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G = C2.D24order 96 = 25·3

2nd central extension by C2 of D24

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C6.5D8, D122C4, C2.2D24, C12.45D4, C6.3SD16, C22.10D12, (C2×C8)⋊2S3, (C2×C24)⋊2C2, C4.8(C4×S3), C4⋊Dic31C2, (C2×C4).71D6, (C2×C6).15D4, C32(D4⋊C4), C12.18(C2×C4), C2.8(D6⋊C4), (C2×D12).1C2, C2.3(C24⋊C2), C4.20(C3⋊D4), C6.7(C22⋊C4), (C2×C12).83C22, SmallGroup(96,28)

Series: Derived Chief Lower central Upper central

C1C12 — C2.D24
C1C3C6C12C2×C12C2×D12 — C2.D24
C3C6C12 — C2.D24
C1C22C2×C4C2×C8

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

12C2
12C2
6C22
6C22
12C22
12C22
12C4
4S3
4S3
2C8
3D4
3D4
6C2×C4
6C23
6D4
2D6
2D6
4Dic3
4D6
4D6
3C4⋊C4
3C2×D4
2C22×S3
2C2×Dic3
2C24
2D12
3D4⋊C4

Character table of C2.D24

 class 12A2B2C2D2E34A4B4C4D6A6B6C8A8B8C8D12A12B12C12D24A24B24C24D24E24F24G24H
 size 1111121222212122222222222222222222
ρ1111111111111111111111111111111    trivial
ρ21111-1-111111111-1-1-1-11111-1-1-1-1-1-1-1-1    linear of order 2
ρ3111111111-1-1111-1-1-1-11111-1-1-1-1-1-1-1-1    linear of order 2
ρ41111-1-1111-1-11111111111111111111    linear of order 2
ρ51-1-11-1111-1i-i-11-1-i-iii1-1-11-iii-iii-i-i    linear of order 4
ρ61-1-111-111-1i-i-11-1ii-i-i1-1-11i-i-ii-i-iii    linear of order 4
ρ71-1-11-1111-1-ii-11-1ii-i-i1-1-11i-i-ii-i-iii    linear of order 4
ρ81-1-111-111-1-ii-11-1-i-iii1-1-11-iii-iii-i-i    linear of order 4
ρ92-2-22002-2200-22-20000-222-200000000    orthogonal lifted from D4
ρ10222200-12200-1-1-12222-1-1-1-1-1-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ11222200-12200-1-1-1-2-2-2-2-1-1-1-111111111    orthogonal lifted from D6
ρ122222002-2-2002220000-2-2-2-200000000    orthogonal lifted from D4
ρ13222200-1-2-200-1-1-100001111-3-3-3333-33    orthogonal lifted from D12
ρ1422-2-20020000-2-22-22-22000022-222-2-2-2    orthogonal lifted from D8
ρ1522-2-200-1000011-12-22-233-3-3ζ83ζ38ζ38ζ83ζ38ζ38ζ83ζ32838ζ32ζ87ζ38785ζ3ζ87ζ38785ζ3ζ87ζ3285ζ3285ζ83ζ32838ζ32ζ87ζ3285ζ3285    orthogonal lifted from D24
ρ1622-2-20020000-2-222-22-20000-2-22-2-2222    orthogonal lifted from D8
ρ1722-2-200-1000011-12-22-2-3-333ζ87ζ38785ζ3ζ87ζ38785ζ3ζ87ζ3285ζ3285ζ83ζ38ζ38ζ83ζ38ζ38ζ83ζ32838ζ32ζ87ζ3285ζ3285ζ83ζ32838ζ32    orthogonal lifted from D24
ρ18222200-1-2-200-1-1-100001111333-3-3-33-3    orthogonal lifted from D12
ρ1922-2-200-1000011-1-22-2233-3-3ζ83ζ32838ζ32ζ83ζ32838ζ32ζ83ζ38ζ38ζ87ζ3285ζ3285ζ87ζ3285ζ3285ζ87ζ38785ζ3ζ83ζ38ζ38ζ87ζ38785ζ3    orthogonal lifted from D24
ρ2022-2-200-1000011-1-22-22-3-333ζ87ζ3285ζ3285ζ87ζ3285ζ3285ζ87ζ38785ζ3ζ83ζ32838ζ32ζ83ζ32838ζ32ζ83ζ38ζ38ζ87ζ38785ζ3ζ83ζ38ζ38    orthogonal lifted from D24
ρ212-2-2200-12-2001-112i2i-2i-2i-111-1-iii-iii-i-i    complex lifted from C4×S3
ρ222-2-2200-12-2001-11-2i-2i2i2i-111-1i-i-ii-i-iii    complex lifted from C4×S3
ρ232-22-200200002-2-2-2--2--2-20000--2-2--2--2-2--2-2-2    complex lifted from SD16
ρ242-22-200-10000-111-2--2--2-23-33-3ζ83ζ3838ζ3ζ87ζ38785ζ3ζ83ζ3838ζ3ζ83ζ32838ζ32ζ87ζ328785ζ32ζ83ζ32838ζ32ζ87ζ38785ζ3ζ87ζ328785ζ32    complex lifted from C24⋊C2
ρ252-2-2200-1-22001-1100001-1-11--3-3-3-3--3--3--3-3    complex lifted from C3⋊D4
ρ262-2-2200-1-22001-1100001-1-11-3--3--3--3-3-3-3--3    complex lifted from C3⋊D4
ρ272-22-200-10000-111--2-2-2--2-33-33ζ87ζ328785ζ32ζ83ζ32838ζ32ζ87ζ328785ζ32ζ87ζ38785ζ3ζ83ζ3838ζ3ζ87ζ38785ζ3ζ83ζ32838ζ32ζ83ζ3838ζ3    complex lifted from C24⋊C2
ρ282-22-200-10000-111--2-2-2--23-33-3ζ87ζ38785ζ3ζ83ζ3838ζ3ζ87ζ38785ζ3ζ87ζ328785ζ32ζ83ζ32838ζ32ζ87ζ328785ζ32ζ83ζ3838ζ3ζ83ζ32838ζ32    complex lifted from C24⋊C2
ρ292-22-200200002-2-2--2-2-2--20000-2--2-2-2--2-2--2--2    complex lifted from SD16
ρ302-22-200-10000-111-2--2--2-2-33-33ζ83ζ32838ζ32ζ87ζ328785ζ32ζ83ζ32838ζ32ζ83ζ3838ζ3ζ87ζ38785ζ3ζ83ζ3838ζ3ζ87ζ328785ζ32ζ87ζ38785ζ3    complex lifted from C24⋊C2

