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G = C241C4order 96 = 25·3

1st semidirect product of C24 and C4 acting via C4/C2=C2

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: C241C4, C6.4D8, C81Dic3, C2.1D24, C6.2Q16, C12.5Q8, C4.5Dic6, C2.2Dic12, C22.9D12, (C2×C8).3S3, C32(C2.D8), C6.6(C4⋊C4), (C2×C24).5C2, (C2×C6).14D4, (C2×C4).69D6, C12.34(C2×C4), C4⋊Dic3.3C2, C4.7(C2×Dic3), C2.4(C4⋊Dic3), (C2×C12).82C22, SmallGroup(96,25)

Series: Derived Chief Lower central Upper central

C1C12 — C241C4
C1C3C6C2×C6C2×C12C4⋊Dic3 — C241C4
C3C6C12 — C241C4
C1C22C2×C4C2×C8

Generators and relations for C241C4
 G = < a,b | a24=b4=1, bab-1=a-1 >

12C4
12C4
6C2×C4
6C2×C4
4Dic3
4Dic3
3C4⋊C4
3C4⋊C4
2C2×Dic3
2C2×Dic3
3C2.D8

Character table of C241C4

 class 12A2B2C34A4B4C4D4E4F6A6B6C8A8B8C8D12A12B12C12D24A24B24C24D24E24F24G24H
 size 1111222121212122222222222222222222
ρ1111111111111111111111111111111    trivial
ρ211111111-11-1111-1-1-1-11111-1-1-1-1-1-1-1-1    linear of order 2
ρ31111111-11-11111-1-1-1-11111-1-1-1-1-1-1-1-1    linear of order 2
ρ41111111-1-1-1-11111111111111111111    linear of order 2
ρ511-1-11-11ii-i-i-11-1-111-1-11-11-111-111-1-1    linear of order 4
ρ611-1-11-11i-i-ii-11-11-1-11-11-111-1-11-1-111    linear of order 4
ρ711-1-11-11-iii-i-11-11-1-11-11-111-1-11-1-111    linear of order 4
ρ811-1-11-11-i-iii-11-1-111-1-11-11-111-111-1-1    linear of order 4
ρ92-22-2-100000011-122-2-2-3-333ζ87ζ38785ζ3ζ83ζ38ζ38ζ87ζ38785ζ3ζ83ζ38ζ38ζ87ζ3285ζ3285ζ83ζ32838ζ32ζ87ζ3285ζ3285ζ83ζ32838ζ32    orthogonal lifted from D24
ρ102222-1220000-1-1-12222-1-1-1-1-1-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ112-22-22000000-2-2222-2-20000-2-2-2-22222    orthogonal lifted from D8
ρ1222222-2-200002220000-2-2-2-200000000    orthogonal lifted from D4
ρ132222-1220000-1-1-1-2-2-2-2-1-1-1-111111111    orthogonal lifted from D6
ρ142222-1-2-20000-1-1-1000011113-33-33-33-3    orthogonal lifted from D12
ρ152-22-22000000-2-22-2-22200002222-2-2-2-2    orthogonal lifted from D8
ρ162-22-2-100000011-122-2-233-3-3ζ83ζ38ζ38ζ87ζ38785ζ3ζ83ζ38ζ38ζ87ζ38785ζ3ζ83ζ32838ζ32ζ87ζ3285ζ3285ζ83ζ32838ζ32ζ87ζ3285ζ3285    orthogonal lifted from D24
ρ172222-1-2-20000-1-1-100001111-33-33-33-33    orthogonal lifted from D12
ρ182-22-2-100000011-1-2-22233-3-3ζ83ζ32838ζ32ζ87ζ3285ζ3285ζ83ζ32838ζ32ζ87ζ3285ζ3285ζ83ζ38ζ38ζ87ζ38785ζ3ζ83ζ38ζ38ζ87ζ38785ζ3    orthogonal lifted from D24
ρ192-22-2-100000011-1-2-222-3-333ζ87ζ3285ζ3285ζ83ζ32838ζ32ζ87ζ3285ζ3285ζ83ζ32838ζ32ζ87ζ38785ζ3ζ83ζ38ζ38ζ87ζ38785ζ3ζ83ζ38ζ38    orthogonal lifted from D24
ρ2022-2-2-1-2200001-112-2-221-11-1-111-111-1-1    symplectic lifted from Dic3, Schur index 2
ρ2122-2-2-1-2200001-11-222-21-11-11-1-11-1-111    symplectic lifted from Dic3, Schur index 2
ρ222-2-2220000002-2-22-22-20000-222-2-2-222    symplectic lifted from Q16, Schur index 2
ρ232-2-2220000002-2-2-22-2200002-2-2222-2-2    symplectic lifted from Q16, Schur index 2
ρ2422-2-2-12-200001-110000-11-11-3-3333-3-33    symplectic lifted from Dic6, Schur index 2
ρ2522-2-222-20000-22-200002-22-200000000    symplectic lifted from Q8, Schur index 2
ρ2622-2-2-12-200001-110000-11-1133-3-3-333-3    symplectic lifted from Dic6, Schur index 2
ρ272-2-22-1000000-111-22-22-333-3ζ83ζ32838ζ32ζ87ζ38785ζ3ζ83ζ38ζ38ζ87ζ3285ζ3285ζ83ζ32838ζ32ζ87ζ3285ζ3285ζ83ζ38ζ38ζ87ζ38785ζ3    symplectic lifted from Dic12, Schur index 2
ρ282-2-22-1000000-1112-22-2-333-3ζ83ζ38ζ38ζ87ζ3285ζ3285ζ83ζ32838ζ32ζ87ζ38785ζ3ζ83ζ38ζ38ζ87ζ38785ζ3ζ83ζ32838ζ32ζ87ζ3285ζ3285    symplectic lifted from Dic12, Schur index 2
ρ292-2-22-1000000-1112-22-23-3-33ζ87ζ38785ζ3ζ83ζ32838ζ32ζ87ζ3285ζ3285ζ83ζ38ζ38ζ87ζ38785ζ3ζ83ζ38ζ38ζ87ζ3285ζ3285ζ83ζ32838ζ32    symplectic lifted from Dic12, Schur index 2
ρ302-2-22-1000000-111-22-223-3-33ζ87ζ3285ζ3285ζ83ζ38ζ38ζ87ζ38785ζ3ζ83ζ32838ζ32ζ87ζ3285ζ3285ζ83ζ32838ζ32ζ87ζ38785ζ3ζ83ζ38ζ38    symplectic lifted from Dic12, Schur index 2

