metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C4○D24, D24⋊7C2, C4○Dic12, C8.17D6, C4.20D12, C12.35D4, Dic12⋊7C2, C22.1D12, C12.30C23, C24.17C22, D12.7C22, Dic6.6C22, (C2×C8)⋊4S3, (C2×C24)⋊6C2, C3⋊1(C4○D8), C4○(C24⋊C2), C24⋊C2⋊7C2, C4○D12⋊1C2, C6.11(C2×D4), (C2×C6).18D4, (C2×C4).81D6, C2.13(C2×D12), C4.28(C22×S3), (C2×C12).99C22, SmallGroup(96,111)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4○D24
G = < a,b,c | a4=c2=1, b12=a2, ab=ba, ac=ca, cbc=a2b11 >
Subgroups: 162 in 62 conjugacy classes, 29 normal (21 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, D4, Q8, Dic3, C12, D6, C2×C6, C2×C8, D8, SD16, Q16, C4○D4, C24, Dic6, C4×S3, D12, C3⋊D4, C2×C12, C4○D8, C24⋊C2, D24, Dic12, C2×C24, C4○D12, C4○D24
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, D12, C22×S3, C4○D8, C2×D12, C4○D24
Character table of C4○D24
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | 4B | 4C | 4D | 4E | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 24A | 24B | 24C | 24D | 24E | 24F | 24G | 24H | |
| size | 1 | 1 | 2 | 12 | 12 | 2 | 1 | 1 | 2 | 12 | 12 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | |
| 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 | 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 | -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 | 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 | -1 | -1 | -1 | linear of order 2 |
| ρ5 | 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 | 1 | -1 | 1 | linear of order 2 |
| ρ6 | 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 | -1 | -1 | -1 | linear of order 2 |
| ρ7 | 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 | 1 | 1 | 1 | linear of order 2 |
| ρ8 | 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 | -1 | 1 | -1 | linear of order 2 |
| ρ9 | 2 | 2 | 2 | 0 | 0 | 2 | -2 | -2 | -2 | 0 | 0 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ10 | 2 | 2 | 2 | 0 | 0 | -1 | 2 | 2 | 2 | 0 | 0 | -1 | -1 | -1 | -2 | -2 | -2 | -2 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | orthogonal lifted from D6 |
| ρ11 | 2 | 2 | -2 | 0 | 0 | -1 | -2 | -2 | 2 | 0 | 0 | -1 | 1 | 1 | 2 | -2 | -2 | 2 | -1 | -1 | 1 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | 1 | -1 | orthogonal lifted from D6 |
