Copied to
clipboard

G = C3×C8.4Q8order 192 = 26·3

Direct product of C3 and C8.4Q8

direct product, metacyclic, nilpotent (class 4), monomial, 2-elementary

Aliases: C3×C8.4Q8, C48.2C4, C16.1C12, C12.70D8, C24.20Q8, C8.5(C3×Q8), (C2×C16).5C6, C4.19(C3×D8), (C2×C6).6Q16, C24.77(C2×C4), (C2×C48).11C2, C8.15(C2×C12), C12.58(C4⋊C4), C8.C4.3C6, (C2×C12).410D4, C6.15(C2.D8), C22.1(C3×Q16), (C2×C24).409C22, C4.9(C3×C4⋊C4), C2.5(C3×C2.D8), (C2×C8).89(C2×C6), (C2×C4).64(C3×D4), (C3×C8.C4).6C2, SmallGroup(192,174)

Series: Derived Chief Lower central Upper central

C1C8 — C3×C8.4Q8
C1C2C4C2×C4C2×C8C2×C24C3×C8.C4 — C3×C8.4Q8
C1C2C4C8 — C3×C8.4Q8
C1C12C2×C12C2×C24 — C3×C8.4Q8

Generators and relations for C3×C8.4Q8
 G = < a,b,c,d | a3=b8=1, c4=b2, d2=bc2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b3, dcd-1=b6c3 >

2C2
2C6
4C8
4C8
2M4(2)
2M4(2)
4C24
4C24
2C3×M4(2)
2C3×M4(2)

Smallest permutation representation of C3×C8.4Q8
On 96 points
Generators in S96
(1 21 37)(2 22 38)(3 23 39)(4 24 40)(5 25 41)(6 26 42)(7 27 43)(8 28 44)(9 29 45)(10 30 46)(11 31 47)(12 32 48)(13 17 33)(14 18 34)(15 19 35)(16 20 36)(49 70 84)(50 71 85)(51 72 86)(52 73 87)(53 74 88)(54 75 89)(55 76 90)(56 77 91)(57 78 92)(58 79 93)(59 80 94)(60 65 95)(61 66 96)(62 67 81)(63 68 82)(64 69 83)
(1 11 5 15 9 3 13 7)(2 12 6 16 10 4 14 8)(17 27 21 31 25 19 29 23)(18 28 22 32 26 20 30 24)(33 43 37 47 41 35 45 39)(34 44 38 48 42 36 46 40)(49 51 53 55 57 59 61 63)(50 52 54 56 58 60 62 64)(65 67 69 71 73 75 77 79)(66 68 70 72 74 76 78 80)(81 83 85 87 89 91 93 95)(82 84 86 88 90 92 94 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 66 13 70 9 74 5 78)(2 65 14 69 10 73 6 77)(3 80 15 68 11 72 7 76)(4 79 16 67 12 71 8 75)(17 84 29 88 25 92 21 96)(18 83 30 87 26 91 22 95)(19 82 31 86 27 90 23 94)(20 81 32 85 28 89 24 93)(33 49 45 53 41 57 37 61)(34 64 46 52 42 56 38 60)(35 63 47 51 43 55 39 59)(36 62 48 50 44 54 40 58)

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

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

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

66 conjugacy classes

class 1 2A2B3A3B4A4B4C6A6B6C6D8A8B8C8D8E8F8G8H12A12B12C12D12E12F16A···16H24A···24H24I···24P48A···48P
order1223344466668888888812121212121216···1624···2424···2448···48
size112111121122222288881111222···22···28···82···2

66 irreducible representations

dim111111112222222222
type+++-++-
imageC1C2C2C3C4C6C6C12Q8D4D8Q16C3×Q8C3×D4C3×D8C3×Q16C8.4Q8C3×C8.4Q8
kernelC3×C8.4Q8C3×C8.C4C2×C48C8.4Q8C48C8.C4C2×C16C16C24C2×C12C12C2×C6C8C2×C4C4C22C3C1
# reps1212442811222244816

Matrix representation of C3×C8.4Q8 in GL2(𝔽97) generated by

610
061
,
470
033
,
700
079
,
01
220
G:=sub<GL(2,GF(97))| [61,0,0,61],[47,0,0,33],[70,0,0,79],[0,22,1,0] >;

C3×C8.4Q8 in GAP, Magma, Sage, TeX

C_3\times C_8._4Q_8
% in TeX

G:=Group("C3xC8.4Q8");
// GroupNames label

G:=SmallGroup(192,174);
// by ID

G=gap.SmallGroup(192,174);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-2,168,197,428,1683,360,172,6053,124]);
// Polycyclic

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

Export

Subgroup lattice of C3×C8.4Q8 in TeX

׿
×
𝔽