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G = D12.32D6order 288 = 25·32

7th non-split extension by D12 of D6 acting via D6/C6=C2

metabelian, supersoluble, monomial

Aliases: D12.32D6, C62.46D4, Dic6.32D6, (C2×Dic6)⋊8S3, (C6×Dic6)⋊2C2, C4○D12.4S3, (C3×C12).65D4, (C2×C12).116D6, C322Q165C2, Dic6⋊S36C2, C34(Q8.14D6), C12.46(C3⋊D4), C12.82(C22×S3), (C3×C12).63C23, (C6×C12).76C22, C326(C8.C22), C12.58D610C2, C4.14(D6⋊S3), C34(Q8.11D6), (C3×D12).33C22, C324C8.5C22, C22.5(D6⋊S3), (C3×Dic6).33C22, (C2×C4).7S32, C4.76(C2×S32), (C3×C6).67(C2×D4), C6.73(C2×C3⋊D4), (C3×C4○D12).7C2, C2.8(C2×D6⋊S3), (C2×C6).18(C3⋊D4), SmallGroup(288,475)

Series: Derived Chief Lower central Upper central

C1C3×C12 — D12.32D6
C1C3C32C3×C6C3×C12C3×D12Dic6⋊S3 — D12.32D6
C32C3×C6C3×C12 — D12.32D6
C1C2C2×C4

Generators and relations for D12.32D6
 G = < a,b,c,d | a12=b2=1, c6=d2=a6, bab=a-1, ac=ca, dad-1=a7, bc=cb, dbd-1=a3b, dcd-1=a6c5 >

Subgroups: 402 in 130 conjugacy classes, 44 normal (34 characteristic)
C1, C2, C2 [×2], C3 [×2], C3, C4 [×2], C4 [×3], C22, C22, S3, C6 [×2], C6 [×5], C8 [×2], C2×C4, C2×C4 [×2], D4 [×2], Q8 [×4], C32, Dic3 [×3], C12 [×4], C12 [×5], D6, C2×C6 [×2], C2×C6 [×2], M4(2), SD16 [×2], Q16 [×2], C2×Q8, C4○D4, C3×S3, C3×C6, C3×C6, C3⋊C8 [×6], Dic6, Dic6 [×2], Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12 [×2], C2×C12 [×3], C3×D4 [×2], C3×Q8 [×4], C8.C22, C3×Dic3 [×3], C3×C12 [×2], S3×C6, C62, C4.Dic3 [×3], D4.S3 [×2], Q82S3 [×2], C3⋊Q16 [×4], C2×Dic6, C4○D12, C6×Q8, C3×C4○D4, C324C8 [×2], C3×Dic6, C3×Dic6 [×2], C3×Dic6, S3×C12, C3×D12, C6×Dic3, C3×C3⋊D4, C6×C12, Q8.11D6, Q8.14D6, Dic6⋊S3 [×2], C322Q16 [×2], C12.58D6, C6×Dic6, C3×C4○D12, D12.32D6
Quotients: C1, C2 [×7], C22 [×7], S3 [×2], D4 [×2], C23, D6 [×6], C2×D4, C3⋊D4 [×4], C22×S3 [×2], C8.C22, S32, C2×C3⋊D4 [×2], D6⋊S3 [×2], C2×S32, Q8.11D6, Q8.14D6, C2×D6⋊S3, D12.32D6

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

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

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

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

39 conjugacy classes

class 1 2A2B2C3A3B3C4A4B4C4D4E6A6B6C6D6E6F6G6H6I6J8A8B12A12B12C···12I12J···12O
order122233344444666666666688121212···1212···12
size11212224221212122222444412123636224···412···12

39 irreducible representations

dim11111122222222244444444
type+++++++++++++-+-+--
imageC1C2C2C2C2C2S3S3D4D4D6D6D6C3⋊D4C3⋊D4C8.C22S32D6⋊S3C2×S32D6⋊S3Q8.11D6Q8.14D6D12.32D6
kernelD12.32D6Dic6⋊S3C322Q16C12.58D6C6×Dic6C3×C4○D12C2×Dic6C4○D12C3×C12C62Dic6D12C2×C12C12C2×C6C32C2×C4C4C4C22C3C3C1
# reps12211111113124411111224

Matrix representation of D12.32D6 in GL8(𝔽73)

172000000
10000000
007200000
000720000
00000100
000072000
000000072
00000010
,
5132000000
1022000000
0030600000
0013430000
000000072
00000010
00000100
000072000
,
10000000
01000000
00010000
0072720000
000007200
00001000
000000072
00000010
,
3013000000
6043000000
0070420000
004530000
0000113000
0000306200
0000003062
0000006243

G:=sub<GL(8,GF(73))| [1,1,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0],[51,10,0,0,0,0,0,0,32,22,0,0,0,0,0,0,0,0,30,13,0,0,0,0,0,0,60,43,0,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,1,72,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0],[30,60,0,0,0,0,0,0,13,43,0,0,0,0,0,0,0,0,70,45,0,0,0,0,0,0,42,3,0,0,0,0,0,0,0,0,11,30,0,0,0,0,0,0,30,62,0,0,0,0,0,0,0,0,30,62,0,0,0,0,0,0,62,43] >;

D12.32D6 in GAP, Magma, Sage, TeX

D_{12}._{32}D_6
% in TeX

G:=Group("D12.32D6");
// GroupNames label

G:=SmallGroup(288,475);
// by ID

G=gap.SmallGroup(288,475);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,141,120,219,100,675,346,80,1356,9414]);
// Polycyclic

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

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