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G = C3×C4.Q8order 96 = 25·3

Direct product of C3 and C4.Q8

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

Aliases: C3×C4.Q8, C82C12, C246C4, C12.9Q8, C6.11SD16, C4⋊C4.2C6, (C2×C8).6C6, C4.1(C3×Q8), C4.6(C2×C12), (C2×C6).48D4, C6.12(C4⋊C4), (C2×C24).16C2, C12.43(C2×C4), C2.3(C3×SD16), C22.10(C3×D4), (C2×C12).117C22, C2.3(C3×C4⋊C4), (C3×C4⋊C4).9C2, (C2×C4).20(C2×C6), SmallGroup(96,56)

Series: Derived Chief Lower central Upper central

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

Generators and relations for C3×C4.Q8
 G = < a,b,c,d | a3=b4=1, c4=b2, d2=b-1c2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c3 >

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

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

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

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

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

C3×C4.Q8 is a maximal subgroup of
C8.Dic6  D248C4  Dic38SD16  Dic129C4  Dic6⋊Q8  C245Q8  C243Q8  Dic6.Q8  C8.8Dic6  (S3×C8)⋊C4  C8⋊(C4×S3)  D6.2SD16  D6.4SD16  C88D12  C247D4  C4.Q8⋊S3  C8.2D12  C6.(C4○D8)  D249C4  D12⋊Q8  D12.Q8  C12×SD16

42 conjugacy classes

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

42 irreducible representations

dim11111111222222
type+++-+
imageC1C2C2C3C4C6C6C12Q8D4SD16C3×Q8C3×D4C3×SD16
kernelC3×C4.Q8C3×C4⋊C4C2×C24C4.Q8C24C4⋊C4C2×C8C8C12C2×C6C6C4C22C2
# reps12124428114228

Matrix representation of C3×C4.Q8 in GL4(𝔽73) generated by

64000
06400
00640
00064
,
1000
0100
00072
0010
,
27000
04600
00667
0066
,
0100
72000
001653
005357
G:=sub<GL(4,GF(73))| [64,0,0,0,0,64,0,0,0,0,64,0,0,0,0,64],[1,0,0,0,0,1,0,0,0,0,0,1,0,0,72,0],[27,0,0,0,0,46,0,0,0,0,6,6,0,0,67,6],[0,72,0,0,1,0,0,0,0,0,16,53,0,0,53,57] >;

C3×C4.Q8 in GAP, Magma, Sage, TeX

C_3\times C_4.Q_8
% in TeX

G:=Group("C3xC4.Q8");
// GroupNames label

G:=SmallGroup(96,56);
// by ID

G=gap.SmallGroup(96,56);
# by ID

G:=PCGroup([6,-2,-2,-3,-2,-2,-2,144,169,79,1443,117]);
// Polycyclic

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

Export

Subgroup lattice of C3×C4.Q8 in TeX

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