metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C48⋊2C2, C16⋊2S3, C6.2D8, C3⋊1SD32, C2.4D24, C8.14D6, C4.2D12, D24.1C2, C12.25D4, Dic12⋊1C2, C24.15C22, SmallGroup(96,7)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C48⋊C2
G = < a,b | a48=b2=1, bab=a23 >
Character table of C48⋊C2
| class | 1 | 2A | 2B | 3 | 4A | 4B | 6 | 8A | 8B | 12A | 12B | 16A | 16B | 16C | 16D | 24A | 24B | 24C | 24D | 48A | 48B | 48C | 48D | 48E | 48F | 48G | 48H | |
| size | 1 | 1 | 24 | 2 | 2 | 24 | 2 | 2 | 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 | 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 | 2 | 2 | 0 | 2 | 2 | 0 | 2 | -2 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | -2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ6 | 2 | 2 | 0 | -1 | 2 | 0 | -1 | 2 | 2 | -1 | -1 | -2 | -2 | -2 | -2 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | orthogonal lifted from D6 |
| ρ7 | 2 | 2 | 0 | -1 | 2 | 0 | -1 | 2 | 2 | -1 | -1 | 2 | 2 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ8 | 2 | 2 | 0 | 2 | -2 | 0 | 2 | 0 | 0 | -2 | -2 | -√2 | √2 | √2 | -√2 | 0 | 0 | 0 | 0 | √2 | -√2 | -√2 | √2 | -√2 | -√2 | √2 | √2 | orthogonal lifted from D8 |
| ρ9 | 2 | 2 | 0 | 2 | -2 | 0 | 2 | 0 | 0 | -2 | -2 | √2 | -√2 | -√2 | √2 | 0 | 0 | 0 | 0 | -√2 | √2 | √2 | -√2 | √2 | √2 | -√2 | -√2 | orthogonal lifted from D8 |
| ρ10 | 2 | 2 | 0 | -1 | 2 | 0 | -1 | -2 | -2 | -1 | -1 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | -√3 | √3 | -√3 | √3 | √3 | -√3 | √3 | -√3 | orthogonal lifted from D12 |
| ρ11 | 2 | 2 | 0 | -1 | 2 | 0 | -1 | -2 | -2 | -1 | -1 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | √3 | -√3 | √3 | -√3 | -√3 | √3 | -√3 | √3 | orthogonal lifted from D12 |
| ρ12 | 2 | 2 | 0 | -1 | -2 | 0 | -1 | 0 | 0 | 1 | 1 | √2 | -√2 | -√2 | √2 | √3 | -√3 | -√3 | √3 | ζ83ζ3+ζ8ζ3+ζ8 | ζ87ζ32+ζ85ζ32+ζ85 | ζ83ζ32+ζ83+ζ8ζ32 | ζ87ζ3+ζ87+ζ85ζ3 | ζ87ζ32+ζ85ζ32+ζ85 | ζ83ζ32+ζ83+ζ8ζ32 | ζ87ζ3+ζ87+ζ85ζ3 | ζ83ζ3+ζ8ζ3+ζ8 | orthogonal lifted from D24 |
| ρ13 | 2 | 2 | 0 | -1 | -2 | 0 | -1 | 0 | 0 | 1 | 1 | √2 | -√2 | -√2 | √2 | -√3 | √3 | √3 | -√3 | ζ87ζ3+ζ87+ζ85ζ3 | ζ83ζ32+ζ83+ζ8ζ32 | ζ87ζ32+ζ85ζ32+ζ85 | ζ83ζ3+ζ8ζ3+ζ8 | ζ83ζ32+ζ83+ζ8ζ32 | ζ87ζ32+ζ85ζ32+ζ85 | ζ83ζ3+ζ8ζ3+ζ8 | ζ87ζ3+ζ87+ζ85ζ3 | orthogonal lifted from D24 |
