 | -1-2i | complex lifted from C42⋊C3 |
| ρ14 | 3 | -1 | -1 | 3 | -1 | -1 | 0 | 0 | 1 | -1-2i | 1 | -1+2i | -1-2i | 1 | 1 | -1-2i | -1+2i | 1 | -1-2i | 1 | 1 | -1+2i | -1+2i | 1 | complex lifted from C42⋊C3 |
| ρ15 | 3 | 3 | -1 | -1 | -1 | -1 | 0 | 0 | 1 | -1+2i | -1+2i | 1 | 1 | -1-2i | 1 | -1-2i | -1+2i | 1 | 1 | -1+2i | -1-2i | 1 | -1-2i | 1 | complex lifted from C42⋊C3 |
| ρ16 | 3 | -1 | -1 | -1 | 3 | -1 | 0 | 0 | -1+2i | 1 | -1+2i | 1 | 1 | -1-2i | 1 | -1+2i | -1-2i | 1 | -1-2i | 1 | 1 | -1+2i | 1 | -1-2i | complex lifted from C42⋊C3 |
| ρ17 | 3 | 3 | -1 | -1 | -1 | -1 | 0 | 0 | -1-2i | 1 | 1 | -1-2i | -1+2i | 1 | -1+2i | 1 | 1 | -1-2i | -1-2i | 1 | 1 | -1+2i | 1 | -1+2i | complex lifted from C42⋊C3 |
| ρ18 | 3 | -1 | -1 | 3 | -1 | -1 | 0 | 0 | -1-2i | 1 | -1+2i | 1 | 1 | -1-2i | -1-2i | 1 | 1 | -1+2i | 1 | -1-2i | -1+2i | 1 | 1 | -1+2i | complex lifted from C42⋊C3 |
| ρ19 | 3 | -1 | -1 | 3 | -1 | -1 | 0 | 0 | -1+2i | 1 | -1-2i | 1 | 1 | -1+2i | -1+2i | 1 | 1 | -1-2i | 1 | -1+2i | -1-2i | 1 | 1 | -1-2i | complex lifted from C42⋊C3 |
| ρ20 | 3 | -1 | -1 | -1 | 3 | -1 | 0 | 0 | 1 | -1+2i | 1 | -1+2i | -1-2i | 1 | -1+2i | 1 | 1 | -1-2i | 1 | -1-2i | -1+2i | 1 | -1-2i | 1 | complex lifted from C42⋊C3 |
| ρ21 | 3 | -1 | 3 | -1 | -1 | -1 | 0 | 0 | -1-2i | 1 | 1 | -1+2i | -1-2i | 1 | 1 | -1+2i | -1-2i | 1 | 1 | -1+2i | -1-2i | 1 | 1 | -1+2i | complex lifted from C42⋊C3 |
| ρ22 | 3 | -1 | -1 | 3 | -1 | -1 | 0 | 0 | 1 | -1+2i | 1 | -1-2i | -1+2i | 1 | 1 | -1+2i | -1-2i | 1 | -1+2i | 1 | 1 | -1-2i | -1-2i | 1 | complex lifted from C42⋊C3 |
| ρ23 | 3 | -1 | 3 | -1 | -1 | -1 | 0 | 0 | 1 | -1+2i | -1-2i | 1 | 1 | -1+2i | -1-2i | 1 | 1 | -1+2i | -1-2i | 1 | 1 | -1+2i | -1-2i | 1 | complex lifted from C42⋊C3 |
| ρ24 | 3 | -1 | -1 | -1 | 3 | -1 | 0 | 0 | 1 | -1-2i | 1 | -1-2i | -1+2i | 1 | -1-2i | 1 | 1 | -1+2i | 1 | -1+2i | -1-2i | 1 | -1+2i | 1 | complex lifted from C42⋊C3 |
In GAP, Magma, Sage, TeX
C_4^2\rtimes_2A_4
% in TeX
G:=Group("C4^2:2A4"); // GroupNames label
G:=SmallGroup(192,1020);
// by ID
G=gap.SmallGroup(192,1020);
# by ID
G:=PCGroup([7,-3,-2,2,-2,2,-2,2,85,680,2207,184,675,1264,4037,7062]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^4=c^2=d^2=e^3=1,a*b=b*a,a*c=c*a,a*d=d*a,e*a*e^-1=a*b^-1,b*c=c*b,b*d=d*b,e*b*e^-1=a^-1*b^2,e*c*e^-1=c*d=d*c,e*d*e^-1=c>;
// generators/relations
×
⋊
ℤ
𝔽
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