| ρ12 | 2 | 2 | -2 | 0 | 0 | -1 | -2 | -2 | 2 | 0 | 0 | -1 | 1 | 1 | -2 | 2 | 2 | -2 | -1 | -1 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | -1 | 1 | orthogonal lifted from D6 |
| ρ13 | 2 | 2 | -2 | 0 | 0 | 2 | 2 | 2 | -2 | 0 | 0 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | -2 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ14 | 2 | 2 | 2 | 0 | 0 | -1 | 2 | 2 | 2 | 0 | 0 | -1 | -1 | -1 | 2 | 2 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ15 | 2 | 2 | -2 | 0 | 0 | -1 | 2 | 2 | -2 | 0 | 0 | -1 | 1 | 1 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | -1 | -√3 | √3 | √3 | -√3 | -√3 | √3 | √3 | -√3 | orthogonal lifted from D12 |
| ρ16 | 2 | 2 | 2 | 0 | 0 | -1 | -2 | -2 | -2 | 0 | 0 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | -√3 | -√3 | √3 | -√3 | √3 | -√3 | √3 | √3 | orthogonal lifted from D12 |
| ρ17 | 2 | 2 | 2 | 0 | 0 | -1 | -2 | -2 | -2 | 0 | 0 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | √3 | √3 | -√3 | √3 | -√3 | √3 | -√3 | -√3 | orthogonal lifted from D12 |
| ρ18 | 2 | 2 | -2 | 0 | 0 | -1 | 2 | 2 | -2 | 0 | 0 | -1 | 1 | 1 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | -1 | √3 | -√3 | -√3 | √3 | √3 | -√3 | -√3 | √3 | orthogonal lifted from D12 |
| ρ19 | 2 | -2 | 0 | 0 | 0 | 2 | -2i | 2i | 0 | 0 | 0 | -2 | 0 | 0 | √-2 | -√2 | √2 | -√-2 | 0 | 0 | 2i | -2i | √2 | -√-2 | -√2 | -√2 | -√-2 | √-2 | √2 | √-2 | complex lifted from C4○D8 |
| ρ20 | 2 | -2 | 0 | 0 | 0 | 2 | 2i | -2i | 0 | 0 | 0 | -2 | 0 | 0 | √-2 | √2 | -√2 | -√-2 | 0 | 0 | -2i | 2i | -√2 | -√-2 | √2 | √2 | -√-2 | √-2 | -√2 | √-2 | complex lifted from C4○D8 |
| ρ21 | 2 | -2 | 0 | 0 | 0 | 2 | 2i | -2i | 0 | 0 | 0 | -2 | 0 | 0 | -√-2 | -√2 | √2 | √-2 | 0 | 0 | -2i | 2i | √2 | √-2 | -√2 | -√2 | √-2 | -√-2 | √2 | -√-2 | complex lifted from C4○D8 |
| ρ22 | 2 | -2 | 0 | 0 | 0 | 2 | -2i | 2i | 0 | 0 | 0 | -2 | 0 | 0 | -√-2 | √2 | -√2 | √-2 | 0 | 0 | 2i | -2i | -√2 | √-2 | √2 | √2 | √-2 | -√-2 | -√2 | -√-2 | complex lifted from C4○D8 |
| ρ23 | 2 | -2 | 0 | 0 | 0 | -1 | -2i | 2i | 0 | 0 | 0 | 1 | -√-3 | √-3 | -√-2 | √2 | -√2 | √-2 | √3 | -√3 | -i | i | ζ83ζ3+ζ8ζ3+ζ8 | ζ87ζ32+ζ87-ζ85ζ32 | ζ87ζ32+ζ85ζ32+ζ85 | ζ83ζ32+ζ83+ζ8ζ32 | ζ87ζ3+ζ87-ζ85ζ3 | ζ83ζ32+ζ83-ζ8ζ32 | ζ87ζ3+ζ87+ζ85ζ3 | ζ83ζ3+ζ83-ζ8ζ3 | complex faithful |
