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G = C12.27D12order 288 = 25·32

27th non-split extension by C12 of D12 acting via D12/C6=C22

metabelian, supersoluble, monomial

Aliases: C12.27D12, C62.30C23, D6⋊Dic32C2, (C4×Dic3)⋊3S3, (C2×D12).3S3, (C6×D12).8C2, C6.75(C2×D12), (C3×C12).75D4, (Dic3×C12)⋊8C2, (C2×C12).130D6, (C22×S3).6D6, C34(C427S3), C6.41(C4○D12), C4.9(C3⋊D12), C12.74(C3⋊D4), (C6×C12).92C22, (C2×Dic3).94D6, C323(C4.4D4), C6.17(D42S3), C31(C23.12D6), C2.10(D125S3), (C6×Dic3).107C22, (C2×C4).74S32, C22.87(C2×S32), (C3×C6).83(C2×D4), C6.11(C2×C3⋊D4), (S3×C2×C6).6C22, (C3×C6).17(C4○D4), C2.15(C2×C3⋊D12), (C2×C6).49(C22×S3), (C2×C324Q8)⋊12C2, (C2×C3⋊Dic3).28C22, SmallGroup(288,508)

Series: Derived Chief Lower central Upper central

C1C62 — C12.27D12
C1C3C32C3×C6C62S3×C2×C6D6⋊Dic3 — C12.27D12
C32C62 — C12.27D12
C1C22C2×C4

Generators and relations for C12.27D12
 G = < a,b,c | a12=b12=1, c2=a6, bab-1=a5, cac-1=a-1, cbc-1=a6b-1 >

Subgroups: 642 in 171 conjugacy classes, 52 normal (22 characteristic)
C1, C2, C2 [×2], C2 [×2], C3 [×2], C3, C4 [×2], C4 [×4], C22, C22 [×6], S3 [×2], C6 [×2], C6 [×4], C6 [×5], C2×C4, C2×C4 [×4], D4 [×2], Q8 [×2], C23 [×2], C32, Dic3 [×10], C12 [×4], C12 [×4], D6 [×6], C2×C6 [×2], C2×C6 [×7], C42, C22⋊C4 [×4], C2×D4, C2×Q8, C3×S3 [×2], C3×C6, C3×C6 [×2], Dic6 [×8], D12 [×2], C2×Dic3 [×2], C2×Dic3 [×6], C2×C12 [×2], C2×C12 [×3], C3×D4 [×2], C22×S3 [×2], C22×C6 [×2], C4.4D4, C3×Dic3 [×2], C3⋊Dic3 [×2], C3×C12 [×2], S3×C6 [×6], C62, C4×Dic3, D6⋊C4 [×4], C6.D4 [×4], C4×C12, C2×Dic6 [×3], C2×D12, C6×D4, C3×D12 [×2], C6×Dic3 [×2], C324Q8 [×2], C2×C3⋊Dic3 [×2], C6×C12, S3×C2×C6 [×2], C427S3, C23.12D6, D6⋊Dic3 [×4], Dic3×C12, C6×D12, C2×C324Q8, C12.27D12
Quotients: C1, C2 [×7], C22 [×7], S3 [×2], D4 [×2], C23, D6 [×6], C2×D4, C4○D4 [×2], D12 [×2], C3⋊D4 [×2], C22×S3 [×2], C4.4D4, S32, C2×D12, C4○D12 [×2], D42S3 [×2], C2×C3⋊D4, C3⋊D12 [×2], C2×S32, C427S3, C23.12D6, D125S3 [×2], C2×C3⋊D12, C12.27D12

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

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

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

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

48 conjugacy classes

class 1 2A2B2C2D2E3A3B3C4A4B4C4D4E4F4G4H6A···6F6G6H6I6J6K6L6M12A12B12C12D12E···12J12K···12R
order122222333444444446···666666661212121212···1212···12
size1111121222422666636362···24441212121222224···46···6

48 irreducible representations

dim11111222222222244444
type+++++++++++++-++-
imageC1C2C2C2C2S3S3D4D6D6D6C4○D4D12C3⋊D4C4○D12S32D42S3C3⋊D12C2×S32D125S3
kernelC12.27D12D6⋊Dic3Dic3×C12C6×D12C2×C324Q8C4×Dic3C2×D12C3×C12C2×Dic3C2×C12C22×S3C3×C6C12C12C6C2×C4C6C4C22C2
# reps14111112222444812214

Matrix representation of C12.27D12 in GL6(𝔽13)

1200000
0120000
00121200
001000
000036
0000710
,
520000
080000
0012000
001100
0000911
0000211
,
990000
740000
001000
00121200
000024
0000211

G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,1,0,0,0,0,12,0,0,0,0,0,0,0,3,7,0,0,0,0,6,10],[5,0,0,0,0,0,2,8,0,0,0,0,0,0,12,1,0,0,0,0,0,1,0,0,0,0,0,0,9,2,0,0,0,0,11,11],[9,7,0,0,0,0,9,4,0,0,0,0,0,0,1,12,0,0,0,0,0,12,0,0,0,0,0,0,2,2,0,0,0,0,4,11] >;

C12.27D12 in GAP, Magma, Sage, TeX

C_{12}._{27}D_{12}
% in TeX

G:=Group("C12.27D12");
// GroupNames label

G:=SmallGroup(288,508);
// by ID

G=gap.SmallGroup(288,508);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,141,176,422,100,1356,9414]);
// Polycyclic

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

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