Copied to
clipboard

G = C12⋊2D12  order 288 = 25·32

2nd semidirect product of C12 and D12 acting via D12/C6=C22

metabelian, supersoluble, monomial

Aliases: C12⋊2D12, C62.86C23, (C3×C12)⋊8D4, (C6×D12)⋊8C2, C6.43(S3×D4), (C2×D12)⋊11S3, C12⋊3(C3⋊D4), C4⋊Dic3⋊15S3, C6.80(C2×D12), C4⋊3(C3⋊D12), C3⋊1(D6⋊3D4), C3⋊5(C12⋊D4), D6⋊Dic3⋊13C2, (C2×C12).142D6, C32⋊10(C4⋊D4), (C2×Dic3).35D6, (C22×S3).21D6, C6.12(D4⋊2S3), C2.18(D6⋊D6), (C6×C12).109C22, C6.36(Q8⋊3S3), C2.19(D12⋊S3), (C6×Dic3).19C22, (C2×C4).120S32, (C2×C3⋊S3)⋊10D4, C6.18(C2×C3⋊D4), (C2×C3⋊D12)⋊6C2, (C3×C4⋊Dic3)⋊19C2, C22.124(C2×S32), (C3×C6).111(C2×D4), (S3×C2×C6).36C22, (C3×C6).53(C4○D4), C2.21(C2×C3⋊D12), (C2×C6).105(C22×S3), (C22×C3⋊S3).74C22, (C2×C3⋊Dic3).139C22, (C2×C4×C3⋊S3)⋊1C2, SmallGroup(288,564)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C62 — C12⋊2D12
Chief series
C1 — C3 — C32 — C3×C6 — C62 — S3×C2×C6 — C2×C3⋊D12 — C12⋊2D12
Lower central
C32 — C62 — C12⋊2D12
Upper central
C1 — C22 — C2×C4

Generators and relations for C12⋊2D12
 G = < a,b,c | a12=b12=c2=1, bab-1=a-1, cac=a5, cbc=b-1 >

Subgroups: 946 in 215 conjugacy classes, 54 normal (32 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, D4, C23, C32, Dic3, C12, C12, D6, C2×C6, C2×C6, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C3×S3, C3⋊S3, C3×C6, C4×S3, D12, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C3×D4, C22×S3, C22×S3, C22×C6, C4⋊D4, C3×Dic3, C3⋊Dic3, C3×C12, S3×C6, C2×C3⋊S3, C2×C3⋊S3, C62, C4⋊Dic3, D6⋊C4, C6.D4, C3×C4⋊C4, S3×C2×C4, C2×D12, C2×D12, C2×C3⋊D4, C6×D4, C3⋊D12, C3×D12, C6×Dic3, C4×C3⋊S3, C2×C3⋊Dic3, C6×C12, S3×C2×C6, C22×C3⋊S3, C12⋊D4, D6⋊3D4, D6⋊Dic3, C3×C4⋊Dic3, C2×C3⋊D12, C6×D12, C2×C4×C3⋊S3, C12⋊2D12
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, D12, C3⋊D4, C22×S3, C4⋊D4, S32, C2×D12, S3×D4, D4⋊2S3, Q8⋊3S3, C2×C3⋊D4, C3⋊D12, C2×S32, C12⋊D4, D6⋊3D4, D12⋊S3, D6⋊D6, C2×C3⋊D12, C12⋊2D12

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

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

42 conjugacy classes

class 1 2A2B2C2D2E2F2G3A3B3C4A4B4C4D4E4F6A···6F6G6H6I6J6K6L6M12A···12H12I12J12K12L
order122222223334444446···6666666612···1212121212
size11111212181822422121218182···2444121212124···412121212

42 irreducible representations

dim111111222222222244444444
type++++++++++++++++-+++
imageC1C2C2C2C2C2S3S3D4D4D6D6D6C4○D4D12C3⋊D4S32S3×D4D4⋊2S3Q8⋊3S3C3⋊D12C2×S32D12⋊S3D6⋊D6
kernelC12⋊2D12D6⋊Dic3C3×C4⋊Dic3C2×C3⋊D12C6×D12C2×C4×C3⋊S3C4⋊Dic3C2×D12C3×C12C2×C3⋊S3C2×Dic3C2×C12C22×S3C3×C6C12C12C2×C4C6C6C6C4C22C2C2
# reps121211112222224412112122

Matrix representation of C12⋊2D12 ►in GL8(𝔽13)

83000000
05000000
001200000
000120000
00001100
000012000
000000120
000000012
,
115000000
22000000
001210000
001200000
00001000
0000121200
0000001211
00000011
,
10000000
01000000
001200000
001210000
00001000
0000121200
000000120
00000011

G:=sub<GL(8,GF(13))| [8,0,0,0,0,0,0,0,3,5,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12],[11,2,0,0,0,0,0,0,5,2,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,1,0,0,0,0,0,0,11,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,1,0,0,0,0,0,0,0,1] >;
 

C12⋊2D12 in GAP, Magma, Sage, TeX 

C_{12}\rtimes_2D_{12}
 
% in TeX
 
G:=Group("C12:2D12");
 
// GroupNames label
 
G:=SmallGroup(288,564);
 
// by ID
 
G=gap.SmallGroup(288,564);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,141,64,422,219,100,1356,9414]);
 
// Polycyclic
 
G:=Group<a,b,c|a^12=b^12=c^2=1,b*a*b^-1=a^-1,c*a*c=a^5,c*b*c=b^-1>;
 
// generators/relations
 

׿
×
⋊
ℤ
𝔽
○
≀
ℚ
◁