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G = C3×C2.D8order 96 = 25·3

Direct product of C3 and C2.D8

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

Aliases: C3×C2.D8, C243C4, C81C12, C6.14D8, C6.7Q16, C12.10Q8, C4⋊C4.3C6, (C2×C8).3C6, C2.2(C3×D8), C4.2(C3×Q8), C4.7(C2×C12), (C2×C24).9C2, (C2×C6).49D4, C6.13(C4⋊C4), C2.2(C3×Q16), C12.44(C2×C4), C22.11(C3×D4), (C2×C12).118C22, C2.4(C3×C4⋊C4), (C3×C4⋊C4).10C2, (C2×C4).21(C2×C6), SmallGroup(96,57)

Series: Derived Chief Lower central Upper central

C1C4 — C3×C2.D8
C1C2C22C2×C4C2×C12C3×C4⋊C4 — C3×C2.D8
C1C2C4 — C3×C2.D8
C1C2×C6C2×C12 — C3×C2.D8

Generators and relations for C3×C2.D8
 G = < a,b,c,d | a3=b2=c8=1, d2=b, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

4C4
4C4
2C2×C4
2C2×C4
4C12
4C12
2C2×C12
2C2×C12

Smallest permutation representation of C3×C2.D8
Regular action on 96 points
Generators in S96
(1 36 9)(2 37 10)(3 38 11)(4 39 12)(5 40 13)(6 33 14)(7 34 15)(8 35 16)(17 53 41)(18 54 42)(19 55 43)(20 56 44)(21 49 45)(22 50 46)(23 51 47)(24 52 48)(25 92 72)(26 93 65)(27 94 66)(28 95 67)(29 96 68)(30 89 69)(31 90 70)(32 91 71)(57 78 85)(58 79 86)(59 80 87)(60 73 88)(61 74 81)(62 75 82)(63 76 83)(64 77 84)
(1 19)(2 20)(3 21)(4 22)(5 23)(6 24)(7 17)(8 18)(9 43)(10 44)(11 45)(12 46)(13 47)(14 48)(15 41)(16 42)(25 79)(26 80)(27 73)(28 74)(29 75)(30 76)(31 77)(32 78)(33 52)(34 53)(35 54)(36 55)(37 56)(38 49)(39 50)(40 51)(57 71)(58 72)(59 65)(60 66)(61 67)(62 68)(63 69)(64 70)(81 95)(82 96)(83 89)(84 90)(85 91)(86 92)(87 93)(88 94)
(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 72 19 58)(2 71 20 57)(3 70 21 64)(4 69 22 63)(5 68 23 62)(6 67 24 61)(7 66 17 60)(8 65 18 59)(9 92 43 86)(10 91 44 85)(11 90 45 84)(12 89 46 83)(13 96 47 82)(14 95 48 81)(15 94 41 88)(16 93 42 87)(25 55 79 36)(26 54 80 35)(27 53 73 34)(28 52 74 33)(29 51 75 40)(30 50 76 39)(31 49 77 38)(32 56 78 37)

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

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

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

C3×C2.D8 is a maximal subgroup of
C6.6D16  C6.SD32  C6.D16  C6.Q32  Dic35D8  Dic35Q16  C242Q8  Dic3.Q16  C244Q8  Dic6.2Q8  C8.6Dic6  C8.27(C4×S3)  C8⋊S3⋊C4  D6.5D8  D62D8  D6.2Q16  C2.D8⋊S3  C83D12  D62Q16  C2.D87S3  C24⋊C2⋊C4  D122Q8  D12.2Q8  C12×D8  C12×Q16

42 conjugacy classes

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

42 irreducible representations

dim1111111122222222
type+++-++-
imageC1C2C2C3C4C6C6C12Q8D4D8Q16C3×Q8C3×D4C3×D8C3×Q16
kernelC3×C2.D8C3×C4⋊C4C2×C24C2.D8C24C4⋊C4C2×C8C8C12C2×C6C6C6C4C22C2C2
# reps1212442811222244

Matrix representation of C3×C2.D8 in GL6(𝔽73)

6400000
0640000
001000
000100
000010
000001
,
100000
010000
0072000
0007200
0000720
0000072
,
010000
7200000
000100
0072000
00001657
00001616
,
49700000
70240000
00325600
00564100
00004362
00006230

G:=sub<GL(6,GF(73))| [64,0,0,0,0,0,0,64,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[0,72,0,0,0,0,1,0,0,0,0,0,0,0,0,72,0,0,0,0,1,0,0,0,0,0,0,0,16,16,0,0,0,0,57,16],[49,70,0,0,0,0,70,24,0,0,0,0,0,0,32,56,0,0,0,0,56,41,0,0,0,0,0,0,43,62,0,0,0,0,62,30] >;

C3×C2.D8 in GAP, Magma, Sage, TeX

C_3\times C_2.D_8
% in TeX

G:=Group("C3xC2.D8");
// GroupNames label

G:=SmallGroup(96,57);
// by ID

G=gap.SmallGroup(96,57);
# by ID

G:=PCGroup([6,-2,-2,-3,-2,-2,-2,144,169,367,1443,117]);
// Polycyclic

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

Export

Subgroup lattice of C3×C2.D8 in TeX

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