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G = C2×C24⋊C2  order 96 = 25·3

Direct product of C2 and C24⋊C2

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

Aliases: C2×C24⋊C2, C8⋊8D6, C6⋊1SD16, C4.6D12, C24⋊9C22, C12.29D4, C12.28C23, Dic6⋊3C22, D12.6C22, C22.12D12, (C2×C8)⋊5S3, (C2×C24)⋊7C2, C6.9(C2×D4), C3⋊1(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

Derived series
C1 — C12 — C2×C24⋊C2
Chief series
C1 — C3 — C6 — C12 — D12 — C2×D12 — C2×C24⋊C2
Lower central
C3 — C6 — C12 — C2×C24⋊C2
Upper central
C1 — C22 — C2×C4 — C2×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, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, D4, Q8, C23, Dic3, C12, D6, C2×C6, C2×C8, SD16, C2×D4, C2×Q8, C24, Dic6, Dic6, D12, D12, C2×Dic3, C2×C12, C22×S3, C2×SD16, C24⋊C2, C2×C24, C2×Dic6, C2×D12, C2×C24⋊C2
Quotients: C1, C2, C22, S3, D4, C23, D6, SD16, C2×D4, D12, C22×S3, C2×SD16, C24⋊C2, 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-√3√3-√3-√3√3√3√3    orthogonal lifted from D12
ρ162-2-2200-1-22001-110000-1-111-√3√3√3-√3√3-√3-√3√3    orthogonal lifted from D12
ρ172-2-2200-1-22001-110000-1-111√3-√3-√3√3-√3√3√3-√3    orthogonal lifted from D12
ρ18222200-1-2-200-1-1-100001111√3√3-√3√3√3-√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√-2√3-√3-√3√3ζ87ζ32+ζ87-ζ85ζ32ζ83ζ32+ζ83-ζ8ζ32ζ83ζ3+ζ83-ζ8ζ3ζ83ζ32+ζ83-ζ8ζ32ζ87ζ32+ζ87-ζ85ζ32ζ83ζ3+ζ83-ζ8ζ3ζ87ζ3+ζ87-ζ85ζ3ζ87ζ3+ζ87-ζ85ζ3    complex lifted from C24⋊C2
ρ242-22-200-10000-111-√-2-√-2√-2√-2-√3√3√3-√3ζ87ζ3+ζ87-ζ85ζ3ζ83ζ3+ζ83-ζ8ζ3ζ83ζ32+ζ83-ζ8ζ32ζ83ζ3+ζ83-ζ8ζ3ζ87ζ3+ζ87-ζ85ζ3ζ83ζ32+ζ83-ζ8ζ32ζ87ζ32+ζ87-ζ85ζ32ζ87ζ32+ζ87-ζ85ζ32    complex lifted from C24⋊C2
ρ2522-2-200-1000011-1-√-2√-2√-2-√-2√3-√3√3-√3ζ83ζ3+ζ83-ζ8ζ3ζ83ζ3+ζ83-ζ8ζ3ζ87ζ32+ζ87-ζ85ζ32ζ87ζ3+ζ87-ζ85ζ3ζ87ζ3+ζ87-ζ85ζ3ζ83ζ32+ζ83-ζ8ζ32ζ87ζ32+ζ87-ζ85ζ32ζ83ζ32+ζ83-ζ8ζ32    complex lifted from C24⋊C2
ρ2622-2-200-1000011-1-√-2√-2√-2-√-2-√3√3-√3√3ζ83ζ32+ζ83-ζ8ζ32ζ83ζ32+ζ83-ζ8ζ32ζ87ζ3+ζ87-ζ85ζ3ζ87ζ32+ζ87-ζ85ζ32ζ87ζ32+ζ87-ζ85ζ32ζ83ζ3+ζ83-ζ8ζ3ζ87ζ3+ζ87-ζ85ζ3ζ83ζ3+ζ83-ζ8ζ3    complex lifted from C24⋊C2
ρ272-22-200-10000-111√-2√-2-√-2-√-2√3-√3-√3√3ζ83ζ32+ζ83-ζ8ζ32ζ87ζ32+ζ87-ζ85ζ32ζ87ζ3+ζ87-ζ85ζ3ζ87ζ32+ζ87-ζ85ζ32ζ83ζ32+ζ83-ζ8ζ32ζ87ζ3+ζ87-ζ85ζ3ζ83ζ3+ζ83-ζ8ζ3ζ83ζ3+ζ83-ζ8ζ3    complex lifted from C24⋊C2
ρ2822-2-200-1000011-1√-2-√-2-√-2√-2√3-√3√3-√3ζ87ζ3+ζ87-ζ85ζ3ζ87ζ3+ζ87-ζ85ζ3ζ83ζ32+ζ83-ζ8ζ32ζ83ζ3+ζ83-ζ8ζ3ζ83ζ3+ζ83-ζ8ζ3ζ87ζ32+ζ87-ζ85ζ32ζ83ζ32+ζ83-ζ8ζ32ζ87ζ32+ζ87-ζ85ζ32    complex lifted from C24⋊C2
ρ2922-2-200-1000011-1√-2-√-2-√-2√-2-√3√3-√3√3ζ87ζ32+ζ87-ζ85ζ32ζ87ζ32+ζ87-ζ85ζ32ζ83ζ3+ζ83-ζ8ζ3ζ83ζ32+ζ83-ζ8ζ32ζ83ζ32+ζ83-ζ8ζ32ζ87ζ3+ζ87-ζ85ζ3ζ83ζ3+ζ83-ζ8ζ3ζ87ζ3+ζ87-ζ85ζ3    complex lifted from C24⋊C2
ρ302-22-200-10000-111√-2√-2-√-2-√-2-√3√3√3-√3ζ83ζ3+ζ83-ζ8ζ3ζ87ζ3+ζ87-ζ85ζ3ζ87ζ32+ζ87-ζ85ζ32ζ87ζ3+ζ87-ζ85ζ3ζ83ζ3+ζ83-ζ8ζ3ζ87ζ32+ζ87-ζ85ζ32ζ83ζ32+ζ83-ζ8ζ32ζ83ζ32+ζ83-ζ8ζ32    complex lifted from C24⋊C2

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

C2×C24⋊C2 is a maximal subgroup of
 C8⋊5D12  C8.8D12  C42.16D6  C8⋊D12  C8.D12  D12.31D4  D12.32D4  D12⋊14D4  Dic6⋊14D4  Dic6⋊2D4  D6⋊5SD16  D4⋊3D12  D12.D4  Dic6.11D4  Q8⋊3D12  Q8.11D12  Dic3⋊SD16  C12⋊SD16  D12.19D4  C42.36D6  Dic6⋊8D4  Dic3⋊8SD16  C8⋊8D12  C8⋊3D12  C24⋊C2⋊C4  C24.42D4  C24⋊30D4  C24⋊2D4  Q8.8D12  C24⋊11D4  C24.43D4  C24⋊15D4  C24.37D4  D4.11D12  C2×S3×SD16  D8⋊11D6
C2×C24⋊C2 is a maximal quotient of
 C24⋊9Q8  C12.14Q16  C8⋊5D12  C4.5D24  C23.39D12  D12.31D4  C23.43D12  Dic6⋊14D4  C12⋊SD16  D12⋊3Q8  Dic6⋊8D4  Dic6⋊4Q8  C24⋊30D4

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

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