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G = C12.6Dic6order 288 = 25·32

6th non-split extension by C12 of Dic6 acting via Dic6/C6=C22

metabelian, supersoluble, monomial

Aliases: C12.6Dic6, C62.27D4, (C3×C12).6Q8, C12.21(C4×S3), C324C81C4, (C2×C12).81D6, C4⋊Dic3.9S3, C326(C4.Q8), C6.6(D4.S3), (C3×C6).14SD16, C6.4(Dic3⋊C4), (C6×C12).34C22, C6.6(Q82S3), C32(C12.Q8), C4.4(C322Q8), C4.6(C6.D6), C2.3(Dic6⋊S3), C22.9(D6⋊S3), C2.3(C62.C22), (C2×C4).102S32, (C3×C6).21(C4⋊C4), (C3×C12).32(C2×C4), (C3×C4⋊Dic3).4C2, (C2×C6).51(C3⋊D4), (C2×C324C8).2C2, SmallGroup(288,222)

Series: Derived Chief Lower central Upper central

C1C3×C12 — C12.6Dic6
C1C3C32C3×C6C62C6×C12C3×C4⋊Dic3 — C12.6Dic6
C32C3×C6C3×C12 — C12.6Dic6
C1C22C2×C4

Generators and relations for C12.6Dic6
 G = < a,b,c | a12=b12=1, c2=a6b6, bab-1=a-1, cac-1=a7, cbc-1=a9b-1 >

Subgroups: 242 in 83 conjugacy classes, 40 normal (14 characteristic)
C1, C2, C2 [×2], C3 [×2], C3, C4 [×2], C4 [×2], C22, C6 [×6], C6 [×3], C8 [×2], C2×C4, C2×C4 [×2], C32, Dic3 [×2], C12 [×4], C12 [×4], C2×C6 [×2], C2×C6, C4⋊C4 [×2], C2×C8, C3×C6, C3×C6 [×2], C3⋊C8 [×6], C2×Dic3 [×2], C2×C12 [×2], C2×C12 [×3], C4.Q8, C3×Dic3 [×2], C3×C12 [×2], C62, C2×C3⋊C8 [×3], C4⋊Dic3 [×2], C3×C4⋊C4 [×2], C324C8 [×2], C6×Dic3 [×2], C6×C12, C12.Q8 [×2], C3×C4⋊Dic3 [×2], C2×C324C8, C12.6Dic6
Quotients: C1, C2 [×3], C4 [×2], C22, S3 [×2], C2×C4, D4, Q8, D6 [×2], C4⋊C4, SD16 [×2], Dic6 [×2], C4×S3 [×2], C3⋊D4 [×2], C4.Q8, S32, Dic3⋊C4 [×2], D4.S3 [×2], Q82S3 [×2], C6.D6, D6⋊S3, C322Q8, C12.Q8 [×2], Dic6⋊S3 [×2], C62.C22, C12.6Dic6

Smallest permutation representation of C12.6Dic6
On 96 points
Generators in S96
(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 39 34 54 9 43 30 58 5 47 26 50)(2 38 35 53 10 42 31 57 6 46 27 49)(3 37 36 52 11 41 32 56 7 45 28 60)(4 48 25 51 12 40 33 55 8 44 29 59)(13 65 96 84 21 69 92 76 17 61 88 80)(14 64 85 83 22 68 93 75 18 72 89 79)(15 63 86 82 23 67 94 74 19 71 90 78)(16 62 87 81 24 66 95 73 20 70 91 77)
(1 61 36 78)(2 68 25 73)(3 63 26 80)(4 70 27 75)(5 65 28 82)(6 72 29 77)(7 67 30 84)(8 62 31 79)(9 69 32 74)(10 64 33 81)(11 71 34 76)(12 66 35 83)(13 51 86 38)(14 58 87 45)(15 53 88 40)(16 60 89 47)(17 55 90 42)(18 50 91 37)(19 57 92 44)(20 52 93 39)(21 59 94 46)(22 54 95 41)(23 49 96 48)(24 56 85 43)

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

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

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

42 conjugacy classes

class 1 2A2B2C3A3B3C4A4B4C4D4E4F6A···6F6G6H6I8A8B8C8D12A···12H12I···12P
order12223334444446···6666888812···1212···12
size111122422121212122···2444181818184···412···12

42 irreducible representations

dim1111222222224444444
type++++-++-+-++--
imageC1C2C2C4S3Q8D4D6SD16Dic6C4×S3C3⋊D4S32D4.S3Q82S3C6.D6C322Q8D6⋊S3Dic6⋊S3
kernelC12.6Dic6C3×C4⋊Dic3C2×C324C8C324C8C4⋊Dic3C3×C12C62C2×C12C3×C6C12C12C2×C6C2×C4C6C6C4C4C22C2
# reps1214211244441221114

Matrix representation of C12.6Dic6 in GL6(𝔽73)

4630000
0270000
000100
0072100
0000720
0000072
,
5830000
22150000
0004600
0046000
00003043
00003060
,
35210000
46380000
0027000
0002700
00001113
0000262

G:=sub<GL(6,GF(73))| [46,0,0,0,0,0,3,27,0,0,0,0,0,0,0,72,0,0,0,0,1,1,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[58,22,0,0,0,0,3,15,0,0,0,0,0,0,0,46,0,0,0,0,46,0,0,0,0,0,0,0,30,30,0,0,0,0,43,60],[35,46,0,0,0,0,21,38,0,0,0,0,0,0,27,0,0,0,0,0,0,27,0,0,0,0,0,0,11,2,0,0,0,0,13,62] >;

C12.6Dic6 in GAP, Magma, Sage, TeX

C_{12}._6{\rm Dic}_6
% in TeX

G:=Group("C12.6Dic6");
// GroupNames label

G:=SmallGroup(288,222);
// by ID

G=gap.SmallGroup(288,222);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,28,141,148,675,346,80,1356,9414]);
// Polycyclic

G:=Group<a,b,c|a^12=b^12=1,c^2=a^6*b^6,b*a*b^-1=a^-1,c*a*c^-1=a^7,c*b*c^-1=a^9*b^-1>;
// generators/relations

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