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G = C2×D24order 96 = 25·3

Direct product of C2 and D24

Aliases: C2×D24, C87D6, C61D8, C4.7D12, C248C22, C12.30D4, D123C22, C12.29C23, C22.13D12, C31(C2×D8), (C2×C8)⋊3S3, (C2×C24)⋊5C2, (C2×D12)⋊5C2, (C2×C4).80D6, (C2×C6).17D4, C6.10(C2×D4), C2.12(C2×D12), C4.27(C22×S3), (C2×C12).89C22, SmallGroup(96,110)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — C2×D24
 Chief series C1 — C3 — C6 — C12 — D12 — C2×D12 — C2×D24
 Lower central C3 — C6 — C12 — C2×D24
 Upper central C1 — C22 — C2×C4 — C2×C8

Generators and relations for C2×D24
G = < a,b,c | a2=b24=c2=1, ab=ba, ac=ca, cbc=b-1 >

Subgroups: 258 in 76 conjugacy classes, 33 normal (15 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, S3, C6, C6, C8, C2×C4, D4, C23, C12, D6, C2×C6, C2×C8, D8, C2×D4, C24, D12, D12, C2×C12, C22×S3, C2×D8, D24, C2×C24, C2×D12, C2×D24
Quotients: C1, C2, C22, S3, D4, C23, D6, D8, C2×D4, D12, C22×S3, C2×D8, D24, C2×D12, C2×D24

