Copied to
clipboard

G = C2×C24⋊C2order 96 = 25·3

Direct product of C2 and C24⋊C2

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

Aliases: C2×C24⋊C2, C88D6, C61SD16, C4.6D12, C249C22, C12.29D4, C12.28C23, Dic63C22, D12.6C22, C22.12D12, (C2×C8)⋊5S3, (C2×C24)⋊7C2, C6.9(C2×D4), C31(C2×SD16), (C2×C4).79D6, (C2×C6).16D4, (C2×Dic6)⋊5C2, (C2×D12).4C2, C2.11(C2×D12), C4.26(C22×S3), (C2×C12).88C22, SmallGroup(96,109)

Series: Derived Chief Lower central Upper central

C1C12 — C2×C24⋊C2
C1C3C6C12D12C2×D12 — C2×C24⋊C2
C3C6C12 — C2×C24⋊C2
C1C22C2×C4C2×C8

Generators and relations for C2×C24⋊C2
 G = < a,b,c | a2=b24=c2=1, ab=ba, ac=ca, cbc=b11 >

Subgroups: 194 in 68 conjugacy classes, 33 normal (17 characteristic)
C1, C2, C2 [×2], C2 [×2], C3, C4 [×2], C4 [×2], C22, C22 [×4], S3 [×2], C6, C6 [×2], C8 [×2], C2×C4, C2×C4, D4 [×3], Q8 [×3], C23, Dic3 [×2], C12 [×2], D6 [×4], C2×C6, C2×C8, SD16 [×4], C2×D4, C2×Q8, C24 [×2], Dic6 [×2], Dic6, D12 [×2], D12, C2×Dic3, C2×C12, C22×S3, C2×SD16, C24⋊C2 [×4], C2×C24, C2×Dic6, C2×D12, C2×C24⋊C2
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D6 [×3], SD16 [×2], C2×D4, D12 [×2], C22×S3, C2×SD16, C24⋊C2 [×2], C2×D12, C2×C24⋊C2

Character table of C2×C24⋊C2

 class 12A2B2C2D2E34A4B4C4D6A6B6C8A8B8C8D12A12B12C12D24A24B24C24D24E24F24G24H
 size 1111121222212122222222222222222222
ρ1111111111111111111111111111111    trivial
ρ21-1-111-111-11-1-11-11-11-1-1-111-11-1-1111-1    linear of order 2
ρ31111-1-111111111-1-1-1-11111-1-1-1-1-1-1-1-1    linear of order 2
ρ41-1-11-1111-11-1-11-1-11-11-1-1111-111-1-1-11    linear of order 2
ρ5111111111-1-1111-1-1-1-11111-1-1-1-1-1-1-1-1    linear of order 2
ρ61-1-111-111-1-11-11-1-11-11-1-1111-111-1-1-11    linear of order 2
ρ71111-1-1111-1-11111111111111111111    linear of order 2
ρ81-1-11-1111-1-11-11-11-11-1-1-111-11-1-1111-1    linear of order 2
ρ92-2-2200-12-2001-11-22-2211-1-1-11-1-1111-1    orthogonal lifted from D6
ρ102-2-22002-2200-22-2000022-2-200000000    orthogonal lifted from D4
ρ11222200-12200-1-1-12222-1-1-1-1-1-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ12222200-12200-1-1-1-2-2-2-2-1-1-1-111111111    orthogonal lifted from D6
ρ132222002-2-2002220000-2-2-2-200000000    orthogonal lifted from D4
ρ142-2-2200-12-2001-112-22-211-1-11-111-1-1-11    orthogonal lifted from D6
ρ15222200-1-2-200-1-1-100001111-3-33-3-3333    orthogonal lifted from D12
ρ162-2-2200-1-22001-110000-1-111-333-33-3-33    orthogonal lifted from D12
ρ172-2-2200-1-22001-110000-1-1113-3-33-333-3    orthogonal lifted from D12
ρ18222200-1-2-200-1-1-10000111133-333-3-3-3    orthogonal lifted from D12
ρ192-22-200200002-2-2-2-2--2--20000--2-2-2-2--2-2--2--2    complex lifted from SD16
ρ202-22-200200002-2-2--2--2-2-20000-2--2--2--2-2--2-2-2    complex lifted from SD16
ρ2122-2-20020000-2-22--2-2-2--20000--2--2-2-2-2--2-2--2    complex lifted from SD16
ρ2222-2-20020000-2-22-2--2--2-20000-2-2--2--2--2-2--2-2    complex lifted from SD16
ρ232-22-200-10000-111--2--2-2-23-3-33ζ87ζ328785ζ32ζ83ζ32838ζ32ζ83ζ3838ζ3ζ83ζ32838ζ32ζ87ζ328785ζ32ζ83ζ3838ζ3ζ87ζ38785ζ3ζ87ζ38785ζ3    complex lifted from C24⋊C2
ρ242-22-200-10000-111--2--2-2-2-333-3ζ87ζ38785ζ3ζ83ζ3838ζ3ζ83ζ32838ζ32ζ83ζ3838ζ3ζ87ζ38785ζ3ζ83ζ32838ζ32ζ87ζ328785ζ32ζ87ζ328785ζ32    complex lifted from C24⋊C2
ρ2522-2-200-1000011-1--2-2-2--23-33-3ζ83ζ3838ζ3ζ83ζ3838ζ3ζ87ζ328785ζ32ζ87ζ38785ζ3ζ87ζ38785ζ3ζ83ζ32838ζ32ζ87ζ328785ζ32ζ83ζ32838ζ32    complex lifted from C24⋊C2
ρ2622-2-200-1000011-1--2-2-2--2-33-33ζ83ζ32838ζ32ζ83ζ32838ζ32ζ87ζ38785ζ3ζ87ζ328785ζ32ζ87ζ328785ζ32ζ83ζ3838ζ3ζ87ζ38785ζ3ζ83ζ3838ζ3    complex lifted from C24⋊C2
ρ272-22-200-10000-111-2-2--2--23-3-33ζ83ζ32838ζ32ζ87ζ328785ζ32ζ87ζ38785ζ3ζ87ζ328785ζ32ζ83ζ32838ζ32ζ87ζ38785ζ3ζ83ζ3838ζ3ζ83ζ3838ζ3    complex lifted from C24⋊C2
ρ2822-2-200-1000011-1-2--2--2-23-33-3ζ87ζ38785ζ3ζ87ζ38785ζ3ζ83ζ32838ζ32ζ83ζ3838ζ3ζ83ζ3838ζ3ζ87ζ328785ζ32ζ83ζ32838ζ32ζ87ζ328785ζ32    complex lifted from C24⋊C2
ρ2922-2-200-1000011-1-2--2--2-2-33-33ζ87ζ328785ζ32ζ87ζ328785ζ32ζ83ζ3838ζ3ζ83ζ32838ζ32ζ83ζ32838ζ32ζ87ζ38785ζ3ζ83ζ3838ζ3ζ87ζ38785ζ3    complex lifted from C24⋊C2
ρ302-22-200-10000-111-2-2--2--2-333-3ζ83ζ3838ζ3ζ87ζ38785ζ3ζ87ζ328785ζ32ζ87ζ38785ζ3ζ83ζ3838ζ3ζ87ζ328785ζ32ζ83ζ32838ζ32ζ83ζ32838ζ32    complex lifted from C24⋊C2

