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G = D12.D6order 288 = 25·32

5th non-split extension by D12 of D6 acting via D6/C3=C22

metabelian, supersoluble, monomial

Aliases: D12.5D6, D44S32, C3⋊C86D6, D4⋊S33S3, (C3×D4)⋊2D6, C6.56(S3×D4), D6⋊D64C2, C33(D8⋊S3), C3⋊Dic3.56D4, (C3×C12).4C23, C12.4(C22×S3), D12.S37C2, C12.31D65C2, C12.D61C2, C3210(C8⋊C22), (D4×C32)⋊4C22, C2.16(Dic3⋊D6), C324Q83C22, (C3×D12).11C22, C4.4(C2×S32), (C3×D4⋊S3)⋊6C2, (C3×C3⋊C8)⋊11C22, (C2×C3⋊S3).21D4, (C3×C6).119(C2×D4), (C4×C3⋊S3).12C22, SmallGroup(288,575)

Series: Derived Chief Lower central Upper central

C1C3×C12 — D12.D6
C1C3C32C3×C6C3×C12C3×D12D6⋊D6 — D12.D6
C32C3×C6C3×C12 — D12.D6
C1C2C4D4

Generators and relations for D12.D6
 G = < a,b,c,d | a12=b2=c6=1, d2=a3, bab=a-1, cac-1=a7, ad=da, cbc-1=dbd-1=a9b, dcd-1=a3c-1 >

Subgroups: 762 in 156 conjugacy classes, 38 normal (16 characteristic)
C1, C2, C2 [×4], C3 [×2], C3, C4, C4 [×2], C22 [×6], S3 [×5], C6 [×2], C6 [×7], C8 [×2], C2×C4 [×2], D4, D4 [×4], Q8, C23, C32, Dic3 [×7], C12 [×2], C12, D6 [×9], C2×C6 [×6], M4(2), D8 [×2], SD16 [×2], C2×D4, C4○D4, C3×S3 [×2], C3⋊S3, C3×C6, C3×C6, C3⋊C8 [×2], C24 [×2], Dic6 [×3], C4×S3 [×3], D12 [×2], C2×Dic3 [×4], C3⋊D4 [×6], C3×D4 [×2], C3×D4 [×3], C22×S3 [×2], C8⋊C22, C3⋊Dic3, C3⋊Dic3, C3×C12, S32 [×2], S3×C6 [×2], C2×C3⋊S3, C62, C8⋊S3 [×2], C24⋊C2 [×2], D4⋊S3 [×2], D4.S3 [×2], C3×D8 [×2], S3×D4 [×2], D42S3 [×3], C3×C3⋊C8 [×2], D6⋊S3, C3×D12 [×2], C324Q8, C4×C3⋊S3, C2×C3⋊Dic3, C327D4, D4×C32, C2×S32, D8⋊S3 [×2], C12.31D6, D12.S3 [×2], C3×D4⋊S3 [×2], D6⋊D6, C12.D6, D12.D6
Quotients: C1, C2 [×7], C22 [×7], S3 [×2], D4 [×2], C23, D6 [×6], C2×D4, C22×S3 [×2], C8⋊C22, S32, S3×D4 [×2], C2×S32, D8⋊S3 [×2], Dic3⋊D6, D12.D6

