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G = C12.73D12order 288 = 25·32

4th non-split extension by C12 of D12 acting via D12/D6=C2

metabelian, supersoluble, monomial

Aliases: C12.73D12, C6.7Dic12, C62.22D4, (C3×C6).5Q16, C12.41(C4×S3), C6.5(D6⋊C4), (C2×C6).52D12, (C3×C12).37D4, C4⋊Dic3.1S3, C324Q86C4, (C2×C12).275D6, C6.2(D4.S3), (C3×C6).12SD16, C6.2(C3⋊Q16), C6.11(C24⋊C2), C31(C6.SD16), C12.33(C3⋊D4), (C6×C12).29C22, C31(C2.Dic12), C4.2(C6.D6), C326(Q8⋊C4), C4.15(C3⋊D12), C2.2(C323Q16), C2.2(D12.S3), C2.6(C6.D12), C22.15(C3⋊D12), (C6×C3⋊C8).2C2, (C2×C4).55S32, (C2×C3⋊C8).2S3, (C3×C12).27(C2×C4), (C3×C4⋊Dic3).2C2, (C2×C6).33(C3⋊D4), (C2×C324Q8).8C2, (C3×C6).34(C22⋊C4), SmallGroup(288,215)

Series: Derived Chief Lower central Upper central

C1C3×C12 — C12.73D12
C1C3C32C3×C6C62C6×C12C3×C4⋊Dic3 — C12.73D12
C32C3×C6C3×C12 — C12.73D12
C1C22C2×C4

Generators and relations for C12.73D12
 G = < a,b,c | a12=b12=1, c2=a6, bab-1=cac-1=a-1, cbc-1=a9b-1 >

Subgroups: 386 in 103 conjugacy classes, 40 normal (38 characteristic)
C1, C2 [×3], C3 [×2], C3, C4 [×2], C4 [×3], C22, C6 [×6], C6 [×3], C8, C2×C4, C2×C4 [×2], Q8 [×3], C32, Dic3 [×9], C12 [×4], C12 [×3], C2×C6 [×2], C2×C6, C4⋊C4, C2×C8, C2×Q8, C3×C6 [×3], C3⋊C8, C24, Dic6 [×10], C2×Dic3 [×5], C2×C12 [×2], C2×C12 [×2], Q8⋊C4, C3×Dic3, C3⋊Dic3 [×2], C3×C12 [×2], C62, C2×C3⋊C8, C4⋊Dic3, C3×C4⋊C4, C2×C24, C2×Dic6 [×3], C3×C3⋊C8, C6×Dic3, C324Q8 [×2], C324Q8, C2×C3⋊Dic3, C6×C12, C6.SD16, C2.Dic12, C6×C3⋊C8, C3×C4⋊Dic3, C2×C324Q8, C12.73D12
Quotients: C1, C2 [×3], C4 [×2], C22, S3 [×2], C2×C4, D4 [×2], D6 [×2], C22⋊C4, SD16, Q16, C4×S3 [×2], D12 [×2], C3⋊D4 [×2], Q8⋊C4, S32, C24⋊C2, Dic12, D6⋊C4 [×2], D4.S3, C3⋊Q16, C6.D6, C3⋊D12 [×2], C6.SD16, C2.Dic12, D12.S3, C323Q16, C6.D12, C12.73D12

Smallest permutation representation of C12.73D12
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 63 18 5 47 67 14 9 43 71 22)(2 38 64 17 6 46 68 13 10 42 72 21)(3 37 65 16 7 45 69 24 11 41 61 20)(4 48 66 15 8 44 70 23 12 40 62 19)(25 86 74 49 33 90 82 53 29 94 78 57)(26 85 75 60 34 89 83 52 30 93 79 56)(27 96 76 59 35 88 84 51 31 92 80 55)(28 95 77 58 36 87 73 50 32 91 81 54)
(1 59 7 53)(2 58 8 52)(3 57 9 51)(4 56 10 50)(5 55 11 49)(6 54 12 60)(13 35 19 29)(14 34 20 28)(15 33 21 27)(16 32 22 26)(17 31 23 25)(18 30 24 36)(37 81 43 75)(38 80 44 74)(39 79 45 73)(40 78 46 84)(41 77 47 83)(42 76 48 82)(61 86 67 92)(62 85 68 91)(63 96 69 90)(64 95 70 89)(65 94 71 88)(66 93 72 87)

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

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

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

48 conjugacy classes

class 1 2A2B2C3A3B3C4A4B4C4D4E4F6A···6F6G6H6I8A8B8C8D12A12B12C12D12E···12J12K12L12M12N24A···24H
order12223334444446···666688881212121212···121212121224···24
size111122422121236362···2444666622224···4121212126···6

48 irreducible representations

dim111112222222222222244444444
type+++++++++-++-+--+++--
imageC1C2C2C2C4S3S3D4D4D6SD16Q16C4×S3D12C3⋊D4D12C3⋊D4C24⋊C2Dic12S32D4.S3C3⋊Q16C6.D6C3⋊D12C3⋊D12D12.S3C323Q16
kernelC12.73D12C6×C3⋊C8C3×C4⋊Dic3C2×C324Q8C324Q8C2×C3⋊C8C4⋊Dic3C3×C12C62C2×C12C3×C6C3×C6C12C12C12C2×C6C2×C6C6C6C2×C4C6C6C4C4C22C2C2
# reps111141111222422224411111122

Matrix representation of C12.73D12 in GL4(𝔽73) generated by

72000
07200
001466
0077
,
02700
464600
00505
005523
,
07200
72000
003013
004343
G:=sub<GL(4,GF(73))| [72,0,0,0,0,72,0,0,0,0,14,7,0,0,66,7],[0,46,0,0,27,46,0,0,0,0,50,55,0,0,5,23],[0,72,0,0,72,0,0,0,0,0,30,43,0,0,13,43] >;

C12.73D12 in GAP, Magma, Sage, TeX

C_{12}._{73}D_{12}
% in TeX

G:=Group("C12.73D12");
// GroupNames label

G:=SmallGroup(288,215);
// by ID

G=gap.SmallGroup(288,215);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,28,141,92,675,80,1356,9414]);
// Polycyclic

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

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