| ρ14 | 2 | 2 | 0 | -1 | -2 | 0 | -1 | 0 | 0 | 1 | 1 | -√2 | √2 | √2 | -√2 | -√3 | √3 | √3 | -√3 | ζ87ζ32+ζ85ζ32+ζ85 | ζ83ζ3+ζ8ζ3+ζ8 | ζ87ζ3+ζ87+ζ85ζ3 | ζ83ζ32+ζ83+ζ8ζ32 | ζ83ζ3+ζ8ζ3+ζ8 | ζ87ζ3+ζ87+ζ85ζ3 | ζ83ζ32+ζ83+ζ8ζ32 | ζ87ζ32+ζ85ζ32+ζ85 | orthogonal lifted from D24 |
| ρ15 | 2 | 2 | 0 | -1 | -2 | 0 | -1 | 0 | 0 | 1 | 1 | -√2 | √2 | √2 | -√2 | √3 | -√3 | -√3 | √3 | ζ83ζ32+ζ83+ζ8ζ32 | ζ87ζ3+ζ87+ζ85ζ3 | ζ83ζ3+ζ8ζ3+ζ8 | ζ87ζ32+ζ85ζ32+ζ85 | ζ87ζ3+ζ87+ζ85ζ3 | ζ83ζ3+ζ8ζ3+ζ8 | ζ87ζ32+ζ85ζ32+ζ85 | ζ83ζ32+ζ83+ζ8ζ32 | orthogonal lifted from D24 |
| ρ16 | 2 | -2 | 0 | 2 | 0 | 0 | -2 | -√2 | √2 | 0 | 0 | ζ1613+ζ1611 | ζ1615+ζ169 | ζ167+ζ16 | ζ165+ζ163 | √2 | √2 | -√2 | -√2 | ζ1615+ζ169 | ζ165+ζ163 | ζ165+ζ163 | ζ1615+ζ169 | ζ1613+ζ1611 | ζ1613+ζ1611 | ζ167+ζ16 | ζ167+ζ16 | complex lifted from SD32 |
| ρ17 | 2 | -2 | 0 | 2 | 0 | 0 | -2 | √2 | -√2 | 0 | 0 | ζ167+ζ16 | ζ1613+ζ1611 | ζ165+ζ163 | ζ1615+ζ169 | -√2 | -√2 | √2 | √2 | ζ1613+ζ1611 | ζ1615+ζ169 | ζ1615+ζ169 | ζ1613+ζ1611 | ζ167+ζ16 | ζ167+ζ16 | ζ165+ζ163 | ζ165+ζ163 | complex lifted from SD32 |
| ρ18 | 2 | -2 | 0 | 2 | 0 | 0 | -2 | -√2 | √2 | 0 | 0 | ζ165+ζ163 | ζ167+ζ16 | ζ1615+ζ169 | ζ1613+ζ1611 | √2 | √2 | -√2 | -√2 | ζ167+ζ16 | ζ1613+ζ1611 | ζ1613+ζ1611 | ζ167+ζ16 | ζ165+ζ163 | ζ165+ζ163 | ζ1615+ζ169 | ζ1615+ζ169 | complex lifted from SD32 |
| ρ19 | 2 | -2 | 0 | 2 | 0 | 0 | -2 | √2 | -√2 | 0 | 0 | ζ1615+ζ169 | ζ165+ζ163 | ζ1613+ζ1611 | ζ167+ζ16 | -√2 | -√2 | √2 | √2 | ζ165+ζ163 | ζ167+ζ16 | ζ167+ζ16 | ζ165+ζ163 | ζ1615+ζ169 | ζ1615+ζ169 | ζ1613+ζ1611 | ζ1613+ζ1611 | complex lifted from SD32 |
| ρ20 | 2 | -2 | 0 | -1 | 0 | 0 | 1 | √2 | -√2 | √3 | -√3 | ζ167+ζ16 | ζ1613+ζ1611 | ζ165+ζ163 | ζ1615+ζ169 | ζ1614ζ3+ζ1614+ζ1610ζ3 | ζ166ζ3+ζ162ζ3+ζ162 | ζ166ζ32+ζ166+ζ162ζ32 | ζ1614ζ32+ζ1610ζ32+ζ1610 | -ζ165ζ3+ζ163ζ3+ζ163 | ζ167ζ32+ζ167-ζ16ζ32 | -ζ167ζ32+ζ16ζ32+ζ16 | ζ165ζ3+ζ165-ζ163ζ3 | ζ1615ζ32+ζ1615-ζ169ζ32 | -ζ1615ζ32+ζ169ζ32+ζ169 | ζ1613ζ3+ζ1613-ζ1611ζ3 | -ζ1613ζ3+ζ1611ζ3+ζ1611 | complex faithful |
