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G = C3×Q8⋊C4order 96 = 25·3

Direct product of C3 and Q8⋊C4

direct product, metabelian, nilpotent (class 3), monomial, 2-elementary

Aliases: C3×Q8⋊C4, Q82C12, C6.6Q16, C12.61D4, C6.10SD16, C4⋊C4.1C6, (C2×C8).1C6, (C3×Q8)⋊4C4, (C2×C24).3C2, C4.2(C2×C12), (C2×C6).47D4, C4.12(C3×D4), (C2×Q8).4C6, (C6×Q8).7C2, C2.1(C3×Q16), C12.29(C2×C4), C2.2(C3×SD16), C22.9(C3×D4), C6.25(C22⋊C4), (C2×C12).115C22, (C3×C4⋊C4).8C2, (C2×C4).18(C2×C6), C2.7(C3×C22⋊C4), SmallGroup(96,53)

Series: Derived Chief Lower central Upper central

C1C4 — C3×Q8⋊C4
C1C2C22C2×C4C2×C12C3×C4⋊C4 — C3×Q8⋊C4
C1C2C4 — C3×Q8⋊C4
C1C2×C6C2×C12 — C3×Q8⋊C4

Generators and relations for C3×Q8⋊C4
 G = < a,b,c,d | a3=b4=d4=1, c2=b2, ab=ba, ac=ca, ad=da, cbc-1=dbd-1=b-1, dcd-1=b-1c >

2C4
2C4
4C4
2C2×C4
2C2×C4
2C8
2Q8
2C12
2C12
4C12
2C2×C12
2C3×Q8
2C24
2C2×C12

Smallest permutation representation of C3×Q8⋊C4
Regular action on 96 points
Generators in S96
(1 24 16)(2 21 13)(3 22 14)(4 23 15)(5 91 83)(6 92 84)(7 89 81)(8 90 82)(9 28 17)(10 25 18)(11 26 19)(12 27 20)(29 45 37)(30 46 38)(31 47 39)(32 48 40)(33 49 41)(34 50 42)(35 51 43)(36 52 44)(53 69 61)(54 70 62)(55 71 63)(56 72 64)(57 74 65)(58 75 66)(59 76 67)(60 73 68)(77 93 85)(78 94 86)(79 95 87)(80 96 88)
(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 34 3 36)(2 33 4 35)(5 71 7 69)(6 70 8 72)(9 32 11 30)(10 31 12 29)(13 41 15 43)(14 44 16 42)(17 40 19 38)(18 39 20 37)(21 49 23 51)(22 52 24 50)(25 47 27 45)(26 46 28 48)(53 83 55 81)(54 82 56 84)(57 80 59 78)(58 79 60 77)(61 91 63 89)(62 90 64 92)(65 88 67 86)(66 87 68 85)(73 93 75 95)(74 96 76 94)
(1 60 11 56)(2 59 12 55)(3 58 9 54)(4 57 10 53)(5 52 96 48)(6 51 93 47)(7 50 94 46)(8 49 95 45)(13 67 20 63)(14 66 17 62)(15 65 18 61)(16 68 19 64)(21 76 27 71)(22 75 28 70)(23 74 25 69)(24 73 26 72)(29 82 33 79)(30 81 34 78)(31 84 35 77)(32 83 36 80)(37 90 41 87)(38 89 42 86)(39 92 43 85)(40 91 44 88)

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

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

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

C3×Q8⋊C4 is a maximal subgroup of
Dic37SD16  C3⋊Q16⋊C4  Dic34Q16  Q82Dic6  Dic3.1Q16  Q83Dic6  (C2×C8).D6  Dic3⋊Q16  Q8.3Dic6  (C2×Q8).36D6  Dic6.11D4  Q8.4Dic6  Q8⋊C4⋊S3  (S3×Q8)⋊C4  Q87(C4×S3)  C4⋊C4.150D6  D6.1SD16  Q83D12  D62SD16  Q8.11D12  D6⋊Q16  Q84D12  D6.Q16  C3⋊(C8⋊D4)  D61Q16  D6⋊C8.C2  C8⋊Dic3⋊C2  C3⋊C8.D4  Q83(C4×S3)  Dic3⋊SD16  D12.12D4  C12×SD16  C12×Q16

42 conjugacy classes

class 1 2A2B2C3A3B4A4B4C4D4E4F6A···6F8A8B8C8D12A12B12C12D12E···12L24A···24H
order1222334444446···688881212121212···1224···24
size1111112244441···1222222224···42···2

42 irreducible representations

dim111111111122222222
type++++++-
imageC1C2C2C2C3C4C6C6C6C12D4D4SD16Q16C3×D4C3×D4C3×SD16C3×Q16
kernelC3×Q8⋊C4C3×C4⋊C4C2×C24C6×Q8Q8⋊C4C3×Q8C4⋊C4C2×C8C2×Q8Q8C12C2×C6C6C6C4C22C2C2
# reps111124222811222244

Matrix representation of C3×Q8⋊C4 in GL3(𝔽73) generated by

6400
080
008
,
100
001
0720
,
100
0659
098
,
4600
0629
02967
G:=sub<GL(3,GF(73))| [64,0,0,0,8,0,0,0,8],[1,0,0,0,0,72,0,1,0],[1,0,0,0,65,9,0,9,8],[46,0,0,0,6,29,0,29,67] >;

C3×Q8⋊C4 in GAP, Magma, Sage, TeX

C_3\times Q_8\rtimes C_4
% in TeX

G:=Group("C3xQ8:C4");
// GroupNames label

G:=SmallGroup(96,53);
// by ID

G=gap.SmallGroup(96,53);
# by ID

G:=PCGroup([6,-2,-2,-3,-2,-2,-2,144,169,295,1443,729,117]);
// Polycyclic

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

Export

Subgroup lattice of C3×Q8⋊C4 in TeX

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