Copied to
clipboard

G = C12⋊7D12  order 288 = 25·32

1st semidirect product of C12 and D12 acting via D12/D6=C2

metabelian, supersoluble, monomial

Aliases: D6⋊3D12, C12⋊7D12, C62.79C23, (C3×C12)⋊5D4, (S3×C6)⋊10D4, C4⋊Dic3⋊9S3, C6.22(S3×D4), C12⋊1(C3⋊D4), C6.23(C2×D12), C2.24(S3×D12), C4⋊2(C3⋊D12), C3⋊3(C12⋊D4), C3⋊1(C12⋊7D4), (C2×C12).139D6, C32⋊7(C4⋊D4), C6.16(C4○D12), C6.D12⋊3C2, (C2×Dic3).32D6, (C22×S3).68D6, (C6×C12).106C22, C6.17(Q8⋊3S3), C2.18(D6.6D6), (C6×Dic3).16C22, (S3×C2×C4)⋊2S3, (S3×C2×C12)⋊4C2, (C2×C4).82S32, C6.16(C2×C3⋊D4), (C2×C3⋊D12)⋊3C2, (C3×C4⋊Dic3)⋊18C2, C22.117(C2×S32), (C3×C6).105(C2×D4), (C2×C12⋊S3)⋊11C2, (S3×C2×C6).81C22, (C3×C6).49(C4○D4), C2.19(C2×C3⋊D12), (C2×C6).98(C22×S3), (C22×C3⋊S3).23C22, SmallGroup(288,557)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C62 — C12⋊7D12
Chief series
C1 — C3 — C32 — C3×C6 — C62 — S3×C2×C6 — C2×C3⋊D12 — C12⋊7D12
Lower central
C32 — C62 — C12⋊7D12
Upper central
C1 — C22 — C2×C4

Generators and relations for C12⋊7D12
 G = < a,b,c | a12=b12=c2=1, bab-1=cac=a-1, cbc=b-1 >

Subgroups: 1058 in 215 conjugacy classes, 56 normal (34 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, D4, C23, C32, Dic3, C12, C12, D6, 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, C22×S3, C22×S3, C22×C6, C4⋊D4, C3×Dic3, C3×C12, S3×C6, S3×C6, C2×C3⋊S3, C62, C4⋊Dic3, D6⋊C4, C3×C4⋊C4, S3×C2×C4, C2×D12, C2×C3⋊D4, C22×C12, C3⋊D12, S3×C12, C6×Dic3, C6×Dic3, C12⋊S3, C6×C12, S3×C2×C6, C22×C3⋊S3, C12⋊D4, C12⋊7D4, C6.D12, C3×C4⋊Dic3, C2×C3⋊D12, S3×C2×C12, C2×C12⋊S3, C12⋊7D12
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, D12, C3⋊D4, C22×S3, C4⋊D4, S32, C2×D12, C4○D12, S3×D4, Q8⋊3S3, C2×C3⋊D4, C3⋊D12, C2×S32, C12⋊D4, C12⋊7D4, D6.6D6, S3×D12, C2×C3⋊D12, C12⋊7D12

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

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

48 conjugacy classes

class 1 2A2B2C2D2E2F2G3A3B3C4A4B4C4D4E4F6A···6F6G6H6I6J6K6L6M12A12B12C12D12E···12J12K12L12M12N12O12P12Q12R
order122222223334444446···666666661212121212···121212121212121212
size1111663636224226612122···2444666622224···4666612121212

48 irreducible representations

dim1111112222222222224444444
type++++++++++++++++++++++
imageC1C2C2C2C2C2S3S3D4D4D6D6D6C4○D4D12C3⋊D4D12C4○D12S32S3×D4Q8⋊3S3C3⋊D12C2×S32D6.6D6S3×D12
kernelC12⋊7D12C6.D12C3×C4⋊Dic3C2×C3⋊D12S3×C2×C12C2×C12⋊S3C4⋊Dic3S3×C2×C4C3×C12S3×C6C2×Dic3C2×C12C22×S3C3×C6C12C12D6C6C2×C4C6C6C4C22C2C2
# reps1212111122321244441112122

Matrix representation of C12⋊7D12 ►in GL8(𝔽13)

43000000
39000000
001200000
000120000
000012000
000001200
0000001212
00000010
,
910000000
54000000
00010000
001200000
000012100
000012000
00000010
0000001212
,
43000000
89000000
00100000
000120000
00001000
000011200
00000010
0000001212

G:=sub<GL(8,GF(13))| [4,3,0,0,0,0,0,0,3,9,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,1,0,0,0,0,0,0,12,0],[9,5,0,0,0,0,0,0,10,4,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,12,0,0,0,0,0,0,0,12],[4,8,0,0,0,0,0,0,3,9,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,12,0,0,0,0,0,0,0,12] >;
 

C12⋊7D12 in GAP, Magma, Sage, TeX 

C_{12}\rtimes_7D_{12}
 
% in TeX
 
G:=Group("C12:7D12");
 
// GroupNames label
 
G:=SmallGroup(288,557);
 
// by ID
 
G=gap.SmallGroup(288,557);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,141,64,219,100,1356,9414]);
 
// Polycyclic
 
G:=Group<a,b,c|a^12=b^12=c^2=1,b*a*b^-1=c*a*c=a^-1,c*b*c=b^-1>;
 
// generators/relations
 

׿
×
⋊
ℤ
𝔽
○
≀
ℚ
◁