| ρ24 | 2 | -2 | 0 | 0 | 0 | -1 | -2i | 2i | 0 | 0 | 0 | 1 | -√-3 | √-3 | √-2 | -√2 | √2 | -√-2 | √3 | -√3 | -i | i | ζ83ζ32+ζ83+ζ8ζ32 | ζ83ζ32+ζ83-ζ8ζ32 | ζ87ζ3+ζ87+ζ85ζ3 | ζ83ζ3+ζ8ζ3+ζ8 | ζ83ζ3+ζ83-ζ8ζ3 | ζ87ζ32+ζ87-ζ85ζ32 | ζ87ζ32+ζ85ζ32+ζ85 | ζ87ζ3+ζ87-ζ85ζ3 | complex faithful |
| ρ25 | 2 | -2 | 0 | 0 | 0 | -1 | -2i | 2i | 0 | 0 | 0 | 1 | √-3 | -√-3 | √-2 | -√2 | √2 | -√-2 | -√3 | √3 | -i | i | ζ87ζ32+ζ85ζ32+ζ85 | ζ83ζ3+ζ83-ζ8ζ3 | ζ83ζ3+ζ8ζ3+ζ8 | ζ87ζ3+ζ87+ζ85ζ3 | ζ83ζ32+ζ83-ζ8ζ32 | ζ87ζ3+ζ87-ζ85ζ3 | ζ83ζ32+ζ83+ζ8ζ32 | ζ87ζ32+ζ87-ζ85ζ32 | complex faithful |
| ρ26 | 2 | -2 | 0 | 0 | 0 | -1 | 2i | -2i | 0 | 0 | 0 | 1 | -√-3 | √-3 | √-2 | √2 | -√2 | -√-2 | -√3 | √3 | i | -i | ζ87ζ3+ζ87+ζ85ζ3 | ζ83ζ3+ζ83-ζ8ζ3 | ζ83ζ32+ζ83+ζ8ζ32 | ζ87ζ32+ζ85ζ32+ζ85 | ζ83ζ32+ζ83-ζ8ζ32 | ζ87ζ3+ζ87-ζ85ζ3 | ζ83ζ3+ζ8ζ3+ζ8 | ζ87ζ32+ζ87-ζ85ζ32 | complex faithful |
| ρ27 | 2 | -2 | 0 | 0 | 0 | -1 | 2i | -2i | 0 | 0 | 0 | 1 | √-3 | -√-3 | √-2 | √2 | -√2 | -√-2 | √3 | -√3 | i | -i | ζ83ζ3+ζ8ζ3+ζ8 | ζ83ζ32+ζ83-ζ8ζ32 | ζ87ζ32+ζ85ζ32+ζ85 | ζ83ζ32+ζ83+ζ8ζ32 | ζ83ζ3+ζ83-ζ8ζ3 | ζ87ζ32+ζ87-ζ85ζ32 | ζ87ζ3+ζ87+ζ85ζ3 | ζ87ζ3+ζ87-ζ85ζ3 | complex faithful |
| ρ28 | 2 | -2 | 0 | 0 | 0 | -1 | 2i | -2i | 0 | 0 | 0 | 1 | √-3 | -√-3 | -√-2 | -√2 | √2 | √-2 | √3 | -√3 | i | -i | ζ83ζ32+ζ83+ζ8ζ32 | ζ87ζ32+ζ87-ζ85ζ32 | ζ87ζ3+ζ87+ζ85ζ3 | ζ83ζ3+ζ8ζ3+ζ8 | ζ87ζ3+ζ87-ζ85ζ3 | ζ83ζ32+ζ83-ζ8ζ32 | ζ87ζ32+ζ85ζ32+ζ85 | ζ83ζ3+ζ83-ζ8ζ3 | complex faithful |
| ρ29 | 2 | -2 | 0 | 0 | 0 | -1 | 2i | -2i | 0 | 0 | 0 | 1 | -√-3 | √-3 | -√-2 | -√2 | √2 | √-2 | -√3 | √3 | i | -i | ζ87ζ32+ζ85ζ32+ζ85 | ζ87ζ3+ζ87-ζ85ζ3 | ζ83ζ3+ζ8ζ3+ζ8 | ζ87ζ3+ζ87+ζ85ζ3 | ζ87ζ32+ζ87-ζ85ζ32 | ζ83ζ3+ζ83-ζ8ζ3 | ζ83ζ32+ζ83+ζ8ζ32 | ζ83ζ32+ζ83-ζ8ζ32 | complex faithful |
| ρ30 | 2 | -2 | 0 | 0 | 0 | -1 | -2i | 2i | 0 | 0 | 0 | 1 | √-3 | -√-3 | -√-2 | √2 | -√2 | √-2 | -√3 | √3 | -i | i | ζ87ζ3+ζ87+ζ85ζ3 | ζ87ζ3+ζ87-ζ85ζ3 | ζ83ζ32+ζ83+ζ8ζ32 | ζ87ζ32+ζ85ζ32+ζ85 | ζ87ζ32+ζ87-ζ85ζ32 | ζ83ζ3+ζ83-ζ8ζ3 | ζ83ζ3+ζ8ζ3+ζ8 | ζ83ζ32+ζ83-ζ8ζ32 | complex faithful |
►On 48 points
(1 45 13 33)(2 46 14 34)(3 47 15 35)(4 48 16 36)(5 25 17 37)(6 26 18 38)(7 27 19 39)(8 28 20 40)(9 29 21 41)(10 30 22 42)(11 31 23 43)(12 32 24 44)
(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 15)(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)(16 24)(17 23)(18 22)(19 21)(25 31)(26 30)(27 29)(32 48)(33 47)(34 46)(35 45)(36 44)(37 43)(38 42)(39 41)