Character table of D12.D6

 class 12A2B2C2D2E3A3B3C4A4B4C6A6B6C6D6E6F6G6H6I8A8B12A12B12C24A24B24C24D
 size 1141212182242183622488882424121244812121212
ρ1111111111111111111111111111111    trivial
ρ2111-11-11111-1-111111111-11-1111-111-1    linear of order 2
ρ311-1-11-11111-11111-1-1-1-11-1-111111-1-11    linear of order 2
ρ411-111111111-1111-1-1-1-111-1-1111-1-1-1-1    linear of order 2
ρ511-11-1-11111-11111-1-1-1-1-111-1111-111-1    linear of order 2
ρ611-1-1-1111111-1111-1-1-1-1-1-1111111111    linear of order 2
ρ7111-1-111111111111111-1-1-1-1111-1-1-1-1    linear of order 2
ρ81111-1-11111-1-11111111-11-111111-1-11    linear of order 2
ρ922-2020-12-1200-12-111-21-10-20-12-10110    orthogonal lifted from D6
ρ102220-20-12-1200-12-1-1-12-110-20-12-10110    orthogonal lifted from D6
ρ11220002222-2-2022200000000-2-2-20000    orthogonal lifted from D4
ρ1222-22002-1-12002-1-1111-20-10-22-1-11001    orthogonal lifted from D6
ρ13222-2002-1-12002-1-1-1-1-12010-22-1-11001    orthogonal lifted from D6
ρ14222020-12-1200-12-1-1-12-1-1020-12-10-1-10    orthogonal lifted from S3
ρ1522000-2222-22022200000000-2-2-20000    orthogonal lifted from D4
ρ162222002-1-12002-1-1-1-1-120-1022-1-1-100-1    orthogonal lifted from S3
ρ1722-20-20-12-1200-12-111-211020-12-10-1-10    orthogonal lifted from D6
ρ1822-2-2002-1-12002-1-1111-201022-1-1-100-1    orthogonal lifted from D6
ρ19440000-2-21-400-2-213-300000022-10000    orthogonal lifted from Dic3⋊D6
ρ20440000-2-21-400-2-21-3300000022-10000    orthogonal lifted from Dic3⋊D6
ρ21440000-24-2-400-24-2000000002-420000    orthogonal lifted from S3×D4
ρ224400004-2-2-4004-2-200000000-4220000    orthogonal lifted from S3×D4
ρ23444000-2-21400-2-2111-2-20000-2-210000    orthogonal lifted from S32
ρ244-40000444000-4-4-4000000000000000    orthogonal lifted from C8⋊C22
ρ2544-4000-2-21400-2-21-1-1220000-2-210000    orthogonal lifted from C2×S32
ρ264-40000-24-20002-42000000000000-6--60    complex lifted from D8⋊S3
ρ274-40000-24-20002-42000000000000--6-60    complex lifted from D8⋊S3
ρ284-400004-2-2000-42200000000000-600--6    complex lifted from D8⋊S3
ρ294-400004-2-2000-42200000000000--600-6    complex lifted from D8⋊S3
ρ308-80000-4-4200044-2000000000000000    symplectic faithful, Schur index 2

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

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

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

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

Matrix representation of D12.D6 in GL8(𝔽73)

172000000
10000000
02110000
7107200000
00000100
000072000
00001112
00000727272
,
717171720000
717172710000
23220000
32220000
0000324600
0000464100
00004141689
0000032465
,
01000000
721000000
484672720000
5048100000
00000010
000072727271
00001000
00000001
,
717172710000
717171720000
5152220000
5251220000
000027275954
00003232564
0000464100
00000464114

G:=sub<GL(8,GF(73))| [1,1,0,71,0,0,0,0,72,0,2,0,0,0,0,0,0,0,1,72,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,72,1,0,0,0,0,0,1,0,1,72,0,0,0,0,0,0,1,72,0,0,0,0,0,0,2,72],[71,71,2,3,0,0,0,0,71,71,3,2,0,0,0,0,71,72,2,2,0,0,0,0,72,71,2,2,0,0,0,0,0,0,0,0,32,46,41,0,0,0,0,0,46,41,41,32,0,0,0,0,0,0,68,46,0,0,0,0,0,0,9,5],[0,72,48,50,0,0,0,0,1,1,46,48,0,0,0,0,0,0,72,1,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,0,72,1,0,0,0,0,0,0,72,0,0,0,0,0,0,1,72,0,0,0,0,0,0,0,71,0,1],[71,71,51,52,0,0,0,0,71,71,52,51,0,0,0,0,72,71,2,2,0,0,0,0,71,72,2,2,0,0,0,0,0,0,0,0,27,32,46,0,0,0,0,0,27,32,41,46,0,0,0,0,59,5,0,41,0,0,0,0,54,64,0,14] >;

D12.D6 in GAP, Magma, Sage, TeX

D_{12}.D_6
% in TeX

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

G:=SmallGroup(288,575);
// by ID

G=gap.SmallGroup(288,575);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,254,303,675,346,185,80,1356,9414]);
// Polycyclic

G:=Group<a,b,c,d|a^12=b^2=c^6=1,d^2=a^3,b*a*b=a^-1,c*a*c^-1=a^7,a*d=d*a,c*b*c^-1=d*b*d^-1=a^9*b,d*c*d^-1=a^3*c^-1>;
// generators/relations

Export

Character table of D12.D6 in TeX

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