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G = C24⋊C4order 96 = 25·3

5th semidirect product of C24 and C4 acting via C4/C2=C2

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: C245C4, C83Dic3, C6.4C42, C6.2M4(2), C3⋊C84C4, (C2×C8).8S3, C32(C8⋊C4), C4.21(C4×S3), (C2×C4).92D6, (C2×C24).15C2, C12.41(C2×C4), C2.4(C4×Dic3), C2.2(C8⋊S3), C22.10(C4×S3), (C4×Dic3).6C2, (C2×Dic3).3C4, C4.13(C2×Dic3), (C2×C12).106C22, (C2×C3⋊C8).10C2, (C2×C6).11(C2×C4), SmallGroup(96,22)

Series: Derived Chief Lower central Upper central

C1C6 — C24⋊C4
C1C3C6C2×C6C2×C12C4×Dic3 — C24⋊C4
C3C6 — C24⋊C4
C1C2×C4C2×C8

Generators and relations for C24⋊C4
 G = < a,b | a24=b4=1, bab-1=a5 >

6C4
6C4
3C8
3C2×C4
3C2×C4
3C8
2Dic3
2Dic3
3C42
3C2×C8
3C8⋊C4

Smallest permutation representation of C24⋊C4
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 40 51 79)(2 45 52 84)(3 26 53 89)(4 31 54 94)(5 36 55 75)(6 41 56 80)(7 46 57 85)(8 27 58 90)(9 32 59 95)(10 37 60 76)(11 42 61 81)(12 47 62 86)(13 28 63 91)(14 33 64 96)(15 38 65 77)(16 43 66 82)(17 48 67 87)(18 29 68 92)(19 34 69 73)(20 39 70 78)(21 44 71 83)(22 25 72 88)(23 30 49 93)(24 35 50 74)

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

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

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

C24⋊C4 is a maximal subgroup of
C48⋊C4  C2412Q8  C4×C8⋊S3  D6.C42  C24⋊Q8  S3×C8⋊C4  D6.4C42  Dic3.M4(2)  C24⋊C4⋊C2  D62M4(2)  C3⋊C826D4  D4.S3⋊C4  C4⋊C4.D6  C12⋊Q8⋊C2  D4⋊S3⋊C4  C3⋊Q16⋊C4  (C2×C8).D6  (C2×Q8).36D6  Q83(C4×S3)  C42.27D6  C42.198D6  C42.202D6  C42.31D6  Dic129C4  C243Q8  D249C4  C244Q8  C24⋊C2⋊C4  D2410C4  C12.12C42  C2433D4  Dic3×M4(2)  C12.7C42  C24⋊D4  C24.54D4  D8⋊Dic3  C2411D4  SD16⋊Dic3  C24.31D4  C249D4  Q16⋊Dic3  C24.37D4  D84Dic3  C72⋊C4  C3⋊C8⋊Dic3  C2.Dic32  C24⋊Dic3  C30.22C42  C30.23C42  C12013C4  C30.4C42  C30.M4(2)  C24⋊F5
C24⋊C4 is a maximal quotient of
C42.279D6  C24⋊C8  C48⋊C4  (C2×C24)⋊5C4  C72⋊C4  C3⋊C8⋊Dic3  C2.Dic32  C24⋊Dic3  C30.22C42  C30.23C42  C12013C4  C30.4C42  C30.M4(2)  C24⋊F5

36 conjugacy classes

class 1 2A2B2C 3 4A4B4C4D4E4F4G4H6A6B6C8A8B8C8D8E8F8G8H12A12B12C12D24A···24H
order1222344444444666888888881212121224···24
size11112111166662222222666622222···2

36 irreducible representations

dim11111112222222
type+++++-+
imageC1C2C2C2C4C4C4S3Dic3D6M4(2)C4×S3C4×S3C8⋊S3
kernelC24⋊C4C2×C3⋊C8C4×Dic3C2×C24C3⋊C8C24C2×Dic3C2×C8C8C2×C4C6C4C22C2
# reps11114441214228

Matrix representation of C24⋊C4 in GL3(𝔽73) generated by

2700
0865
0816
,
2700
05468
01419
G:=sub<GL(3,GF(73))| [27,0,0,0,8,8,0,65,16],[27,0,0,0,54,14,0,68,19] >;

C24⋊C4 in GAP, Magma, Sage, TeX

C_{24}\rtimes C_4
% in TeX

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

G:=SmallGroup(96,22);
// by ID

G=gap.SmallGroup(96,22);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-3,24,217,55,69,2309]);
// Polycyclic

G:=Group<a,b|a^24=b^4=1,b*a*b^-1=a^5>;
// generators/relations

Export

Subgroup lattice of C24⋊C4 in TeX

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