Smallest permutation representation of C241C4
Regular action on 96 points
Generators in S96
(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)(49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72)(73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96)
(1 76 66 41)(2 75 67 40)(3 74 68 39)(4 73 69 38)(5 96 70 37)(6 95 71 36)(7 94 72 35)(8 93 49 34)(9 92 50 33)(10 91 51 32)(11 90 52 31)(12 89 53 30)(13 88 54 29)(14 87 55 28)(15 86 56 27)(16 85 57 26)(17 84 58 25)(18 83 59 48)(19 82 60 47)(20 81 61 46)(21 80 62 45)(22 79 63 44)(23 78 64 43)(24 77 65 42)

G:=sub<Sym(96)| (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)(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96), (1,76,66,41)(2,75,67,40)(3,74,68,39)(4,73,69,38)(5,96,70,37)(6,95,71,36)(7,94,72,35)(8,93,49,34)(9,92,50,33)(10,91,51,32)(11,90,52,31)(12,89,53,30)(13,88,54,29)(14,87,55,28)(15,86,56,27)(16,85,57,26)(17,84,58,25)(18,83,59,48)(19,82,60,47)(20,81,61,46)(21,80,62,45)(22,79,63,44)(23,78,64,43)(24,77,65,42)>;

G:=Group( (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)(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96), (1,76,66,41)(2,75,67,40)(3,74,68,39)(4,73,69,38)(5,96,70,37)(6,95,71,36)(7,94,72,35)(8,93,49,34)(9,92,50,33)(10,91,51,32)(11,90,52,31)(12,89,53,30)(13,88,54,29)(14,87,55,28)(15,86,56,27)(16,85,57,26)(17,84,58,25)(18,83,59,48)(19,82,60,47)(20,81,61,46)(21,80,62,45)(22,79,63,44)(23,78,64,43)(24,77,65,42) );

G=PermutationGroup([(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),(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72),(73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96)], [(1,76,66,41),(2,75,67,40),(3,74,68,39),(4,73,69,38),(5,96,70,37),(6,95,71,36),(7,94,72,35),(8,93,49,34),(9,92,50,33),(10,91,51,32),(11,90,52,31),(12,89,53,30),(13,88,54,29),(14,87,55,28),(15,86,56,27),(16,85,57,26),(17,84,58,25),(18,83,59,48),(19,82,60,47),(20,81,61,46),(21,80,62,45),(22,79,63,44),(23,78,64,43),(24,77,65,42)])

C241C4 is a maximal subgroup of
C6.6D16  C6.SD32  C2.Dic24  C485C4  C486C4  C2.D48  D81Dic3  C6.5Q32  C248Q8  C24.13Q8  C4×D24  C4×Dic12  C8⋊Dic6  C42.16D6  C23.40D12  C23.15D12  C22.D24  C23.18D12  Dic3.D8  D4.2Dic6  D6.D8  C241C4⋊C2  Q83Dic6  Q8.3Dic6  D6.Q16  D6⋊C8.C2  Dic6.3Q8  D124Q8  D12.3Q8  Dic63Q8  C243Q8  C8⋊(C4×S3)  C242Q8  C8.6Dic6  S3×C2.D8  C8.27(C4×S3)  C23.27D12  C2429D4  C24.82D4  C23.52D12  C242D4  Dic3×D8  D63D8  SD16⋊Dic3  C248D4  Dic3×Q16  D63Q16  C721C4  C6.18D24  C241Dic3  C60.8Q8  C1209C4  D5.D24
C241C4 is a maximal quotient of
C241C8  C485C4  C486C4  C48.C4  C12.9C42  C721C4  C6.18D24  C241Dic3  C60.8Q8  C1209C4  D5.D24

Matrix representation of C241C4 in GL4(𝔽73) generated by

1100
72000
005055
001868
,
187100
535500
005355
00220
G:=sub<GL(4,GF(73))| [1,72,0,0,1,0,0,0,0,0,50,18,0,0,55,68],[18,53,0,0,71,55,0,0,0,0,53,2,0,0,55,20] >;

C241C4 in GAP, Magma, Sage, TeX

C_{24}\rtimes_1C_4
% in TeX

G:=Group("C24:1C4");
// GroupNames label

G:=SmallGroup(96,25);
// by ID

G=gap.SmallGroup(96,25);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-3,24,121,151,579,69,2309]);
// Polycyclic

G:=Group<a,b|a^24=b^4=1,b*a*b^-1=a^-1>;
// generators/relations

Export

Subgroup lattice of C241C4 in TeX
Character table of C241C4 in TeX

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