Copied to
clipboard

G = C12.30D12  order 288 = 25·32

30th non-split extension by C12 of D12 acting via D12/C6=C22

metabelian, supersoluble, monomial

Aliases: C12.30D12, C62.41C23, C6.7(S3xQ8), C4:Dic3:12S3, (C6xDic6):8C2, C6.78(C2xD12), (C3xC12).79D4, C3:2(D6:3Q8), C3:4(C4.D12), (C2xDic6):11S3, (C2xC12).133D6, C6.5(D4:2S3), C32:7(C22:Q8), C12.54(C3:D4), (C6xC12).97C22, (C2xDic3).19D6, Dic3:Dic3:33C2, C4.25(C3:D12), C6.27(Q8:3S3), C6.D12.4C2, C2.12(D12:S3), C2.9(Dic3.D6), (C6xDic3).79C22, (C2xC3:S3):6Q8, (C2xC4).114S32, C22.98(C2xS32), (C3xC6).88(C2xD4), C6.14(C2xC3:D4), (C3xC6).23(C2xQ8), (C3xC4:Dic3):10C2, (C3xC6).25(C4oD4), C2.18(C2xC3:D12), (C2xC6).60(C22xS3), (C22xC3:S3).66C22, (C2xC3:Dic3).122C22, (C2xC4xC3:S3).3C2, SmallGroup(288,519)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C62 — C12.30D12
Chief series
C1 — C3 — C32 — C3xC6 — C62 — C6xDic3 — Dic3:Dic3 — C12.30D12
Lower central
C32 — C62 — C12.30D12
Upper central
C1 — C22 — C2xC4

Generators and relations for C12.30D12
 G = < a,b,c | a12=b12=1, c2=a6, bab-1=a-1, cac-1=a5, cbc-1=b-1 >

Subgroups: 690 in 175 conjugacy classes, 54 normal (32 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C22, S3, C6, C6, C2xC4, C2xC4, Q8, C23, C32, Dic3, C12, C12, D6, C2xC6, C2xC6, C22:C4, C4:C4, C22xC4, C2xQ8, C3:S3, C3xC6, Dic6, C4xS3, C2xDic3, C2xDic3, C2xC12, C2xC12, C3xQ8, C22xS3, C22:Q8, C3xDic3, C3:Dic3, C3xC12, C2xC3:S3, C2xC3:S3, C62, Dic3:C4, C4:Dic3, C4:Dic3, D6:C4, C3xC4:C4, C2xDic6, S3xC2xC4, C6xQ8, C3xDic6, C6xDic3, C4xC3:S3, C2xC3:Dic3, C6xC12, C22xC3:S3, C4.D12, D6:3Q8, C6.D12, Dic3:Dic3, C3xC4:Dic3, C6xDic6, C2xC4xC3:S3, C12.30D12
Quotients: C1, C2, C22, S3, D4, Q8, C23, D6, C2xD4, C2xQ8, C4oD4, D12, C3:D4, C22xS3, C22:Q8, S32, C2xD12, D4:2S3, S3xQ8, Q8:3S3, C2xC3:D4, C3:D12, C2xS32, C4.D12, D6:3Q8, D12:S3, Dic3.D6, C2xC3:D12, C12.30D12

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

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

42 conjugacy classes

class 1 2A2B2C2D2E3A3B3C4A4B4C4D4E4F4G4H6A···6F6G6H6I12A···12H12I···12P
order122222333444444446···666612···1212···12
size11111818224221212121218182···24444···412···12

42 irreducible representations

dim11111122222222244444444
type+++++++++-++++--+++
imageC1C2C2C2C2C2S3S3D4Q8D6D6C4oD4D12C3:D4S32D4:2S3S3xQ8Q8:3S3C3:D12C2xS32D12:S3Dic3.D6
kernelC12.30D12C6.D12Dic3:Dic3C3xC4:Dic3C6xDic6C2xC4xC3:S3C4:Dic3C2xDic6C3xC12C2xC3:S3C2xDic3C2xC12C3xC6C12C12C2xC4C6C6C6C4C22C2C2
# reps12211111224224411212122

Matrix representation of C12.30D12 ►in GL8(F13)

01000000
120000000
00100000
00010000
0000121200
00001000
00000010
00000001
,
93000000
34000000
000120000
00110000
000012000
00001100
00000001
000000120
,
01000000
120000000
00010000
00100000
00001000
0000121200
000000120
00000001

G:=sub<GL(8,GF(13))| [0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,1,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[9,3,0,0,0,0,0,0,3,4,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,12,1,0,0,0,0,0,0,0,0,12,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0],[0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1] >;
 

C12.30D12 in GAP, Magma, Sage, TeX 

C_{12}._{30}D_{12}
 
% in TeX
 
G:=Group("C12.30D12");
 
// GroupNames label
 
G:=SmallGroup(288,519);
 
// by ID
 
G=gap.SmallGroup(288,519);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,112,141,64,422,219,100,1356,9414]);
 
// Polycyclic
 
G:=Group<a,b,c|a^12=b^12=1,c^2=a^6,b*a*b^-1=a^-1,c*a*c^-1=a^5,c*b*c^-1=b^-1>;
 
// generators/relations
 

׿
x
:
Z
F
o
wr
Q
<