Copied to
clipboard

G = C3×C4.Dic6order 288 = 25·32

Direct product of C3 and C4.Dic6

direct product, metabelian, supersoluble, monomial

Aliases: C3×C4.Dic6, C12.18Dic6, C62.181C23, C6.5(C6×Q8), C12.3(C3×Q8), C4⋊Dic3.7C6, (C3×C12).15Q8, C4.3(C3×Dic6), C2.8(C6×Dic6), (C2×C12).235D6, Dic3⋊C4.3C6, (C4×Dic3).2C6, C6.52(C2×Dic6), (C6×C12).192C22, C6.56(Q83S3), (Dic3×C12).11C2, C6.118(D42S3), C3212(C42.C2), (C6×Dic3).124C22, (C3×C4⋊C4).7C6, C4⋊C4.6(C3×S3), (C3×C4⋊C4).29S3, (C2×C4).42(S3×C6), C6.25(C3×C4○D4), C22.48(S3×C2×C6), (C3×C6).48(C2×Q8), C33(C3×C42.C2), (C2×C12).22(C2×C6), C2.4(C3×Q83S3), C2.12(C3×D42S3), (C32×C4⋊C4).10C2, (C3×C4⋊Dic3).26C2, (C2×C6).36(C22×C6), (C3×C6).132(C4○D4), (C3×Dic3⋊C4).11C2, (C2×C6).314(C22×S3), (C2×Dic3).27(C2×C6), SmallGroup(288,661)

Series: Derived Chief Lower central Upper central

C1C2×C6 — C3×C4.Dic6
C1C3C6C2×C6C62C6×Dic3Dic3×C12 — C3×C4.Dic6
C3C2×C6 — C3×C4.Dic6
C1C2×C6C3×C4⋊C4

Generators and relations for C3×C4.Dic6
 G = < a,b,c,d | a3=b12=c4=1, d2=c2, ab=ba, ac=ca, ad=da, cbc-1=b7, dbd-1=b5, dcd-1=b6c-1 >

Subgroups: 242 in 125 conjugacy classes, 66 normal (38 characteristic)
C1, C2 [×3], C3 [×2], C3, C4 [×2], C4 [×6], C22, C6 [×6], C6 [×3], C2×C4, C2×C4 [×2], C2×C4 [×4], C32, Dic3 [×4], C12 [×4], C12 [×12], C2×C6 [×2], C2×C6, C42, C4⋊C4, C4⋊C4 [×5], C3×C6 [×3], C2×Dic3 [×2], C2×Dic3 [×2], C2×C12 [×2], C2×C12 [×4], C2×C12 [×7], C42.C2, C3×Dic3 [×4], C3×C12 [×2], C3×C12 [×2], C62, C4×Dic3, Dic3⋊C4 [×2], C4⋊Dic3, C4⋊Dic3 [×2], C4×C12, C3×C4⋊C4 [×2], C3×C4⋊C4 [×6], C6×Dic3 [×2], C6×Dic3 [×2], C6×C12, C6×C12 [×2], C4.Dic6, C3×C42.C2, Dic3×C12, C3×Dic3⋊C4 [×2], C3×C4⋊Dic3, C3×C4⋊Dic3 [×2], C32×C4⋊C4, C3×C4.Dic6
Quotients: C1, C2 [×7], C3, C22 [×7], S3, C6 [×7], Q8 [×2], C23, D6 [×3], C2×C6 [×7], C2×Q8, C4○D4 [×2], C3×S3, Dic6 [×2], C3×Q8 [×2], C22×S3, C22×C6, C42.C2, S3×C6 [×3], C2×Dic6, D42S3, Q83S3, C6×Q8, C3×C4○D4 [×2], C3×Dic6 [×2], S3×C2×C6, C4.Dic6, C3×C42.C2, C6×Dic6, C3×D42S3, C3×Q83S3, C3×C4.Dic6

