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G = D12⋊15D6  order 288 = 25·32

9th semidirect product of D12 and D6 acting via D6/S3=C2

metabelian, supersoluble, monomial

Aliases: D12⋊15D6, Q8⋊7S32, (C4×S3)⋊10D6, (C3×Q8)⋊12D6, Q8⋊3S3⋊9S3, D6⋊D6⋊13C2, C6.26(S3×C23), (C3×C6).26C24, D12⋊5S3⋊12C2, (S3×C12)⋊10C22, D6⋊S3⋊6C22, (C3×D12)⋊17C22, (S3×C6).14C23, C12.38(C22×S3), (C3×C12).38C23, D6.14(C22×S3), (S3×Dic3)⋊15C22, C3⋊Dic3.26C23, (Q8×C32)⋊11C22, C32⋊4Q8⋊12C22, Dic3.27(C22×S3), (C3×Dic3).27C23, C6.D6.15C22, (C4×S32)⋊8C2, C4.38(C2×S32), C3⋊6(S3×C4○D4), (Q8×C3⋊S3)⋊8C2, C3⋊S3⋊3(C4○D4), C2.28(C22×S32), C32⋊10(C2×C4○D4), (C2×S32).14C22, (C3×Q8⋊3S3)⋊9C2, (C4×C3⋊S3).47C22, (C2×C3⋊S3).48C23, SmallGroup(288,967)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C3×C6 — D12⋊15D6
Chief series
C1 — C3 — C32 — C3×C6 — S3×C6 — C2×S32 — C4×S32 — D12⋊15D6
Lower central
C32 — C3×C6 — D12⋊15D6
Upper central
C1 — C2 — Q8

Generators and relations for D12⋊15D6
 G = < a,b,c,d | a12=b2=c6=d2=1, bab=a-1, cac-1=a5, ad=da, cbc-1=a10b, dbd=a6b, dcd=c-1 >

Subgroups: 1250 in 347 conjugacy classes, 110 normal (10 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C2×C4, D4, Q8, Q8, C23, C32, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C22×C4, C2×D4, C2×Q8, C4○D4, C3×S3, C3⋊S3, C3×C6, Dic6, C4×S3, C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C3×Q8, C3×Q8, C22×S3, C2×C4○D4, C3×Dic3, C3⋊Dic3, C3×C12, S32, S3×C6, C2×C3⋊S3, S3×C2×C4, C4○D12, S3×D4, D4⋊2S3, S3×Q8, Q8⋊3S3, C3×C4○D4, S3×Dic3, C6.D6, D6⋊S3, S3×C12, C3×D12, C32⋊4Q8, C4×C3⋊S3, Q8×C32, C2×S32, S3×C4○D4, D12⋊5S3, C4×S32, D6⋊D6, C3×Q8⋊3S3, Q8×C3⋊S3, D12⋊15D6
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, C24, C22×S3, C2×C4○D4, S32, S3×C23, C2×S32, S3×C4○D4, C22×S32, D12⋊15D6

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

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

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

45 conjugacy classes

class 1 2A2B···2G2H2I3A3B3C4A4B4C4D4E4F4G4H4I4J6A6B6C6D···6I12A···12F12G12H12I12J12K12L12M
order122···22233344444444446666···612···1212121212121212
size116···699224222333318181822412···124···46666888

45 irreducible representations

dim111111222224448
type++++++++++++-
imageC1C2C2C2C2C2S3D6D6D6C4○D4S32C2×S32S3×C4○D4D12⋊15D6
kernelD12⋊15D6D12⋊5S3C4×S32D6⋊D6C3×Q8⋊3S3Q8×C3⋊S3Q8⋊3S3C4×S3D12C3×Q8C3⋊S3Q8C4C3C1
# reps163321266241341

Matrix representation of D12⋊15D6 ►in GL6(𝔽13)

1210000
1200000
00121100
001100
0000120
0000012
,
1200000
1210000
001200
0001200
000010
000001
,
010000
100000
008300
005500
000001
00001212
,
100000
010000
0051000
008800
000001
000010

G:=sub<GL(6,GF(13))| [12,12,0,0,0,0,1,0,0,0,0,0,0,0,12,1,0,0,0,0,11,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[12,12,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,2,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,8,5,0,0,0,0,3,5,0,0,0,0,0,0,0,12,0,0,0,0,1,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,5,8,0,0,0,0,10,8,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;
 

D12⋊15D6 in GAP, Magma, Sage, TeX 

D_{12}\rtimes_{15}D_6
 
% in TeX
 
G:=Group("D12:15D6");
 
// GroupNames label
 
G:=SmallGroup(288,967);
 
// by ID
 
G=gap.SmallGroup(288,967);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,100,675,346,185,80,1356,9414]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^12=b^2=c^6=d^2=1,b*a*b=a^-1,c*a*c^-1=a^5,a*d=d*a,c*b*c^-1=a^10*b,d*b*d=a^6*b,d*c*d=c^-1>;
 
// generators/relations
 

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