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G = D12.15D6order 288 = 25·32

15th non-split extension by D12 of D6 acting via D6/C3=C22

metabelian, supersoluble, monomial

Aliases: D12.15D6, Dic6.13D6, C3⋊C8.11D6, Q8.17S32, C3⋊Q166S3, C6.66(S3×D4), Q82S36S3, (C3×Q8).35D6, C34(D4.D6), C33(Q16⋊S3), C3⋊Dic3.24D4, D12⋊S3.2C2, C12.31D62C2, (C3×C12).28C23, C12.28(C22×S3), C323Q1614C2, D12.S310C2, C2.26(Dic3⋊D6), (C3×D12).24C22, C3215(C8.C22), (C3×Dic6).23C22, (Q8×C32).10C22, C324Q8.15C22, C4.28(C2×S32), (Q8×C3⋊S3)⋊2C2, (C2×C3⋊S3).61D4, (C3×C3⋊Q16)⋊8C2, (C3×Q82S3)⋊4C2, (C3×C6).143(C2×D4), (C3×C3⋊C8).13C22, (C4×C3⋊S3).22C22, SmallGroup(288,599)

Series: Derived Chief Lower central Upper central

C1C3×C12 — D12.15D6
C1C3C32C3×C6C3×C12C3×D12D12⋊S3 — D12.15D6
C32C3×C6C3×C12 — D12.15D6
C1C2C4Q8

Generators and relations for D12.15D6
 G = < a,b,c,d | a12=b2=1, c6=a6, d2=a9, bab=a-1, cac-1=a7, ad=da, cbc-1=dbd-1=a3b, dcd-1=a9c5 >

Subgroups: 578 in 139 conjugacy classes, 38 normal (all characteristic)
C1, C2, C2 [×2], C3 [×2], C3, C4, C4 [×4], C22 [×2], S3 [×5], C6 [×2], C6 [×2], C8 [×2], C2×C4 [×3], D4 [×2], Q8, Q8 [×3], C32, Dic3 [×8], C12 [×2], C12 [×6], D6 [×4], C2×C6, M4(2), SD16 [×2], Q16 [×2], C2×Q8, C4○D4, C3×S3, C3⋊S3, C3×C6, C3⋊C8 [×2], C24 [×2], Dic6, Dic6 [×7], C4×S3 [×8], D12, D12, C2×Dic3, C3⋊D4, C3×D4, C3×Q8 [×2], C3×Q8 [×2], C8.C22, C3×Dic3, C3⋊Dic3, C3⋊Dic3, C3×C12, C3×C12, S3×C6, C2×C3⋊S3, C8⋊S3 [×2], C24⋊C2, Dic12, D4.S3, Q82S3, C3⋊Q16, C3⋊Q16, C3×SD16, C3×Q16, D42S3, S3×Q8 [×3], Q83S3, C3×C3⋊C8 [×2], S3×Dic3, C3⋊D12, C3×Dic6, C3×D12, C324Q8, C324Q8, C4×C3⋊S3, C4×C3⋊S3, Q8×C32, D4.D6, Q16⋊S3, C12.31D6, D12.S3, C323Q16, C3×Q82S3, C3×C3⋊Q16, D12⋊S3, Q8×C3⋊S3, D12.15D6
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, D4.D6, Q16⋊S3, Dic3⋊D6, D12.15D6