| ρ21 | 2 | -2 | 0 | -1 | 0 | 0 | 1 | -√2 | √2 | -√3 | √3 | ζ165+ζ163 | ζ167+ζ16 | ζ1615+ζ169 | ζ1613+ζ1611 | ζ166ζ32+ζ166+ζ162ζ32 | ζ1614ζ32+ζ1610ζ32+ζ1610 | ζ1614ζ3+ζ1614+ζ1610ζ3 | ζ166ζ3+ζ162ζ3+ζ162 | ζ1615ζ32+ζ1615-ζ169ζ32 | -ζ165ζ3+ζ163ζ3+ζ163 | ζ165ζ3+ζ165-ζ163ζ3 | -ζ1615ζ32+ζ169ζ32+ζ169 | -ζ1613ζ3+ζ1611ζ3+ζ1611 | ζ1613ζ3+ζ1613-ζ1611ζ3 | -ζ167ζ32+ζ16ζ32+ζ16 | ζ167ζ32+ζ167-ζ16ζ32 | complex faithful |
| ρ22 | 2 | -2 | 0 | -1 | 0 | 0 | 1 | -√2 | √2 | √3 | -√3 | ζ1613+ζ1611 | ζ1615+ζ169 | ζ167+ζ16 | ζ165+ζ163 | ζ1614ζ32+ζ1610ζ32+ζ1610 | ζ166ζ32+ζ166+ζ162ζ32 | ζ166ζ3+ζ162ζ3+ζ162 | ζ1614ζ3+ζ1614+ζ1610ζ3 | -ζ167ζ32+ζ16ζ32+ζ16 | ζ1613ζ3+ζ1613-ζ1611ζ3 | -ζ1613ζ3+ζ1611ζ3+ζ1611 | ζ167ζ32+ζ167-ζ16ζ32 | ζ165ζ3+ζ165-ζ163ζ3 | -ζ165ζ3+ζ163ζ3+ζ163 | ζ1615ζ32+ζ1615-ζ169ζ32 | -ζ1615ζ32+ζ169ζ32+ζ169 | complex faithful |
| ρ23 | 2 | -2 | 0 | -1 | 0 | 0 | 1 | √2 | -√2 | -√3 | √3 | ζ167+ζ16 | ζ1613+ζ1611 | ζ165+ζ163 | ζ1615+ζ169 | ζ166ζ3+ζ162ζ3+ζ162 | ζ1614ζ3+ζ1614+ζ1610ζ3 | ζ1614ζ32+ζ1610ζ32+ζ1610 | ζ166ζ32+ζ166+ζ162ζ32 | ζ165ζ3+ζ165-ζ163ζ3 | -ζ167ζ32+ζ16ζ32+ζ16 | ζ167ζ32+ζ167-ζ16ζ32 | -ζ165ζ3+ζ163ζ3+ζ163 | -ζ1615ζ32+ζ169ζ32+ζ169 | ζ1615ζ32+ζ1615-ζ169ζ32 | -ζ1613ζ3+ζ1611ζ3+ζ1611 | ζ1613ζ3+ζ1613-ζ1611ζ3 | complex faithful |
| ρ24 | 2 | -2 | 0 | -1 | 0 | 0 | 1 | -√2 | √2 | √3 | -√3 | ζ165+ζ163 | ζ167+ζ16 | ζ1615+ζ169 | ζ1613+ζ1611 | ζ1614ζ32+ζ1610ζ32+ζ1610 | ζ166ζ32+ζ166+ζ162ζ32 | ζ166ζ3+ζ162ζ3+ζ162 | ζ1614ζ3+ζ1614+ζ1610ζ3 | -ζ1615ζ32+ζ169ζ32+ζ169 | ζ165ζ3+ζ165-ζ163ζ3 | -ζ165ζ3+ζ163ζ3+ζ163 | ζ1615ζ32+ζ1615-ζ169ζ32 | ζ1613ζ3+ζ1613-ζ1611ζ3 | -ζ1613ζ3+ζ1611ζ3+ζ1611 | ζ167ζ32+ζ167-ζ16ζ32 | -ζ167ζ32+ζ16ζ32+ζ16 | complex faithful |
| ρ25 | 2 | -2 | 0 | -1 | 0 | 0 | 1 | -√2 | √2 | -√3 | √3 | ζ1613+ζ1611 | ζ1615+ζ169 | ζ167+ζ16 | ζ165+ζ163 | ζ166ζ32+ζ166+ζ162ζ32 | ζ1614ζ32+ζ1610ζ32+ζ1610 | ζ1614ζ3+ζ1614+ζ1610ζ3 | ζ166ζ3+ζ162ζ3+ζ162 | ζ167ζ32+ζ167-ζ16ζ32 | -ζ1613ζ3+ζ1611ζ3+ζ1611 | ζ1613ζ3+ζ1613-ζ1611ζ3 | -ζ167ζ32+ζ16ζ32+ζ16 | -ζ165ζ3+ζ163ζ3+ζ163 | ζ165ζ3+ζ165-ζ163ζ3 | -ζ1615ζ32+ζ169ζ32+ζ169 | ζ1615ζ32+ζ1615-ζ169ζ32 | complex faithful |
| ρ26 | 2 | -2 | 0 | -1 | 0 | 0 | 1 | √2 | -√2 | -√3 | √3 | ζ1615+ζ169 | ζ165+ζ163 | ζ1613+ζ1611 | ζ167+ζ16 | ζ166ζ3+ζ162ζ3+ζ162 | ζ1614ζ3+ζ1614+ζ1610ζ3 | ζ1614ζ32+ζ1610ζ32+ζ1610 | ζ166ζ32+ζ166+ζ162ζ32 | ζ1613ζ3+ζ1613-ζ1611ζ3 | -ζ1615ζ32+ζ169ζ32+ζ169 | ζ1615ζ32+ζ1615-ζ169ζ32 | -ζ1613ζ3+ζ1611ζ3+ζ1611 | -ζ167ζ32+ζ16ζ32+ζ16 | ζ167ζ32+ζ167-ζ16ζ32 | -ζ165ζ3+ζ163ζ3+ζ163 | ζ165ζ3+ζ165-ζ163ζ3 | complex faithful |
