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G = C2×D24order 96 = 25·3

Direct product of C2 and D24

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

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

C1C12 — C2×D24
C1C3C6C12D12C2×D12 — C2×D24
C3C6C12 — C2×D24
C1C22C2×C4C2×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 [×2], C2 [×4], C3, C4 [×2], C22, C22 [×8], S3 [×4], C6, C6 [×2], C8 [×2], C2×C4, D4 [×6], C23 [×2], C12 [×2], D6 [×8], C2×C6, C2×C8, D8 [×4], C2×D4 [×2], C24 [×2], D12 [×4], D12 [×2], C2×C12, C22×S3 [×2], C2×D8, D24 [×4], C2×C24, C2×D12 [×2], C2×D24
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D6 [×3], D8 [×2], C2×D4, D12 [×2], C22×S3, C2×D8, D24 [×2], C2×D12, C2×D24

Character table of C2×D24

 class 12A2B2C2D2E2F2G34A4B6A6B6C8A8B8C8D12A12B12C12D24A24B24C24D24E24F24G24H
 size 1111121212122222222222222222222222
ρ1111111111111111111111111111111    trivial
ρ21-1-1111-1-11-11-1-11-111-1-1-111-1-1111-1-11    linear of order 2
ρ311111-11-1111111-1-1-1-11111-1-1-1-1-1-1-1-1    linear of order 2
ρ41-1-111-1-111-11-1-111-1-11-1-11111-1-1-111-1    linear of order 2
ρ51111-11-11111111-1-1-1-11111-1-1-1-1-1-1-1-1    linear of order 2
ρ61-1-11-111-11-11-1-111-1-11-1-11111-1-1-111-1    linear of order 2
ρ71-1-11-1-1111-11-1-11-111-1-1-111-1-1111-1-11    linear of order 2
ρ81111-1-1-1-11111111111111111111111    linear of order 2
ρ922220000-122-1-1-1-2-2-2-2-1-1-1-111111111    orthogonal lifted from D6
ρ102-2-220000-1-2211-12-2-2211-1-1-1-1111-1-11    orthogonal lifted from D6
ρ11222200002-2-22220000-2-2-2-200000000    orthogonal lifted from D4
ρ122-2-22000022-2-2-22000022-2-200000000    orthogonal lifted from D4
ρ1322220000-122-1-1-12222-1-1-1-1-1-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ142-2-220000-1-2211-1-222-211-1-111-1-1-111-1    orthogonal lifted from D6
ρ1522-2-200002002-2-2-2-2220000-2-222-222-2    orthogonal lifted from D8
ρ162-22-20000200-22-22-22-200002222-2-2-2-2    orthogonal lifted from D8
ρ172-22-20000200-22-2-22-220000-2-2-2-22222    orthogonal lifted from D8
ρ1822-2-200002002-2-222-2-2000022-2-22-2-22    orthogonal lifted from D8
ρ1922220000-1-2-2-1-1-1000011113-33-333-3-3    orthogonal lifted from D12
ρ202-2-220000-12-211-10000-1-111-333-33-33-3    orthogonal lifted from D12
ρ2122220000-1-2-2-1-1-100001111-33-33-3-333    orthogonal lifted from D12
ρ222-2-220000-12-211-10000-1-1113-3-33-33-33    orthogonal lifted from D12
ρ232-22-20000-1001-112-22-2-33-33ζ87ζ385ζ385ζ83ζ3838ζ3ζ87ζ385ζ385ζ83ζ3838ζ3ζ87ζ328785ζ32ζ87ζ328785ζ32ζ83ζ328ζ328ζ83ζ328ζ328    orthogonal lifted from D24
ρ242-22-20000-1001-11-22-223-33-3ζ83ζ328ζ328ζ87ζ328785ζ32ζ83ζ328ζ328ζ87ζ328785ζ32ζ83ζ3838ζ3ζ83ζ3838ζ3ζ87ζ385ζ385ζ87ζ385ζ385    orthogonal lifted from D24
ρ2522-2-20000-100-111-2-2223-3-33ζ87ζ328785ζ32ζ83ζ328ζ328ζ87ζ385ζ385ζ83ζ3838ζ3ζ87ζ328785ζ32ζ87ζ385ζ385ζ83ζ3838ζ3ζ83ζ328ζ328    orthogonal lifted from D24
ρ262-22-20000-1001-11-22-22-33-33ζ87ζ328785ζ32ζ83ζ328ζ328ζ87ζ328785ζ32ζ83ζ328ζ328ζ87ζ385ζ385ζ87ζ385ζ385ζ83ζ3838ζ3ζ83ζ3838ζ3    orthogonal lifted from D24
ρ2722-2-20000-100-11122-2-23-3-33ζ87ζ385ζ385ζ83ζ3838ζ3ζ87ζ328785ζ32ζ83ζ328ζ328ζ87ζ385ζ385ζ87ζ328785ζ32ζ83ζ328ζ328ζ83ζ3838ζ3    orthogonal lifted from D24
ρ282-22-20000-1001-112-22-23-33-3ζ83ζ3838ζ3ζ87ζ385ζ385ζ83ζ3838ζ3ζ87ζ385ζ385ζ83ζ328ζ328ζ83ζ328ζ328ζ87ζ328785ζ32ζ87ζ328785ζ32    orthogonal lifted from D24
ρ2922-2-20000-100-11122-2-2-333-3ζ83ζ3838ζ3ζ87ζ385ζ385ζ83ζ328ζ328ζ87ζ328785ζ32ζ83ζ3838ζ3ζ83ζ328ζ328ζ87ζ328785ζ32ζ87ζ385ζ385    orthogonal lifted from D24
ρ3022-2-20000-100-111-2-222-333-3ζ83ζ328ζ328ζ87ζ328785ζ32ζ83ζ3838ζ3ζ87ζ385ζ385ζ83ζ328ζ328ζ83ζ3838ζ3ζ87ζ385ζ385ζ87ζ328785ζ32    orthogonal lifted from D24

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

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

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

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

C2×D24 is a maximal subgroup of
C6.D16  D24.C4  C2.D48  M5(2)⋊S3  C124D8  C8.8D12  D24⋊C4  C8⋊D12  D1213D4  D1214D4  D4⋊D12  D123D4  Q84D12  D12.12D4  C4⋊D24  D12.19D4  C247D4  D249C4  Dic35D8  D62D8  C24.19D4  C16⋊D6  C2429D4  C243D4  Q8.9D12  C245D4  C249D4  C24.28D4  Q16⋊D6  D4.12D12  C2×S3×D8  D815D6
C2×D24 is a maximal quotient of
C248Q8  C4.5D24  C124D8  D1213D4  C22.D24  C4⋊D24  D124Q8  D487C2  C16⋊D6  C16.D6  C2429D4

Matrix representation of C2×D24 in GL5(𝔽73)

720000
01000
00100
000720
000072
,
720000
00100
0727200
000048
0003841
,
720000
00100
01000
00010
0004872

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

Export

Character table of C2×D24 in TeX

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