G:=sub<Sym(48)| (1,45,13,33)(2,46,14,34)(3,47,15,35)(4,48,16,36)(5,25,17,37)(6,26,18,38)(7,27,19,39)(8,28,20,40)(9,29,21,41)(10,30,22,42)(11,31,23,43)(12,32,24,44), (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), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(16,24)(17,23)(18,22)(19,21)(25,31)(26,30)(27,29)(32,48)(33,47)(34,46)(35,45)(36,44)(37,43)(38,42)(39,41)>;
G:=Group( (1,45,13,33)(2,46,14,34)(3,47,15,35)(4,48,16,36)(5,25,17,37)(6,26,18,38)(7,27,19,39)(8,28,20,40)(9,29,21,41)(10,30,22,42)(11,31,23,43)(12,32,24,44), (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), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(16,24)(17,23)(18,22)(19,21)(25,31)(26,30)(27,29)(32,48)(33,47)(34,46)(35,45)(36,44)(37,43)(38,42)(39,41) );
G=PermutationGroup([[(1,45,13,33),(2,46,14,34),(3,47,15,35),(4,48,16,36),(5,25,17,37),(6,26,18,38),(7,27,19,39),(8,28,20,40),(9,29,21,41),(10,30,22,42),(11,31,23,43),(12,32,24,44)], [(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)], [(1,15),(2,14),(3,13),(4,12),(5,11),(6,10),(7,9),(16,24),(17,23),(18,22),(19,21),(25,31),(26,30),(27,29),(32,48),(33,47),(34,46),(35,45),(36,44),(37,43),(38,42),(39,41)]])
C4○D24 is a maximal subgroup of
D24⋊8C4 Dic12.C4 D24.1C4 D24⋊2C4 D24⋊11C4 D24⋊4C4 D24⋊10C4 D24⋊7C4 D48⋊7C2 C16⋊D6 C16.D6 D8.D6 C24.27C23 Q16.D6 C24.9C23 D4.11D12 D4.12D12 D4.13D12 D8⋊13D6 SD16⋊13D6 D12.30D4 S3×C4○D8 SD16⋊D6 D72⋊7C2 D6.1D12 D24⋊7S3 D6.3D12 D12.27D6 C24.78D6 C40.31D6 D24⋊7D5 D120⋊C2 C20.60D12 C40.69D6
C4○D24 is a maximal quotient of
C24.13Q8 C4×C24⋊C2 C4×D24 C8.8D12 C42.264D6 C4×Dic12 C23.15D12 D12.32D4 D12⋊14D4 C23.18D12 Dic6.3Q8 D12.19D4 C42.36D6 D12.3Q8 C23.27D12 C23.28D12 C24⋊30D4 C24⋊29D4 C24.82D4 D72⋊7C2 D6.1D12 D24⋊7S3 D6.3D12 D12.27D6 C24.78D6 C40.31D6 D24⋊7D5 D120⋊C2 C20.60D12 C40.69D6
Matrix representation of C4○D24 ►in GL2(𝔽73) generated by
| 27 | 0 |
| 0 | 27 |
| 55 | 50 |
| 23 | 5 |
| 66 | 7 |
| 14 | 7 |
G:=sub<GL(2,GF(73))| [27,0,0,27],[55,23,50,5],[66,14,7,7] >;
C4○D24 in GAP, Magma, Sage, TeX
C_4\circ D_{24} % in TeX
G:=Group("C4oD24"); // GroupNames label
G:=SmallGroup(96,111);
// by ID
G=gap.SmallGroup(96,111);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-3,103,218,50,579,69,2309]);
// Polycyclic
G:=Group<a,b,c|a^4=c^2=1,b^12=a^2,a*b=b*a,a*c=c*a,c*b*c=a^2*b^11>;
// generators/relations
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