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G = C24.C4order 96 = 25·3

1st non-split extension by C24 of C4 acting via C4/C2=C2

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: C24.1C4, C12.34D4, C4.18D12, C8.1Dic3, C22.2Dic6, (C2×C8).5S3, C6.7(C4⋊C4), (C2×C6).3Q8, (C2×C24).7C2, (C2×C4).70D6, C31(C8.C4), C12.35(C2×C4), C4.8(C2×Dic3), C2.5(C4⋊Dic3), C4.Dic3.1C2, (C2×C12).97C22, SmallGroup(96,26)

Series: Derived Chief Lower central Upper central

C1C12 — C24.C4
C1C3C6C12C2×C12C4.Dic3 — C24.C4
C3C6C12 — C24.C4
C1C4C2×C4C2×C8

Generators and relations for C24.C4
 G = < a,b,c | a8=1, b6=a4, c2=a4b3, ab=ba, cac-1=a-1, cbc-1=b5 >

2C2
2C6
6C8
6C8
3M4(2)
3M4(2)
2C3⋊C8
2C3⋊C8
3C8.C4

Character table of C24.C4

 class 12A2B34A4B4C6A6B6C8A8B8C8D8E8F8G8H12A12B12C12D24A24B24C24D24E24F24G24H
 size 1122112222222212121212222222222222
ρ1111111111111111111111111111111    trivial
ρ21111111111-1-1-1-11-11-11111-1-1-1-1-1-1-1-1    linear of order 2
ρ311111111111111-1-1-1-1111111111111    linear of order 2
ρ41111111111-1-1-1-1-11-111111-1-1-1-1-1-1-1-1    linear of order 2
ρ511-11-1-111-1-111-1-1ii-i-i1-11-1-111-1-111-1    linear of order 4
ρ611-11-1-111-1-1-1-111i-i-ii1-11-11-1-111-1-11    linear of order 4
ρ711-11-1-111-1-111-1-1-i-iii1-11-1-111-1-111-1    linear of order 4
ρ811-11-1-111-1-1-1-111-iii-i1-11-11-1-111-1-11    linear of order 4
ρ9222-1222-1-1-122220000-1-1-1-1-1-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ1022-2222-22-2-200000000-22-2200000000    orthogonal lifted from D4
ρ1122-2-122-2-111000000001-11-1-33-33-3-333    orthogonal lifted from D12
ρ12222-1222-1-1-1-2-2-2-20000-1-1-1-111111111    orthogonal lifted from D6
ρ1322-2-122-2-111000000001-11-13-33-333-3-3    orthogonal lifted from D12
ρ142222-2-2-222200000000-2-2-2-200000000    symplectic lifted from Q8, Schur index 2
ρ1522-2-1-2-22-11122-2-20000-11-111-1-111-1-11    symplectic lifted from Dic3, Schur index 2
ρ1622-2-1-2-22-111-2-2220000-11-11-111-1-111-1    symplectic lifted from Dic3, Schur index 2
ρ17222-1-2-2-2-1-1-100000000111133-3-33-33-3    symplectic lifted from Dic6, Schur index 2
ρ18222-1-2-2-2-1-1-1000000001111-3-333-33-33    symplectic lifted from Dic6, Schur index 2
ρ192-202-2i2i0-200--2-2-2200000-2i02i2-2-22-2--2--2-2    complex lifted from C8.C4
ρ202-2022i-2i0-200--2-22-2000002i0-2i-2-2-2-22--2--22    complex lifted from C8.C4
ρ212-202-2i2i0-200-2--22-200000-2i02i-2--2--2-22-2-22    complex lifted from C8.C4
ρ222-2022i-2i0-200-2--2-22000002i0-2i2--2--22-2-2-2-2    complex lifted from C8.C4
ρ232-20-12i-2i01--3-3--2-22-20000-3-i3iζ83ζ38ζ38ζ87ζ328785ζ32ζ87ζ38785ζ3ζ87ζ38785ζ3ζ83ζ32838ζ32ζ83ζ3838ζ3ζ83ζ32838ζ32ζ87ζ3285ζ3285    complex faithful
ρ242-20-1-2i2i01--3-3-2--22-200003i-3-iζ87ζ38785ζ3ζ83ζ3838ζ3ζ83ζ32838ζ32ζ83ζ38ζ38ζ87ζ3285ζ3285ζ87ζ328785ζ32ζ87ζ38785ζ3ζ83ζ32838ζ32    complex faithful
ρ252-20-1-2i2i01-3--3-2--22-20000-3i3-iζ83ζ38ζ38ζ83ζ32838ζ32ζ83ζ3838ζ3ζ87ζ38785ζ3ζ83ζ32838ζ32ζ87ζ38785ζ3ζ87ζ328785ζ32ζ87ζ3285ζ3285    complex faithful
ρ262-20-12i-2i01-3--3-2--2-2200003-i-3iζ87ζ3285ζ3285ζ83ζ3838ζ3ζ83ζ32838ζ32ζ83ζ32838ζ32ζ87ζ38785ζ3ζ87ζ328785ζ32ζ87ζ38785ζ3ζ83ζ38ζ38    complex faithful
ρ272-20-12i-2i01--3-3-2--2-220000-3-i3iζ83ζ32838ζ32ζ83ζ32838ζ32ζ83ζ3838ζ3ζ87ζ3285ζ3285ζ83ζ38ζ38ζ87ζ38785ζ3ζ87ζ328785ζ32ζ87ζ38785ζ3    complex faithful
ρ282-20-1-2i2i01-3--3--2-2-220000-3i3-iζ83ζ32838ζ32ζ87ζ328785ζ32ζ87ζ38785ζ3ζ87ζ3285ζ3285ζ83ζ38ζ38ζ83ζ3838ζ3ζ83ζ32838ζ32ζ87ζ38785ζ3    complex faithful
ρ292-20-1-2i2i01--3-3--2-2-2200003i-3-iζ87ζ3285ζ3285ζ87ζ38785ζ3ζ87ζ328785ζ32ζ83ζ32838ζ32ζ87ζ38785ζ3ζ83ζ32838ζ32ζ83ζ3838ζ3ζ83ζ38ζ38    complex faithful
ρ302-20-12i-2i01-3--3--2-22-200003-i-3iζ87ζ38785ζ3ζ87ζ38785ζ3ζ87ζ328785ζ32ζ83ζ38ζ38ζ87ζ3285ζ3285ζ83ζ32838ζ32ζ83ζ3838ζ3ζ83ζ32838ζ32    complex faithful

