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G = D12.2D6order 288 = 25·32

2nd non-split extension by D12 of D6 acting via D6/C3=C22

metabelian, supersoluble, monomial

Aliases: D12.2D6, C24.39D6, Dic6.2D6, C8.18S32, C24⋊C26S3, C6.32(S3×D4), C326(C4○D8), C322D84C2, D12⋊S32C2, C3⋊Dic3.42D4, C322Q163C2, C2.9(D6⋊D6), C32(Q8.7D6), C12.74(C22×S3), (C3×C24).37C22, (C3×C12).54C23, (C3×D12).7C22, (C3×Dic6).7C22, C324C8.22C22, (C8×C3⋊S3)⋊7C2, C4.69(C2×S32), (C2×C3⋊S3).43D4, (C3×C6).38(C2×D4), (C3×C24⋊C2)⋊12C2, (C4×C3⋊S3).68C22, SmallGroup(288,457)

Series: Derived Chief Lower central Upper central

C1C3×C12 — D12.2D6
C1C3C32C3×C6C3×C12C3×D12D12⋊S3 — D12.2D6
C32C3×C6C3×C12 — D12.2D6
C1C2C4C8

Generators and relations for D12.2D6
 G = < a,b,c,d | a12=b2=1, c6=d2=a6, bab=cac-1=a-1, dad-1=a5, cbc-1=ab, dbd-1=a4b, dcd-1=c5 >

Subgroups: 562 in 135 conjugacy classes, 38 normal (16 characteristic)
C1, C2, C2 [×3], C3 [×2], C3, C4, C4 [×3], C22 [×3], S3 [×6], C6 [×2], C6 [×3], C8, C8, C2×C4 [×3], D4 [×4], Q8 [×2], C32, Dic3 [×5], C12 [×2], C12 [×3], D6 [×5], C2×C6 [×2], C2×C8, D8, SD16 [×2], Q16, C4○D4 [×2], C3×S3 [×2], C3⋊S3, C3×C6, C3⋊C8 [×3], C24 [×2], C24, Dic6 [×2], C4×S3 [×5], D12 [×2], D12 [×2], C2×Dic3 [×2], C3⋊D4 [×2], C3×D4 [×2], C3×Q8 [×2], C4○D8, C3×Dic3 [×2], C3⋊Dic3, C3×C12, S3×C6 [×2], C2×C3⋊S3, S3×C8 [×3], C24⋊C2 [×2], D4⋊S3 [×2], C3⋊Q16 [×2], C3×SD16 [×2], D42S3 [×2], Q83S3 [×2], C324C8, C3×C24, S3×Dic3 [×2], C3⋊D12 [×2], C3×Dic6 [×2], C3×D12 [×2], C4×C3⋊S3, Q8.7D6 [×2], C322D8, C322Q16, C3×C24⋊C2 [×2], C8×C3⋊S3, D12⋊S3 [×2], D12.2D6
Quotients: C1, C2 [×7], C22 [×7], S3 [×2], D4 [×2], C23, D6 [×6], C2×D4, C22×S3 [×2], C4○D8, S32, S3×D4 [×2], C2×S32, Q8.7D6 [×2], D6⋊D6, D12.2D6

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

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

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

36 conjugacy classes

class 1 2A2B2C2D3A3B3C4A4B4C4D4E6A6B6C6D6E8A8B8C8D12A12B12C12D12E12F24A···24H
order122223334444466666888812121212121224···24
size1112121822429912122242424221818444424244···4

36 irreducible representations

dim1111112222222444444
type+++++++++++++++
imageC1C2C2C2C2C2S3D4D4D6D6D6C4○D8S32S3×D4C2×S32Q8.7D6D6⋊D6D12.2D6
kernelD12.2D6C322D8C322Q16C3×C24⋊C2C8×C3⋊S3D12⋊S3C24⋊C2C3⋊Dic3C2×C3⋊S3C24Dic6D12C32C8C6C4C3C2C1
# reps1112122112224121424

Matrix representation of D12.2D6 in GL6(𝔽73)

4600000
0270000
001000
000100
000001
00007272
,
0100000
2200000
0072000
0007200
000001
000010
,
010000
7200000
0017200
001000
000010
00007272
,
2700000
0270000
0072000
0072100
000010
00007272

G:=sub<GL(6,GF(73))| [46,0,0,0,0,0,0,27,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,72,0,0,0,0,1,72],[0,22,0,0,0,0,10,0,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[0,72,0,0,0,0,1,0,0,0,0,0,0,0,1,1,0,0,0,0,72,0,0,0,0,0,0,0,1,72,0,0,0,0,0,72],[27,0,0,0,0,0,0,27,0,0,0,0,0,0,72,72,0,0,0,0,0,1,0,0,0,0,0,0,1,72,0,0,0,0,0,72] >;

D12.2D6 in GAP, Magma, Sage, TeX

D_{12}._2D_6
% in TeX

G:=Group("D12.2D6");
// GroupNames label

G:=SmallGroup(288,457);
// by ID

G=gap.SmallGroup(288,457);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,120,254,135,58,675,346,80,1356,9414]);
// Polycyclic

G:=Group<a,b,c,d|a^12=b^2=1,c^6=d^2=a^6,b*a*b=c*a*c^-1=a^-1,d*a*d^-1=a^5,c*b*c^-1=a*b,d*b*d^-1=a^4*b,d*c*d^-1=c^5>;
// generators/relations

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