Smallest permutation representation of C3×C4.Dic6
On 96 points
Generators in S96
(1 5 9)(2 6 10)(3 7 11)(4 8 12)(13 17 21)(14 18 22)(15 19 23)(16 20 24)(25 29 33)(26 30 34)(27 31 35)(28 32 36)(37 45 41)(38 46 42)(39 47 43)(40 48 44)(49 57 53)(50 58 54)(51 59 55)(52 60 56)(61 65 69)(62 66 70)(63 67 71)(64 68 72)(73 81 77)(74 82 78)(75 83 79)(76 84 80)(85 93 89)(86 94 90)(87 95 91)(88 96 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)
(1 35 70 19)(2 30 71 14)(3 25 72 21)(4 32 61 16)(5 27 62 23)(6 34 63 18)(7 29 64 13)(8 36 65 20)(9 31 66 15)(10 26 67 22)(11 33 68 17)(12 28 69 24)(37 78 52 85)(38 73 53 92)(39 80 54 87)(40 75 55 94)(41 82 56 89)(42 77 57 96)(43 84 58 91)(44 79 59 86)(45 74 60 93)(46 81 49 88)(47 76 50 95)(48 83 51 90)
(1 80 70 87)(2 73 71 92)(3 78 72 85)(4 83 61 90)(5 76 62 95)(6 81 63 88)(7 74 64 93)(8 79 65 86)(9 84 66 91)(10 77 67 96)(11 82 68 89)(12 75 69 94)(13 54 29 39)(14 59 30 44)(15 52 31 37)(16 57 32 42)(17 50 33 47)(18 55 34 40)(19 60 35 45)(20 53 36 38)(21 58 25 43)(22 51 26 48)(23 56 27 41)(24 49 28 46)

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

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

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

72 conjugacy classes

class 1 2A2B2C3A3B3C3D3E4A4B4C4D4E4F4G4H4I4J6A···6F6G···6O12A12B12C12D12E···12Z12AA···12AH12AI12AJ12AK12AL
order12223333344444444446···66···61212121212···1212···1212121212
size1111112222244666612121···12···222224···46···612121212

72 irreducible representations

dim111111111122222222224444
type++++++-+--+
imageC1C2C2C2C2C3C6C6C6C6S3Q8D6C4○D4C3×S3Dic6C3×Q8S3×C6C3×C4○D4C3×Dic6D42S3Q83S3C3×D42S3C3×Q83S3
kernelC3×C4.Dic6Dic3×C12C3×Dic3⋊C4C3×C4⋊Dic3C32×C4⋊C4C4.Dic6C4×Dic3Dic3⋊C4C4⋊Dic3C3×C4⋊C4C3×C4⋊C4C3×C12C2×C12C3×C6C4⋊C4C12C12C2×C4C6C4C6C6C2C2
# reps112312246212342446881122

Matrix representation of C3×C4.Dic6 in GL4(𝔽13) generated by

1000
0100
0090
0009
,
01200
1000
0090
0023
,
0100
1000
0080
0015
,
0800
5000
0082
0005
G:=sub<GL(4,GF(13))| [1,0,0,0,0,1,0,0,0,0,9,0,0,0,0,9],[0,1,0,0,12,0,0,0,0,0,9,2,0,0,0,3],[0,1,0,0,1,0,0,0,0,0,8,1,0,0,0,5],[0,5,0,0,8,0,0,0,0,0,8,0,0,0,2,5] >;

C3×C4.Dic6 in GAP, Magma, Sage, TeX

C_3\times C_4.{\rm Dic}_6
% in TeX

G:=Group("C3xC4.Dic6");
// GroupNames label

G:=SmallGroup(288,661);
// by ID

G=gap.SmallGroup(288,661);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,168,344,1598,555,142,9414]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^12=c^4=1,d^2=c^2,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^7,d*b*d^-1=b^5,d*c*d^-1=b^6*c^-1>;
// generators/relations

׿
×
𝔽