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G = C3×C8⋊C4order 96 = 25·3

Direct product of C3 and C8⋊C4

direct product, metacyclic, nilpotent (class 2), monomial, 2-elementary

Aliases: C3×C8⋊C4, C247C4, C83C12, C6.7C42, C42.1C6, C6.7M4(2), (C2×C8).7C6, (C2×C4).2C12, (C2×C12).5C4, (C4×C12).1C2, C2.2(C4×C12), (C2×C24).17C2, C4.11(C2×C12), C12.48(C2×C4), C22.8(C2×C12), C2.1(C3×M4(2)), (C2×C12).134C22, (C2×C6).37(C2×C4), (C2×C4).30(C2×C6), SmallGroup(96,47)

Series: Derived Chief Lower central Upper central

C1C2 — C3×C8⋊C4
C1C2C22C2×C4C2×C12C2×C24 — C3×C8⋊C4
C1C2 — C3×C8⋊C4
C1C2×C12 — C3×C8⋊C4

Generators and relations for C3×C8⋊C4
 G = < a,b,c | a3=b8=c4=1, ab=ba, ac=ca, cbc-1=b5 >

2C4
2C4
2C12
2C12

Smallest permutation representation of C3×C8⋊C4
Regular action on 96 points
Generators in S96
(1 50 26)(2 51 27)(3 52 28)(4 53 29)(5 54 30)(6 55 31)(7 56 32)(8 49 25)(9 67 83)(10 68 84)(11 69 85)(12 70 86)(13 71 87)(14 72 88)(15 65 81)(16 66 82)(17 80 91)(18 73 92)(19 74 93)(20 75 94)(21 76 95)(22 77 96)(23 78 89)(24 79 90)(33 45 60)(34 46 61)(35 47 62)(36 48 63)(37 41 64)(38 42 57)(39 43 58)(40 44 59)
(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 45 17 67)(2 42 18 72)(3 47 19 69)(4 44 20 66)(5 41 21 71)(6 46 22 68)(7 43 23 65)(8 48 24 70)(9 26 33 91)(10 31 34 96)(11 28 35 93)(12 25 36 90)(13 30 37 95)(14 27 38 92)(15 32 39 89)(16 29 40 94)(49 63 79 86)(50 60 80 83)(51 57 73 88)(52 62 74 85)(53 59 75 82)(54 64 76 87)(55 61 77 84)(56 58 78 81)

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

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

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

C3×C8⋊C4 is a maximal subgroup of
C42.D6  C42.2D6  C12.15C42  C24⋊Q8  C8⋊Dic6  C42.14D6  C42.182D6  C89D12  Dic35M4(2)  D6.4C42  C42.185D6  C42.16D6  D24⋊C4  C8⋊D12  C42.19D6  C42.20D6  C8.D12  Dic12⋊C4  D244C4  C12×M4(2)

60 conjugacy classes

class 1 2A2B2C3A3B4A4B4C4D4E4F4G4H6A···6F8A···8H12A···12H12I···12P24A···24P
order122233444444446···68···812···1212···1224···24
size111111111122221···12···21···12···22···2

60 irreducible representations

dim111111111122
type+++
imageC1C2C2C3C4C4C6C6C12C12M4(2)C3×M4(2)
kernelC3×C8⋊C4C4×C12C2×C24C8⋊C4C24C2×C12C42C2×C8C8C2×C4C6C2
# reps1122842416848

Matrix representation of C3×C8⋊C4 in GL4(𝔽73) generated by

64000
0100
0010
0001
,
1000
0100
005954
005414
,
72000
04600
0001
00720
G:=sub<GL(4,GF(73))| [64,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,59,54,0,0,54,14],[72,0,0,0,0,46,0,0,0,0,0,72,0,0,1,0] >;

C3×C8⋊C4 in GAP, Magma, Sage, TeX

C_3\times C_8\rtimes C_4
% in TeX

G:=Group("C3xC8:C4");
// GroupNames label

G:=SmallGroup(96,47);
// by ID

G=gap.SmallGroup(96,47);
# by ID

G:=PCGroup([6,-2,-2,-3,-2,-2,-2,72,601,151,117]);
// Polycyclic

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

Export

Subgroup lattice of C3×C8⋊C4 in TeX

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