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G = D1215D6order 288 = 25·32

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

metabelian, supersoluble, monomial

Aliases: D1215D6, Q87S32, (C4×S3)⋊10D6, (C3×Q8)⋊12D6, Q83S39S3, D6⋊D613C2, C6.26(S3×C23), (C3×C6).26C24, D125S312C2, (S3×C12)⋊10C22, D6⋊S36C22, (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, C324Q812C22, Dic3.27(C22×S3), (C3×Dic3).27C23, C6.D6.15C22, (C4×S32)⋊8C2, C4.38(C2×S32), C36(S3×C4○D4), (Q8×C3⋊S3)⋊8C2, C3⋊S33(C4○D4), C2.28(C22×S32), C3210(C2×C4○D4), (C2×S32).14C22, (C3×Q83S3)⋊9C2, (C4×C3⋊S3).47C22, (C2×C3⋊S3).48C23, SmallGroup(288,967)

Series: Derived Chief Lower central Upper central

C1C3×C6 — D1215D6
C1C3C32C3×C6S3×C6C2×S32C4×S32 — D1215D6
C32C3×C6 — D1215D6
C1C2Q8

Generators and relations for D1215D6
 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 [×8], C3 [×2], C3, C4 [×3], C4 [×5], C22 [×13], S3 [×12], C6 [×2], C6 [×7], C2×C4 [×16], D4 [×12], Q8, Q8 [×3], C23 [×3], C32, Dic3 [×2], Dic3 [×9], C12 [×6], C12 [×5], D6 [×6], D6 [×15], C2×C6 [×6], C22×C4 [×3], C2×D4 [×3], C2×Q8, C4○D4 [×8], C3×S3 [×6], C3⋊S3 [×2], C3×C6, Dic6 [×9], C4×S3 [×6], C4×S3 [×17], D12 [×6], C2×Dic3 [×6], C3⋊D4 [×12], C2×C12 [×6], C3×D4 [×6], C3×Q8 [×2], C3×Q8, C22×S3 [×6], C2×C4○D4, C3×Dic3 [×2], C3⋊Dic3 [×3], C3×C12 [×3], S32 [×6], S3×C6 [×6], C2×C3⋊S3, S3×C2×C4 [×6], C4○D12 [×6], S3×D4 [×6], D42S3 [×6], S3×Q8 [×3], Q83S3 [×2], C3×C4○D4 [×2], S3×Dic3 [×6], C6.D6, D6⋊S3 [×6], S3×C12 [×6], C3×D12 [×6], C324Q8 [×3], C4×C3⋊S3 [×3], Q8×C32, C2×S32 [×3], S3×C4○D4 [×2], D125S3 [×6], C4×S32 [×3], D6⋊D6 [×3], C3×Q83S3 [×2], Q8×C3⋊S3, D1215D6
Quotients: C1, C2 [×15], C22 [×35], S3 [×2], C23 [×15], D6 [×14], C4○D4 [×2], C24, C22×S3 [×14], C2×C4○D4, S32, S3×C23 [×2], C2×S32 [×3], S3×C4○D4 [×2], C22×S32, D1215D6

Smallest permutation representation of D1215D6
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 43)(14 42)(15 41)(16 40)(17 39)(18 38)(19 37)(20 48)(21 47)(22 46)(23 45)(24 44)
(1 38 5 46 9 42)(2 43 6 39 10 47)(3 48 7 44 11 40)(4 41 8 37 12 45)(13 36 17 32 21 28)(14 29 18 25 22 33)(15 34 19 30 23 26)(16 27 20 35 24 31)
(1 48)(2 37)(3 38)(4 39)(5 40)(6 41)(7 42)(8 43)(9 44)(10 45)(11 46)(12 47)(13 26)(14 27)(15 28)(16 29)(17 30)(18 31)(19 32)(20 33)(21 34)(22 35)(23 36)(24 25)

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

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

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

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○D4D1215D6
kernelD1215D6D125S3C4×S32D6⋊D6C3×Q83S3Q8×C3⋊S3Q83S3C4×S3D12C3×Q8C3⋊S3Q8C4C3C1
# reps163321266241341

Matrix representation of D1215D6 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] >;

D1215D6 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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