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G = C12.15Dic6order 288 = 25·32

2nd non-split extension by C12 of Dic6 acting via Dic6/Dic3=C2

metabelian, supersoluble, monomial

Aliases: C12.15Dic6, C6.5(S3×C8), C327(C4⋊C8), C3⋊Dic34C8, (C3×C12).18Q8, C6.3(C8⋊S3), C31(Dic3⋊C8), (C3×C12).113D4, (C2×C12).298D6, C62.32(C2×C4), (C3×C6).8M4(2), C12.96(C3⋊D4), C6.2(Dic3⋊C4), C4.8(C322Q8), C4.18(D6⋊S3), (C6×C12).203C22, C2.3(C12.31D6), C2.5(C12.29D6), C2.1(C62.C22), C22.10(C6.D6), (C6×C3⋊C8).4C2, (C2×C4).131S32, (C2×C3⋊C8).10S3, (C3×C6).21(C2×C8), (C2×C6).28(C4×S3), (C3×C6).19(C4⋊C4), (C2×C3⋊Dic3).9C4, (C4×C3⋊Dic3).11C2, SmallGroup(288,220)

Series: Derived Chief Lower central Upper central

C1C3×C6 — C12.15Dic6
C1C3C32C3×C6C3×C12C6×C12C6×C3⋊C8 — C12.15Dic6
C32C3×C6 — C12.15Dic6
C1C2×C4

Generators and relations for C12.15Dic6
 G = < a,b,c | a12=1, b12=a6, c2=a6b6, bab-1=a5, ac=ca, cbc-1=a3b11 >

Subgroups: 266 in 91 conjugacy classes, 42 normal (20 characteristic)
C1, C2 [×3], C3 [×2], C3, C4 [×2], C4 [×3], C22, C6 [×6], C6 [×3], C8 [×2], C2×C4, C2×C4 [×2], C32, Dic3 [×10], C12 [×4], C12 [×2], C2×C6 [×2], C2×C6, C42, C2×C8 [×2], C3×C6 [×3], C3⋊C8 [×2], C24 [×2], C2×Dic3 [×6], C2×C12 [×2], C2×C12, C4⋊C8, C3⋊Dic3 [×2], C3⋊Dic3, C3×C12 [×2], C62, C2×C3⋊C8 [×2], C4×Dic3 [×3], C2×C24 [×2], C3×C3⋊C8 [×2], C2×C3⋊Dic3 [×2], C6×C12, Dic3⋊C8 [×2], C6×C3⋊C8 [×2], C4×C3⋊Dic3, C12.15Dic6
Quotients: C1, C2 [×3], C4 [×2], C22, S3 [×2], C8 [×2], C2×C4, D4, Q8, D6 [×2], C4⋊C4, C2×C8, M4(2), Dic6 [×2], C4×S3 [×2], C3⋊D4 [×2], C4⋊C8, S32, S3×C8 [×2], C8⋊S3 [×2], Dic3⋊C4 [×2], C6.D6, D6⋊S3, C322Q8, Dic3⋊C8 [×2], C12.29D6, C12.31D6, C62.C22, C12.15Dic6

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

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

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

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

60 conjugacy classes

class 1 2A2B2C3A3B3C4A4B4C4D4E4F4G4H6A···6F6G6H6I8A···8H12A···12H12I12J12K12L24A···24P
order1222333444444446···66668···812···121212121224···24
size11112241111181818182···24446···62···244446···6

60 irreducible representations

dim111112222222222444444
type+++++-+-+--+
imageC1C2C2C4C8S3D4Q8D6M4(2)Dic6C3⋊D4C4×S3S3×C8C8⋊S3S32D6⋊S3C322Q8C6.D6C12.29D6C12.31D6
kernelC12.15Dic6C6×C3⋊C8C4×C3⋊Dic3C2×C3⋊Dic3C3⋊Dic3C2×C3⋊C8C3×C12C3×C12C2×C12C3×C6C12C12C2×C6C6C6C2×C4C4C4C22C2C2
# reps121482112244488111122

Matrix representation of C12.15Dic6 in GL6(𝔽73)

100000
010000
0046000
0004600
0000072
0000172
,
010000
7200000
0002200
00512200
000001
000010
,
23280000
28500000
0005100
0051000
000010
000001

G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,46,0,0,0,0,0,0,46,0,0,0,0,0,0,0,1,0,0,0,0,72,72],[0,72,0,0,0,0,1,0,0,0,0,0,0,0,0,51,0,0,0,0,22,22,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[23,28,0,0,0,0,28,50,0,0,0,0,0,0,0,51,0,0,0,0,51,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

C12.15Dic6 in GAP, Magma, Sage, TeX

C_{12}._{15}{\rm Dic}_6
% in TeX

G:=Group("C12.15Dic6");
// GroupNames label

G:=SmallGroup(288,220);
// by ID

G=gap.SmallGroup(288,220);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,28,141,36,100,1356,9414]);
// Polycyclic

G:=Group<a,b,c|a^12=1,b^12=a^6,c^2=a^6*b^6,b*a*b^-1=a^5,a*c=c*a,c*b*c^-1=a^3*b^11>;
// generators/relations

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