Smallest permutation representation of C24.C4
On 48 points
Generators in S48
(1 20 10 17 7 14 4 23)(2 21 11 18 8 15 5 24)(3 22 12 19 9 16 6 13)(25 46 28 37 31 40 34 43)(26 47 29 38 32 41 35 44)(27 48 30 39 33 42 36 45)
(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 10 34 7 31 4 28)(2 30 11 27 8 36 5 33)(3 35 12 32 9 29 6 26)(13 44 22 41 19 38 16 47)(14 37 23 46 20 43 17 40)(15 42 24 39 21 48 18 45)

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

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

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

C24.C4 is a maximal subgroup of
C8.Dic6  C24.7Q8  C48.C4  D24.1C4  C24.Q8  M5(2)⋊S3  C12.4D8  D8.Dic3  Q16.Dic3  C24.41D4  D2411C4  D244C4  S3×C8.C4  M4(2).25D6  C23.9Dic6  Q8.8D12  Q8.9D12  Q8.10D12  C24.23D4  C24.44D4  C24.29D4  D85Dic3  D84Dic3  C72.C4  C12.82D12  C12.59D12  C60.105D4  C4.18D60  C40.Dic3  C24.1F5
C24.C4 is a maximal quotient of
C242C8  C241C8  C12.10C42  C72.C4  C12.82D12  C12.59D12  C60.105D4  C4.18D60  C40.Dic3  C24.1F5

Matrix representation of C24.C4 in GL2(𝔽73) generated by

630
2851
,
490
2470
,
7263
121
G:=sub<GL(2,GF(73))| [63,28,0,51],[49,24,0,70],[72,12,63,1] >;

C24.C4 in GAP, Magma, Sage, TeX

C_{24}.C_4
% in TeX

G:=Group("C24.C4");
// GroupNames label

G:=SmallGroup(96,26);
// by ID

G=gap.SmallGroup(96,26);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-3,24,121,55,86,579,69,2309]);
// Polycyclic

G:=Group<a,b,c|a^8=1,b^6=a^4,c^2=a^4*b^3,a*b=b*a,c*a*c^-1=a^-1,c*b*c^-1=b^5>;
// generators/relations

Export

Subgroup lattice of C24.C4 in TeX
Character table of C24.C4 in TeX

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