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G = C3×C4⋊C8order 96 = 25·3

Direct product of C3 and C4⋊C8

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

Aliases: C3×C4⋊C8, C4⋊C24, C123C8, C12.67D4, C42.2C6, C12.12Q8, C6.9M4(2), (C2×C8).2C6, C4.4(C3×Q8), C2.2(C2×C24), (C2×C24).4C2, (C4×C12).8C2, C6.12(C2×C8), (C2×C4).4C12, C4.18(C3×D4), C6.11(C4⋊C4), (C2×C12).13C4, C2.3(C3×M4(2)), C22.10(C2×C12), (C2×C12).136C22, C2.2(C3×C4⋊C4), (C2×C6).39(C2×C4), (C2×C4).32(C2×C6), SmallGroup(96,55)

Series: Derived Chief Lower central Upper central

C1C2 — C3×C4⋊C8
C1C2C4C2×C4C2×C12C2×C24 — C3×C4⋊C8
C1C2 — C3×C4⋊C8
C1C2×C12 — C3×C4⋊C8

Generators and relations for C3×C4⋊C8
 G = < a,b,c | a3=b4=c8=1, ab=ba, ac=ca, cbc-1=b-1 >

2C4
2C8
2C8
2C12
2C24
2C24

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

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

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

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

C3×C4⋊C8 is a maximal subgroup of
C12.53D8  C12.39SD16  C4.Dic12  C12.47D8  D122C8  Dic62C8  C4.D24  C12.2D8  C42.27D6  Dic6.3Q8  Dic6⋊C8  C42.198D6  C42.200D6  D12⋊C8  C42.202D6  D63M4(2)  C12⋊M4(2)  C122M4(2)  C42.30D6  C42.31D6  C12⋊SD16  D123Q8  C4⋊D24  D12.19D4  C42.36D6  D124Q8  D12.3Q8  Dic68D4  C4⋊Dic12  Dic63Q8  Dic64Q8  D4×C24  Q8×C24

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

dim1111111111222222
type++++-
imageC1C2C2C3C4C6C6C8C12C24D4Q8M4(2)C3×D4C3×Q8C3×M4(2)
kernelC3×C4⋊C8C4×C12C2×C24C4⋊C8C2×C12C42C2×C8C12C2×C4C4C12C12C6C4C4C2
# reps11224248816112224

Matrix representation of C3×C4⋊C8 in GL3(𝔽73) generated by

100
0640
0064
,
7200
0072
010
,
5100
0368
0837
G:=sub<GL(3,GF(73))| [1,0,0,0,64,0,0,0,64],[72,0,0,0,0,1,0,72,0],[51,0,0,0,36,8,0,8,37] >;

C3×C4⋊C8 in GAP, Magma, Sage, TeX

C_3\times C_4\rtimes C_8
% in TeX

G:=Group("C3xC4:C8");
// GroupNames label

G:=SmallGroup(96,55);
// by ID

G=gap.SmallGroup(96,55);
# by ID

G:=PCGroup([6,-2,-2,-3,-2,-2,-2,144,169,79,88]);
// Polycyclic

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

Export

Subgroup lattice of C3×C4⋊C8 in TeX

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