Smallest permutation representation of C2×C24⋊C2
On 48 points
Generators in S48
(1 25)(2 26)(3 27)(4 28)(5 29)(6 30)(7 31)(8 32)(9 33)(10 34)(11 35)(12 36)(13 37)(14 38)(15 39)(16 40)(17 41)(18 42)(19 43)(20 44)(21 45)(22 46)(23 47)(24 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)
(1 25)(2 36)(3 47)(4 34)(5 45)(6 32)(7 43)(8 30)(9 41)(10 28)(11 39)(12 26)(13 37)(14 48)(15 35)(16 46)(17 33)(18 44)(19 31)(20 42)(21 29)(22 40)(23 27)(24 38)

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

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

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

C2×C24⋊C2 is a maximal subgroup of
C85D12  C8.8D12  C42.16D6  C8⋊D12  C8.D12  D12.31D4  D12.32D4  D1214D4  Dic614D4  Dic62D4  D65SD16  D43D12  D12.D4  Dic6.11D4  Q83D12  Q8.11D12  Dic3⋊SD16  C12⋊SD16  D12.19D4  C42.36D6  Dic68D4  Dic38SD16  C88D12  C83D12  C24⋊C2⋊C4  C24.42D4  C2430D4  C242D4  Q8.8D12  C2411D4  C24.43D4  C2415D4  C24.37D4  D4.11D12  C2×S3×SD16  D811D6
C2×C24⋊C2 is a maximal quotient of
C249Q8  C12.14Q16  C85D12  C4.5D24  C23.39D12  D12.31D4  C23.43D12  Dic614D4  C12⋊SD16  D123Q8  Dic68D4  Dic64Q8  C2430D4

Matrix representation of C2×C24⋊C2 in GL3(𝔽73) generated by

7200
0720
0072
,
100
06237
03625
,
100
0720
011
G:=sub<GL(3,GF(73))| [72,0,0,0,72,0,0,0,72],[1,0,0,0,62,36,0,37,25],[1,0,0,0,72,1,0,0,1] >;

C2×C24⋊C2 in GAP, Magma, Sage, TeX

C_2\times C_{24}\rtimes C_2
% in TeX

G:=Group("C2xC24:C2");
// GroupNames label

G:=SmallGroup(96,109);
// by ID

G=gap.SmallGroup(96,109);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-3,218,50,579,69,2309]);
// 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^11>;
// generators/relations

Export

Character table of C2×C24⋊C2 in TeX

׿
×
𝔽