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G = C12.78D12order 288 = 25·32

9th non-split extension by C12 of D12 acting via D12/D6=C2

metabelian, supersoluble, monomial

Aliases: C12.78D12, C6.4(S3×C8), C31(D6⋊C8), C6.1(D6⋊C4), C6.2(C8⋊S3), (C3×C12).108D4, (C2×C12).296D6, C325(C22⋊C8), C62.26(C2×C4), (C3×C6).6M4(2), C12.77(C3⋊D4), C4.27(C3⋊D12), (C6×C12).201C22, C2.4(C12.29D6), C2.2(C12.31D6), C2.1(C6.D12), C22.9(C6.D6), (C6×C3⋊C8)⋊2C2, (C2×C3⋊S3)⋊3C8, (C2×C3⋊C8)⋊10S3, (C2×C4).129S32, (C2×C6).27(C4×S3), (C3×C6).19(C2×C8), (C22×C3⋊S3).6C4, (C2×C3⋊Dic3).8C4, (C3×C6).24(C22⋊C4), (C2×C4×C3⋊S3).11C2, SmallGroup(288,205)

Series: Derived Chief Lower central Upper central

C1C3×C6 — C12.78D12
C1C3C32C3×C6C3×C12C6×C12C6×C3⋊C8 — C12.78D12
C32C3×C6 — C12.78D12
C1C2×C4

Generators and relations for C12.78D12
 G = < a,b,c | a12=1, b12=c2=a6, bab-1=cac-1=a5, cbc-1=a3b11 >

Subgroups: 530 in 127 conjugacy classes, 42 normal (16 characteristic)
C1, C2 [×3], C2 [×2], C3 [×2], C3, C4 [×2], C4, C22, C22 [×4], S3 [×8], C6 [×6], C6 [×3], C8 [×2], C2×C4, C2×C4 [×3], C23, C32, Dic3 [×4], C12 [×4], C12 [×2], D6 [×14], C2×C6 [×2], C2×C6, C2×C8 [×2], C22×C4, C3⋊S3 [×2], C3×C6 [×3], C3⋊C8 [×2], C24 [×2], C4×S3 [×8], C2×Dic3 [×3], C2×C12 [×2], C2×C12, C22×S3 [×3], C22⋊C8, C3⋊Dic3, C3×C12 [×2], C2×C3⋊S3 [×2], C2×C3⋊S3 [×2], C62, C2×C3⋊C8 [×2], C2×C24 [×2], S3×C2×C4 [×3], C3×C3⋊C8 [×2], C4×C3⋊S3 [×2], C2×C3⋊Dic3, C6×C12, C22×C3⋊S3, D6⋊C8 [×2], C6×C3⋊C8 [×2], C2×C4×C3⋊S3, C12.78D12
Quotients: C1, C2 [×3], C4 [×2], C22, S3 [×2], C8 [×2], C2×C4, D4 [×2], D6 [×2], C22⋊C4, C2×C8, M4(2), C4×S3 [×2], D12 [×2], C3⋊D4 [×2], C22⋊C8, S32, S3×C8 [×2], C8⋊S3 [×2], D6⋊C4 [×2], C6.D6, C3⋊D12 [×2], D6⋊C8 [×2], C12.29D6, C12.31D6, C6.D12, C12.78D12

Smallest permutation representation of C12.78D12
On 48 points
Generators in S48
(1 39 5 43 9 47 13 27 17 31 21 35)(2 48 22 44 18 40 14 36 10 32 6 28)(3 41 7 45 11 25 15 29 19 33 23 37)(4 26 24 46 20 42 16 38 12 34 8 30)
(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)
(1 3 13 15)(2 32 14 44)(4 42 16 30)(5 23 17 11)(6 28 18 40)(7 9 19 21)(8 38 20 26)(10 48 22 36)(12 34 24 46)(25 27 37 39)(29 47 41 35)(31 33 43 45)

G:=sub<Sym(48)| (1,39,5,43,9,47,13,27,17,31,21,35)(2,48,22,44,18,40,14,36,10,32,6,28)(3,41,7,45,11,25,15,29,19,33,23,37)(4,26,24,46,20,42,16,38,12,34,8,30), (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), (1,3,13,15)(2,32,14,44)(4,42,16,30)(5,23,17,11)(6,28,18,40)(7,9,19,21)(8,38,20,26)(10,48,22,36)(12,34,24,46)(25,27,37,39)(29,47,41,35)(31,33,43,45)>;

G:=Group( (1,39,5,43,9,47,13,27,17,31,21,35)(2,48,22,44,18,40,14,36,10,32,6,28)(3,41,7,45,11,25,15,29,19,33,23,37)(4,26,24,46,20,42,16,38,12,34,8,30), (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), (1,3,13,15)(2,32,14,44)(4,42,16,30)(5,23,17,11)(6,28,18,40)(7,9,19,21)(8,38,20,26)(10,48,22,36)(12,34,24,46)(25,27,37,39)(29,47,41,35)(31,33,43,45) );

G=PermutationGroup([(1,39,5,43,9,47,13,27,17,31,21,35),(2,48,22,44,18,40,14,36,10,32,6,28),(3,41,7,45,11,25,15,29,19,33,23,37),(4,26,24,46,20,42,16,38,12,34,8,30)], [(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)], [(1,3,13,15),(2,32,14,44),(4,42,16,30),(5,23,17,11),(6,28,18,40),(7,9,19,21),(8,38,20,26),(10,48,22,36),(12,34,24,46),(25,27,37,39),(29,47,41,35),(31,33,43,45)])

60 conjugacy classes

class 1 2A2B2C2D2E3A3B3C4A4B4C4D4E4F6A···6F6G6H6I8A···8H12A···12H12I12J12K12L24A···24P
order1222223334444446···66668···812···121212121224···24
size11111818224111118182···24446···62···244446···6

60 irreducible representations

dim11111122222222244444
type++++++++++
imageC1C2C2C4C4C8S3D4D6M4(2)D12C3⋊D4C4×S3S3×C8C8⋊S3S32C3⋊D12C6.D6C12.29D6C12.31D6
kernelC12.78D12C6×C3⋊C8C2×C4×C3⋊S3C2×C3⋊Dic3C22×C3⋊S3C2×C3⋊S3C2×C3⋊C8C3×C12C2×C12C3×C6C12C12C2×C6C6C6C2×C4C4C22C2C2
# reps12122822224448812122

Matrix representation of C12.78D12 in GL6(𝔽73)

110000
7200000
0072000
0007200
0000270
0000027
,
4600000
27270000
00467100
00722700
00006310
0000630
,
7200000
110000
001000
00467200
0000027
0000270

G:=sub<GL(6,GF(73))| [1,72,0,0,0,0,1,0,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,27,0,0,0,0,0,0,27],[46,27,0,0,0,0,0,27,0,0,0,0,0,0,46,72,0,0,0,0,71,27,0,0,0,0,0,0,63,63,0,0,0,0,10,0],[72,1,0,0,0,0,0,1,0,0,0,0,0,0,1,46,0,0,0,0,0,72,0,0,0,0,0,0,0,27,0,0,0,0,27,0] >;

C12.78D12 in GAP, Magma, Sage, TeX

C_{12}._{78}D_{12}
% in TeX

G:=Group("C12.78D12");
// GroupNames label

G:=SmallGroup(288,205);
// by ID

G=gap.SmallGroup(288,205);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,28,141,92,100,1356,9414]);
// Polycyclic

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

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