Character table of D12.15D6

 class 12A2B2C3A3B3C4A4B4C4D4E6A6B6C6D8A8B12A12B12C12D12E12F12G12H24A24B24C24D
 size 1112182242412183622424121244888882412121212
ρ1111111111111111111111111111111    trivial
ρ2111-111111-1-1-11111-111111111-111-1-1    linear of order 2
ρ311-1-11111-11-11111-1-11111-1-1-1-1111-1-1    linear of order 2
ρ411-111111-1-11-1111-111111-1-1-1-1-11111    linear of order 2
ρ511111111-111-11111-1-1111-1-1-1-11-1-1-1-1    linear of order 2
ρ6111-11111-1-1-1111111-1111-1-1-1-1-1-1-111    linear of order 2
ρ711-1-1111111-1-1111-11-111111111-1-111    linear of order 2
ρ811-1111111-111111-1-1-11111111-1-1-1-1-1    linear of order 2
ρ922-202-1-122000-12-110-22-1-12-1-1-101100    orthogonal lifted from D6
ρ102200-12-1222002-1-1020-12-1-12-1-1-100-1-1    orthogonal lifted from S3
ρ112200-12-122-2002-1-10-20-12-1-12-1-110011    orthogonal lifted from D6
ρ122202222-200-20222000-2-2-2000000000    orthogonal lifted from D4
ρ1322-202-1-12-2000-12-11022-1-1-21110-1-100    orthogonal lifted from D6
ρ14220-2222-20020222000-2-2-2000000000    orthogonal lifted from D4
ρ1522202-1-12-2000-12-1-10-22-1-1-211101100    orthogonal lifted from D6
ρ162200-12-12-22002-1-10-20-12-11-211-10011    orthogonal lifted from D6
ρ172200-12-12-2-2002-1-1020-12-11-211100-1-1    orthogonal lifted from D6
ρ1822202-1-122000-12-1-1022-1-12-1-1-10-1-100    orthogonal lifted from S3
ρ194400-2-21-40000-2-2100022-100-3300000    orthogonal lifted from Dic3⋊D6
ρ204400-2-214-4000-2-21000-2-2122-1-100000    orthogonal lifted from C2×S32
ρ214400-2-21-40000-2-2100022-1003-300000    orthogonal lifted from Dic3⋊D6
ρ224400-24-2-400004-2-20002-42000000000    orthogonal lifted from S3×D4
ρ234400-2-2144000-2-21000-2-21-2-21100000    orthogonal lifted from S32
ρ2444004-2-2-40000-24-2000-422000000000    orthogonal lifted from S3×D4
ρ254-40044400000-4-4-4000000000000000    symplectic lifted from C8.C22, Schur index 2
ρ264-4004-2-2000002-42000000000006-600    symplectic lifted from D4.D6, Schur index 2
ρ274-4004-2-2000002-4200000000000-6600    symplectic lifted from D4.D6, Schur index 2
ρ284-400-24-200000-4220000000000000--6-6    complex lifted from Q16⋊S3
ρ294-400-24-200000-4220000000000000-6--6    complex lifted from Q16⋊S3
ρ308-800-4-420000044-2000000000000000    symplectic faithful, Schur index 2

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

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

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

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

Matrix representation of D12.15D6 in GL8(𝔽73)

072000000
172000000
50072720000
023100000
000001071
0000727222
000001072
0000727211
,
2323120000
2323210000
121350500000
131250500000
0000006368
000000510
0000683400
000039500
,
072000000
172000000
023010000
50072720000
0000460540
0000046054
000000270
000000027
,
505072710000
001720000
30310230000
30300230000
000062511122
000022115162
0000316200
0000114200

G:=sub<GL(8,GF(73))| [0,1,50,0,0,0,0,0,72,72,0,23,0,0,0,0,0,0,72,1,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,0,72,0,72,0,0,0,0,1,72,1,72,0,0,0,0,0,2,0,1,0,0,0,0,71,2,72,1],[23,23,12,13,0,0,0,0,23,23,13,12,0,0,0,0,1,2,50,50,0,0,0,0,2,1,50,50,0,0,0,0,0,0,0,0,0,0,68,39,0,0,0,0,0,0,34,5,0,0,0,0,63,5,0,0,0,0,0,0,68,10,0,0],[0,1,0,50,0,0,0,0,72,72,23,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,1,72,0,0,0,0,0,0,0,0,46,0,0,0,0,0,0,0,0,46,0,0,0,0,0,0,54,0,27,0,0,0,0,0,0,54,0,27],[50,0,30,30,0,0,0,0,50,0,31,30,0,0,0,0,72,1,0,0,0,0,0,0,71,72,23,23,0,0,0,0,0,0,0,0,62,22,31,11,0,0,0,0,51,11,62,42,0,0,0,0,11,51,0,0,0,0,0,0,22,62,0,0] >;

D12.15D6 in GAP, Magma, Sage, TeX

D_{12}._{15}D_6
% in TeX

G:=Group("D12.15D6");
// GroupNames label

G:=SmallGroup(288,599);
// by ID

G=gap.SmallGroup(288,599);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,120,254,303,100,675,346,185,80,1356,9414]);
// Polycyclic

G:=Group<a,b,c,d|a^12=b^2=1,c^6=a^6,d^2=a^9,b*a*b=a^-1,c*a*c^-1=a^7,a*d=d*a,c*b*c^-1=d*b*d^-1=a^3*b,d*c*d^-1=a^9*c^5>;
// generators/relations

Export

Character table of D12.15D6 in TeX

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