Character table of C2×D24

 class 1 2A 2B 2C 2D 2E 2F 2G 3 4A 4B 6A 6B 6C 8A 8B 8C 8D 12A 12B 12C 12D 24A 24B 24C 24D 24E 24F 24G 24H size 1 1 1 1 12 12 12 12 2 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 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 2 0 0 0 0 -1 2 2 -1 -1 -1 -2 -2 -2 -2 -1 -1 -1 -1 1 1 1 1 1 1 1 1 orthogonal lifted from D6 ρ10 2 -2 -2 2 0 0 0 0 -1 -2 2 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 2 0 0 0 0 2 -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 ρ12 2 -2 -2 2 0 0 0 0 2 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 ρ13 2 2 2 2 0 0 0 0 -1 2 2 -1 -1 -1 2 2 2 2 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ14 2 -2 -2 2 0 0 0 0 -1 -2 2 1 1 -1 -2 2 2 -2 1 1 -1 -1 1 1 -1 -1 -1 1 1 -1 orthogonal lifted from D6 ρ15 2 2 -2 -2 0 0 0 0 2 0 0 2 -2 -2 -√2 -√2 √2 √2 0 0 0 0 -√2 -√2 √2 √2 -√2 √2 √2 -√2 orthogonal lifted from D8 ρ16 2 -2 2 -2 0 0 0 0 2 0 0 -2 2 -2 √2 -√2 √2 -√2 0 0 0 0 √2 √2 √2 √2 -√2 -√2 -√2 -√2 orthogonal lifted from D8 ρ17 2 -2 2 -2 0 0 0 0 2 0 0 -2 2 -2 -√2 √2 -√2 √2 0 0 0 0 -√2 -√2 -√2 -√2 √2 √2 √2 √2 orthogonal lifted from D8 ρ18 2 2 -2 -2 0 0 0 0 2 0 0 2 -2 -2 √2 √2 -√2 -√2 0 0 0 0 √2 √2 -√2 -√2 √2 -√2 -√2 √2 orthogonal lifted from D8 ρ19 2 2 2 2 0 0 0 0 -1 -2 -2 -1 -1 -1 0 0 0 0 1 1 1 1 √3 -√3 √3 -√3 √3 √3 -√3 -√3 orthogonal lifted from D12 ρ20 2 -2 -2 2 0 0 0 0 -1 2 -2 1 1 -1 0 0 0 0 -1 -1 1 1 -√3 √3 √3 -√3 √3 -√3 √3 -√3 orthogonal lifted from D12 ρ21 2 2 2 2 0 0 0 0 -1 -2 -2 -1 -1 -1 0 0 0 0 1 1 1 1 -√3 √3 -√3 √3 -√3 -√3 √3 √3 orthogonal lifted from D12 ρ22 2 -2 -2 2 0 0 0 0 -1 2 -2 1 1 -1 0 0 0 0 -1 -1 1 1 √3 -√3 -√3 √3 -√3 √3 -√3 √3 orthogonal lifted from D12 ρ23 2 -2 2 -2 0 0 0 0 -1 0 0 1 -1 1 √2 -√2 √2 -√2 -√3 √3 -√3 √3 ζ87ζ3+ζ85ζ3+ζ85 ζ83ζ3+ζ83+ζ8ζ3 ζ87ζ3+ζ85ζ3+ζ85 ζ83ζ3+ζ83+ζ8ζ3 ζ87ζ32+ζ87+ζ85ζ32 ζ87ζ32+ζ87+ζ85ζ32 ζ83ζ32+ζ8ζ32+ζ8 ζ83ζ32+ζ8ζ32+ζ8 orthogonal lifted from D24 ρ24 2 -2 2 -2 0 0 0 0 -1 0 0 1 -1 1 -√2 √2 -√2 √2 √3 -√3 √3 -√3 ζ83ζ32+ζ8ζ32+ζ8 ζ87ζ32+ζ87+ζ85ζ32 ζ83ζ32+ζ8ζ32+ζ8 ζ87ζ32+ζ87+ζ85ζ32 ζ83ζ3+ζ83+ζ8ζ3 ζ83ζ3+ζ83+ζ8ζ3 ζ87ζ3+ζ85ζ3+ζ85 ζ87ζ3+ζ85ζ3+ζ85 orthogonal lifted from D24 ρ25 2 2 -2 -2 0 0 0 0 -1 0 0 -1 1 1 -√2 -√2 √2 √2 √3 -√3 -√3 √3 ζ87ζ32+ζ87+ζ85ζ32 ζ83ζ32+ζ8ζ32+ζ8 ζ87ζ3+ζ85ζ3+ζ85 ζ83ζ3+ζ83+ζ8ζ3 ζ87ζ32+ζ87+ζ85ζ32 ζ87ζ3+ζ85ζ3+ζ85 ζ83ζ3+ζ83+ζ8ζ3 ζ83ζ32+ζ8ζ32+ζ8 orthogonal lifted from D24 ρ26 2 -2 2 -2 0 0 0 0 -1 0 0 1 -1 1 -√2 √2 -√2 √2 -√3 √3 -√3 √3 ζ87ζ32+ζ87+ζ85ζ32 ζ83ζ32+ζ8ζ32+ζ8 ζ87ζ32+ζ87+ζ85ζ32 ζ83ζ32+ζ8ζ32+ζ8 ζ87ζ3+ζ85ζ3+ζ85 ζ87ζ3+ζ85ζ3+ζ85 ζ83ζ3+ζ83+ζ8ζ3 ζ83ζ3+ζ83+ζ8ζ3 orthogonal lifted from D24 ρ27 2 2 -2 -2 0 0 0 0 -1 0 0 -1 1 1 √2 √2 -√2 -√2 √3 -√3 -√3 √3 ζ87ζ3+ζ85ζ3+ζ85 ζ83ζ3+ζ83+ζ8ζ3 ζ87ζ32+ζ87+ζ85ζ32 ζ83ζ32+ζ8ζ32+ζ8 ζ87ζ3+ζ85ζ3+ζ85 ζ87ζ32+ζ87+ζ85ζ32 ζ83ζ32+ζ8ζ32+ζ8 ζ83ζ3+ζ83+ζ8ζ3 orthogonal lifted from D24 ρ28 2 -2 2 -2 0 0 0 0 -1 0 0 1 -1 1 √2 -√2 √2 -√2 √3 -√3 √3 -√3 ζ83ζ3+ζ83+ζ8ζ3 ζ87ζ3+ζ85ζ3+ζ85 ζ83ζ3+ζ83+ζ8ζ3 ζ87ζ3+ζ85ζ3+ζ85 ζ83ζ32+ζ8ζ32+ζ8 ζ83ζ32+ζ8ζ32+ζ8 ζ87ζ32+ζ87+ζ85ζ32 ζ87ζ32+ζ87+ζ85ζ32 orthogonal lifted from D24 ρ29 2 2 -2 -2 0 0 0 0 -1 0 0 -1 1 1 √2 √2 -√2 -√2 -√3 √3 √3 -√3 ζ83ζ3+ζ83+ζ8ζ3 ζ87ζ3+ζ85ζ3+ζ85 ζ83ζ32+ζ8ζ32+ζ8 ζ87ζ32+ζ87+ζ85ζ32 ζ83ζ3+ζ83+ζ8ζ3 ζ83ζ32+ζ8ζ32+ζ8 ζ87ζ32+ζ87+ζ85ζ32 ζ87ζ3+ζ85ζ3+ζ85 orthogonal lifted from D24 ρ30 2 2 -2 -2 0 0 0 0 -1 0 0 -1 1 1 -√2 -√2 √2 √2 -√3 √3 √3 -√3 ζ83ζ32+ζ8ζ32+ζ8 ζ87ζ32+ζ87+ζ85ζ32 ζ83ζ3+ζ83+ζ8ζ3 ζ87ζ3+ζ85ζ3+ζ85 ζ83ζ32+ζ8ζ32+ζ8 ζ83ζ3+ζ83+ζ8ζ3 ζ87ζ3+ζ85ζ3+ζ85 ζ87ζ32+ζ87+ζ85ζ32 orthogonal lifted from D24