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

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

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

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

C2.D24 is a maximal subgroup of
C4×C24⋊C2  C4×D24  C4.5D24  C42.264D6  C42.16D6  D24⋊C4  C42.19D6  C42.20D6  D12.31D4  D1213D4  D12.32D4  D1214D4  C23.43D12  C22.D24  C23.18D12  Dic34D8  Dic3.SD16  C4⋊C4.D6  S3×D4⋊C4  C4⋊C419D6  D6⋊D8  C3⋊C8⋊D4  D4⋊S3⋊C4  Dic37SD16  (C2×C8).D6  Q8⋊C4⋊S3  Q87(C4×S3)  C4⋊C4.150D6  D62SD16  C3⋊(C8⋊D4)  Q83(C4×S3)  D123Q8  C4⋊D24  D12.19D4  C42.36D6  D124Q8  D12.3Q8  Dic68D4  D6.4SD16  C4.Q8⋊S3  D12⋊Q8  D12.Q8  D6.5D8  C2.D8⋊S3  D122Q8  D12.2Q8  C23.28D12  C2430D4  C2429D4  C23.53D12  C23.54D12  C242D4  C243D4  Dic3⋊D8  D12⋊D4  Dic35SD16  (C3×D4).D4  D66SD16  D127D4  (C2×Q16)⋊S3  D12.17D4  C2.D72  C6.16D24  C6.17D24  C62.84D4  C10.D24  D6015C4  D608C4  D12⋊F5
C2.D24 is a maximal quotient of
C4.17D24  C22.2D24  C4.D24  C12.2D8  C2.Dic24  C2.D48  D24.1C4  M5(2)⋊S3  C12.4D8  D242C4  C12.9C42  C2.D72  C6.16D24  C6.17D24  C62.84D4  C10.D24  D6015C4  D608C4  D12⋊F5

Matrix representation of C2.D24 in GL5(𝔽73)

720000
01000
00100
00010
00001
,
460000
007200
01100
000025
0003532
,
270000
007200
072000
000025
000380

G:=sub<GL(5,GF(73))| [72,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[46,0,0,0,0,0,0,1,0,0,0,72,1,0,0,0,0,0,0,35,0,0,0,25,32],[27,0,0,0,0,0,0,72,0,0,0,72,0,0,0,0,0,0,0,38,0,0,0,25,0] >;

C2.D24 in GAP, Magma, Sage, TeX

C_2.D_{24}
% in TeX

G:=Group("C2.D24");
// GroupNames label

G:=SmallGroup(96,28);
// by ID

G=gap.SmallGroup(96,28);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-3,73,79,362,86,2309]);
// Polycyclic

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

Export

Subgroup lattice of C2.D24 in TeX
Character table of C2.D24 in TeX

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