| ρ27 | 2 | -2 | 0 | -1 | 0 | 0 | 1 | √2 | -√2 | √3 | -√3 | ζ1615+ζ169 | ζ165+ζ163 | ζ1613+ζ1611 | ζ167+ζ16 | ζ1614ζ3+ζ1614+ζ1610ζ3 | ζ166ζ3+ζ162ζ3+ζ162 | ζ166ζ32+ζ166+ζ162ζ32 | ζ1614ζ32+ζ1610ζ32+ζ1610 | -ζ1613ζ3+ζ1611ζ3+ζ1611 | ζ1615ζ32+ζ1615-ζ169ζ32 | -ζ1615ζ32+ζ169ζ32+ζ169 | ζ1613ζ3+ζ1613-ζ1611ζ3 | ζ167ζ32+ζ167-ζ16ζ32 | -ζ167ζ32+ζ16ζ32+ζ16 | ζ165ζ3+ζ165-ζ163ζ3 | -ζ165ζ3+ζ163ζ3+ζ163 | complex faithful |
►On 48 points
Generators in S48
(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 24)(3 47)(4 22)(5 45)(6 20)(7 43)(8 18)(9 41)(10 16)(11 39)(12 14)(13 37)(15 35)(17 33)(19 31)(21 29)(23 27)(26 48)(28 46)(30 44)(32 42)(34 40)(36 38)
G:=sub<Sym(48)| (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), (2,24)(3,47)(4,22)(5,45)(6,20)(7,43)(8,18)(9,41)(10,16)(11,39)(12,14)(13,37)(15,35)(17,33)(19,31)(21,29)(23,27)(26,48)(28,46)(30,44)(32,42)(34,40)(36,38)>;
G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48), (2,24)(3,47)(4,22)(5,45)(6,20)(7,43)(8,18)(9,41)(10,16)(11,39)(12,14)(13,37)(15,35)(17,33)(19,31)(21,29)(23,27)(26,48)(28,46)(30,44)(32,42)(34,40)(36,38) );
G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48)], [(2,24),(3,47),(4,22),(5,45),(6,20),(7,43),(8,18),(9,41),(10,16),(11,39),(12,14),(13,37),(15,35),(17,33),(19,31),(21,29),(23,27),(26,48),(28,46),(30,44),(32,42),(34,40),(36,38)]])
C48⋊C2 is a maximal subgroup of
D48⋊7C2 C16⋊D6 C16.D6 D8⋊D6 S3×SD32 D6.2D8 Q32⋊S3 C144⋊C2 C32⋊3SD32 C24.49D6 C6.D24 D24.D5 Dic12⋊D5 C48⋊D5
C48⋊C2 is a maximal quotient of
C2.Dic24 C48⋊6C4 C2.D48 C144⋊C2 C32⋊3SD32 C24.49D6 C6.D24 D24.D5 Dic12⋊D5 C48⋊D5
Matrix representation of C48⋊C2 ►in GL2(𝔽23) generated by
| 0 | 1 |
| 1 | 22 |
| 12 | 9 |
| 2 | 11 |
G:=sub<GL(2,GF(23))| [0,1,1,22],[12,2,9,11] >;
C48⋊C2 in GAP, Magma, Sage, TeX
C_{48}\rtimes C_2 % in TeX
G:=Group("C48:C2"); // GroupNames label
G:=SmallGroup(96,7);
// by ID
G=gap.SmallGroup(96,7);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-3,73,79,506,50,579,69,2309]);
// Polycyclic
G:=Group<a,b|a^48=b^2=1,b*a*b=a^23>;
// generators/relations
Export
Subgroup lattice of C48⋊C2 in TeX
Character table of C48⋊C2 in TeX