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G = D6⋊D12order 288 = 25·32

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

metabelian, supersoluble, monomial

Aliases: D61D12, C62.76C23, D6⋊C43S3, (S3×C6)⋊8D4, C6.20(S3×D4), C6.21(C2×D12), C2.22(S3×D12), C34(Dic3⋊D4), C32(C127D4), (C3×Dic3)⋊11D4, (C2×C12).262D6, C325(C4⋊D4), Dic3⋊Dic38C2, C6.33(C4○D12), Dic34(C3⋊D4), (C2×Dic3).30D6, (C22×S3).12D6, C6.11D1217C2, (C6×C12).237C22, C2.17(D6.D6), (C6×Dic3).15C22, (S3×C2×C12)⋊1C2, (S3×C2×C4)⋊13S3, (C2×C4).51S32, (C3×D6⋊C4)⋊6C2, C6.38(C2×C3⋊D4), C2.17(S3×C3⋊D4), (C2×D6⋊S3)⋊2C2, (C2×C3⋊D12)⋊2C2, C22.114(C2×S32), (C3×C6).103(C2×D4), (S3×C2×C6).27C22, (C3×C6).46(C4○D4), (C2×C6).95(C22×S3), (C22×C3⋊S3).21C22, (C2×C3⋊Dic3).53C22, SmallGroup(288,554)

Series: Derived Chief Lower central Upper central

C1C62 — D6⋊D12
C1C3C32C3×C6C62S3×C2×C6C2×D6⋊S3 — D6⋊D12
C32C62 — D6⋊D12
C1C22C2×C4

Generators and relations for D6⋊D12
 G = < a,b,c,d | a6=b2=c12=d2=1, bab=cac-1=dad=a-1, cbc-1=a4b, dbd=ab, dcd=c-1 >

Subgroups: 922 in 205 conjugacy classes, 52 normal (44 characteristic)
C1, C2 [×3], C2 [×4], C3 [×2], C3, C4 [×5], C22, C22 [×10], S3 [×7], C6 [×6], C6 [×6], C2×C4, C2×C4 [×5], D4 [×6], C23 [×3], C32, Dic3 [×2], Dic3 [×4], C12 [×6], D6 [×2], D6 [×15], C2×C6 [×2], C2×C6 [×8], C22⋊C4 [×2], C4⋊C4, C22×C4, C2×D4 [×3], C3×S3 [×3], C3⋊S3, C3×C6 [×3], C4×S3 [×2], D12 [×4], C2×Dic3 [×2], C2×Dic3 [×3], C3⋊D4 [×8], C2×C12 [×2], C2×C12 [×5], C22×S3 [×2], C22×S3 [×3], C22×C6 [×2], C4⋊D4, C3×Dic3 [×2], C3×Dic3, C3⋊Dic3, C3×C12, S3×C6 [×2], S3×C6 [×5], C2×C3⋊S3 [×3], C62, Dic3⋊C4, C4⋊Dic3, D6⋊C4, D6⋊C4 [×3], C3×C22⋊C4, S3×C2×C4, C2×D12 [×2], C2×C3⋊D4 [×4], C22×C12, D6⋊S3 [×2], C3⋊D12 [×4], S3×C12 [×2], C6×Dic3 [×2], C2×C3⋊Dic3, C6×C12, S3×C2×C6 [×2], C22×C3⋊S3, Dic3⋊D4, C127D4, Dic3⋊Dic3, C3×D6⋊C4, C6.11D12, C2×D6⋊S3, C2×C3⋊D12 [×2], S3×C2×C12, D6⋊D12
Quotients: C1, C2 [×7], C22 [×7], S3 [×2], D4 [×4], C23, D6 [×6], C2×D4 [×2], C4○D4, D12 [×2], C3⋊D4 [×2], C22×S3 [×2], C4⋊D4, S32, C2×D12, C4○D12 [×2], S3×D4 [×2], C2×C3⋊D4, C2×S32, Dic3⋊D4, C127D4, D6.D6, S3×D12, S3×C3⋊D4, D6⋊D12

Smallest permutation representation of D6⋊D12
On 48 points
Generators in S48
(1 30 5 34 9 26)(2 27 10 35 6 31)(3 32 7 36 11 28)(4 29 12 25 8 33)(13 41 17 45 21 37)(14 38 22 46 18 42)(15 43 19 47 23 39)(16 40 24 48 20 44)
(1 42)(2 39)(3 44)(4 41)(5 46)(6 43)(7 48)(8 45)(9 38)(10 47)(11 40)(12 37)(13 29)(14 26)(15 31)(16 28)(17 33)(18 30)(19 35)(20 32)(21 25)(22 34)(23 27)(24 36)
(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 36)(2 35)(3 34)(4 33)(5 32)(6 31)(7 30)(8 29)(9 28)(10 27)(11 26)(12 25)(13 17)(14 16)(18 24)(19 23)(20 22)(37 45)(38 44)(39 43)(40 42)(46 48)

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

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

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

48 conjugacy classes

class 1 2A2B2C2D2E2F2G3A3B3C4A4B4C4D4E4F6A···6F6G6H6I6J6K6L6M6N6O12A12B12C12D12E···12J12K12L12M12N12O12P
order122222223334444446···66666666661212121212···12121212121212
size1111661236224226612362···24446666121222224···466661212

48 irreducible representations

dim111111122222222222444444
type+++++++++++++++++++
imageC1C2C2C2C2C2C2S3S3D4D4D6D6D6C4○D4C3⋊D4D12C4○D12S32S3×D4C2×S32D6.D6S3×D12S3×C3⋊D4
kernelD6⋊D12Dic3⋊Dic3C3×D6⋊C4C6.11D12C2×D6⋊S3C2×C3⋊D12S3×C2×C12D6⋊C4S3×C2×C4C3×Dic3S3×C6C2×Dic3C2×C12C22×S3C3×C6Dic3D6C6C2×C4C6C22C2C2C2
# reps111112111222222448121222

Matrix representation of D6⋊D12 in GL8(𝔽13)

120000000
012000000
00100000
00010000
000012100
000012000
00000010
00000001
,
96000000
44000000
001200000
000120000
00001000
000011200
00000010
00000001
,
10000000
01000000
00010000
001200000
000001200
000012000
00000001
0000001212
,
120000000
31000000
00100000
000120000
00000100
00001000
00000001
00000010

G:=sub<GL(8,GF(13))| [12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[9,4,0,0,0,0,0,0,6,4,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,1,12],[12,3,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0] >;

D6⋊D12 in GAP, Magma, Sage, TeX

D_6\rtimes D_{12}
% in TeX

G:=Group("D6:D12");
// GroupNames label

G:=SmallGroup(288,554);
// by ID

G=gap.SmallGroup(288,554);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,141,219,142,1356,9414]);
// Polycyclic

G:=Group<a,b,c,d|a^6=b^2=c^12=d^2=1,b*a*b=c*a*c^-1=d*a*d=a^-1,c*b*c^-1=a^4*b,d*b*d=a*b,d*c*d=c^-1>;
// generators/relations

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