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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(S3xC8), C3:1(D6:C8), C6.1(D6:C4), C6.2(C8:S3), (C3xC12).108D4, (C2xC12).296D6, C32:5(C22:C8), C62.26(C2xC4), (C3xC6).6M4(2), C12.77(C3:D4), C4.27(C3:D12), (C6xC12).201C22, C2.4(C12.29D6), C2.2(C12.31D6), C2.1(C6.D12), C22.9(C6.D6), (C6xC3:C8):2C2, (C2xC3:S3):3C8, (C2xC3:C8):10S3, (C2xC4).129S32, (C2xC6).27(C4xS3), (C3xC6).19(C2xC8), (C22xC3:S3).6C4, (C2xC3:Dic3).8C4, (C3xC6).24(C22:C4), (C2xC4xC3:S3).11C2, SmallGroup(288,205)

Series: Derived Chief Lower central Upper central

C1C3xC6 — C12.78D12
C1C3C32C3xC6C3xC12C6xC12C6xC3:C8 — C12.78D12
C32C3xC6 — C12.78D12
C1C2xC4

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, C2, C3, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2xC4, C2xC4, C23, C32, Dic3, C12, C12, D6, C2xC6, C2xC6, C2xC8, C22xC4, C3:S3, C3xC6, C3:C8, C24, C4xS3, C2xDic3, C2xC12, C2xC12, C22xS3, C22:C8, C3:Dic3, C3xC12, C2xC3:S3, C2xC3:S3, C62, C2xC3:C8, C2xC24, S3xC2xC4, C3xC3:C8, C4xC3:S3, C2xC3:Dic3, C6xC12, C22xC3:S3, D6:C8, C6xC3:C8, C2xC4xC3:S3, C12.78D12
Quotients: C1, C2, C4, C22, S3, C8, C2xC4, D4, D6, C22:C4, C2xC8, M4(2), C4xS3, D12, C3:D4, C22:C8, S32, S3xC8, C8:S3, D6:C4, C6.D6, C3:D12, D6:C8, C12.29D6, C12.31D6, C6.D12, C12.78D12

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

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

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

G=PermutationGroup([[(1,27,5,31,9,35,13,39,17,43,21,47),(2,36,22,32,18,28,14,48,10,44,6,40),(3,29,7,33,11,37,15,41,19,45,23,25),(4,38,24,34,20,30,16,26,12,46,8,42)], [(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),(2,44,14,32),(4,30,16,42),(5,23,17,11),(6,40,18,28),(7,9,19,21),(8,26,20,38),(10,36,22,48),(12,46,24,34),(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:D4C4xS3S3xC8C8:S3S32C3:D12C6.D6C12.29D6C12.31D6
kernelC12.78D12C6xC3:C8C2xC4xC3:S3C2xC3:Dic3C22xC3:S3C2xC3:S3C2xC3:C8C3xC12C2xC12C3xC6C12C12C2xC6C6C6C2xC4C4C22C2C2
# 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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