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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(S3×Q8), C4⋊Dic3⋊12S3, (C6×Dic6)⋊8C2, C6.78(C2×D12), (C3×C12).79D4, C3⋊2(D6⋊3Q8), C3⋊4(C4.D12), (C2×Dic6)⋊11S3, (C2×C12).133D6, C6.5(D4⋊2S3), C32⋊7(C22⋊Q8), C12.54(C3⋊D4), (C6×C12).97C22, (C2×Dic3).19D6, Dic3⋊Dic3⋊33C2, C4.25(C3⋊D12), C6.27(Q8⋊3S3), C6.D12.4C2, C2.12(D12⋊S3), C2.9(Dic3.D6), (C6×Dic3).79C22, (C2×C3⋊S3)⋊6Q8, (C2×C4).114S32, C22.98(C2×S32), (C3×C6).88(C2×D4), C6.14(C2×C3⋊D4), (C3×C6).23(C2×Q8), (C3×C4⋊Dic3)⋊10C2, (C3×C6).25(C4○D4), C2.18(C2×C3⋊D12), (C2×C6).60(C22×S3), (C22×C3⋊S3).66C22, (C2×C3⋊Dic3).122C22, (C2×C4×C3⋊S3).3C2, SmallGroup(288,519)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C62 — C12.30D12
Chief series
C1 — C3 — C32 — C3×C6 — C62 — C6×Dic3 — Dic3⋊Dic3 — C12.30D12
Lower central
C32 — C62 — C12.30D12
Upper central
C1 — C22 — C2×C4

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, C2×C4, C2×C4, Q8, C23, C32, Dic3, C12, C12, D6, C2×C6, C2×C6, C22⋊C4, C4⋊C4, C22×C4, C2×Q8, C3⋊S3, C3×C6, Dic6, C4×S3, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C3×Q8, C22×S3, C22⋊Q8, C3×Dic3, C3⋊Dic3, C3×C12, C2×C3⋊S3, C2×C3⋊S3, C62, Dic3⋊C4, C4⋊Dic3, C4⋊Dic3, D6⋊C4, C3×C4⋊C4, C2×Dic6, S3×C2×C4, C6×Q8, C3×Dic6, C6×Dic3, C4×C3⋊S3, C2×C3⋊Dic3, C6×C12, C22×C3⋊S3, C4.D12, D6⋊3Q8, C6.D12, Dic3⋊Dic3, C3×C4⋊Dic3, C6×Dic6, C2×C4×C3⋊S3, C12.30D12
Quotients: C1, C2, C22, S3, D4, Q8, C23, D6, C2×D4, C2×Q8, C4○D4, D12, C3⋊D4, C22×S3, C22⋊Q8, S32, C2×D12, D4⋊2S3, S3×Q8, Q8⋊3S3, C2×C3⋊D4, C3⋊D12, C2×S32, C4.D12, D6⋊3Q8, D12⋊S3, Dic3.D6, C2×C3⋊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+++++++++-++++--+++
imageC1C2C2C2C2C2S3S3D4Q8D6D6C4○D4D12C3⋊D4S32D4⋊2S3S3×Q8Q8⋊3S3C3⋊D12C2×S32D12⋊S3Dic3.D6
kernelC12.30D12C6.D12Dic3⋊Dic3C3×C4⋊Dic3C6×Dic6C2×C4×C3⋊S3C4⋊Dic3C2×Dic6C3×C12C2×C3⋊S3C2×Dic3C2×C12C3×C6C12C12C2×C4C6C6C6C4C22C2C2
# 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
 

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