Smallest permutation representation of C2×D24
On 48 points
Generators in S48
(1 26)(2 27)(3 28)(4 29)(5 30)(6 31)(7 32)(8 33)(9 34)(10 35)(11 36)(12 37)(13 38)(14 39)(15 40)(16 41)(17 42)(18 43)(19 44)(20 45)(21 46)(22 47)(23 48)(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)
(1 9)(2 8)(3 7)(4 6)(10 24)(11 23)(12 22)(13 21)(14 20)(15 19)(16 18)(25 35)(26 34)(27 33)(28 32)(29 31)(36 48)(37 47)(38 46)(39 45)(40 44)(41 43)

G:=sub<Sym(48)| (1,26)(2,27)(3,28)(4,29)(5,30)(6,31)(7,32)(8,33)(9,34)(10,35)(11,36)(12,37)(13,38)(14,39)(15,40)(16,41)(17,42)(18,43)(19,44)(20,45)(21,46)(22,47)(23,48)(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), (1,9)(2,8)(3,7)(4,6)(10,24)(11,23)(12,22)(13,21)(14,20)(15,19)(16,18)(25,35)(26,34)(27,33)(28,32)(29,31)(36,48)(37,47)(38,46)(39,45)(40,44)(41,43)>;

G:=Group( (1,26)(2,27)(3,28)(4,29)(5,30)(6,31)(7,32)(8,33)(9,34)(10,35)(11,36)(12,37)(13,38)(14,39)(15,40)(16,41)(17,42)(18,43)(19,44)(20,45)(21,46)(22,47)(23,48)(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), (1,9)(2,8)(3,7)(4,6)(10,24)(11,23)(12,22)(13,21)(14,20)(15,19)(16,18)(25,35)(26,34)(27,33)(28,32)(29,31)(36,48)(37,47)(38,46)(39,45)(40,44)(41,43) );

G=PermutationGroup([[(1,26),(2,27),(3,28),(4,29),(5,30),(6,31),(7,32),(8,33),(9,34),(10,35),(11,36),(12,37),(13,38),(14,39),(15,40),(16,41),(17,42),(18,43),(19,44),(20,45),(21,46),(22,47),(23,48),(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)], [(1,9),(2,8),(3,7),(4,6),(10,24),(11,23),(12,22),(13,21),(14,20),(15,19),(16,18),(25,35),(26,34),(27,33),(28,32),(29,31),(36,48),(37,47),(38,46),(39,45),(40,44),(41,43)]])

Matrix representation of C2×D24 in GL5(𝔽73)

 72 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 72 0 0 0 0 0 72
,
 72 0 0 0 0 0 0 1 0 0 0 72 72 0 0 0 0 0 0 48 0 0 0 38 41
,
 72 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 48 72

G:=sub<GL(5,GF(73))| [72,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,72,0,0,0,0,0,72],[72,0,0,0,0,0,0,72,0,0,0,1,72,0,0,0,0,0,0,38,0,0,0,48,41],[72,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1,48,0,0,0,0,72] >;

C2×D24 in GAP, Magma, Sage, TeX

C_2\times D_{24}
% in TeX

G:=Group("C2xD24");
// GroupNames label

G:=SmallGroup(96,110);
// by ID

G=gap.SmallGroup(96,110);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-3,218,122,579,69,2309]);
// Polycyclic

G:=Group<a,b,c|a^2=b^24=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

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