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G = D65D12order 288 = 25·32

2nd semidirect product of D6 and D12 acting via D12/D6=C2

metabelian, supersoluble, monomial

Aliases: D65D12, C62.93C23, D6⋊C47S3, (S3×C6)⋊4D4, (C2×C12)⋊1D6, C6.27(S3×D4), (C6×C12)⋊1C22, (C2×Dic3)⋊2D6, C6.29(C2×D12), C2.29(S3×D12), C323C22≀C2, C31(D6⋊D4), C6.D124C2, (C6×Dic3)⋊2C22, (C22×S3).46D6, C2.14(Dic3⋊D6), (C2×C4)⋊2S32, (C2×C3⋊S3)⋊3D4, (C3×D6⋊C4)⋊1C2, (C22×S32)⋊2C2, (C2×C12⋊S3)⋊2C2, (C2×C3⋊D12)⋊8C2, C22.128(C2×S32), (C3×C6).115(C2×D4), (S3×C2×C6).38C22, (C22×C3⋊S3)⋊1C22, (C2×C6).112(C22×S3), SmallGroup(288,571)

Series: Derived Chief Lower central Upper central

C1C62 — D65D12
C1C3C32C3×C6C62S3×C2×C6C22×S32 — D65D12
C32C62 — D65D12
C1C22C2×C4

Generators and relations for D65D12
 G = < a,b,c,d | a6=b2=c12=d2=1, bab=dad=a-1, ac=ca, cbc-1=a3b, dbd=ab, dcd=c-1 >

Subgroups: 1538 in 287 conjugacy classes, 54 normal (14 characteristic)
C1, C2, C2 [×2], C2 [×7], C3 [×2], C3, C4 [×3], C22, C22 [×23], S3 [×14], C6 [×6], C6 [×7], C2×C4, C2×C4 [×2], D4 [×6], C23 [×10], C32, Dic3 [×2], C12 [×6], D6 [×4], D6 [×42], C2×C6 [×2], C2×C6 [×9], C22⋊C4 [×3], C2×D4 [×3], C24, C3×S3 [×4], C3⋊S3 [×3], C3×C6, C3×C6 [×2], D12 [×12], C2×Dic3 [×2], C3⋊D4 [×4], C2×C12 [×2], C2×C12 [×3], C22×S3 [×2], C22×S3 [×18], C22×C6 [×2], C22≀C2, C3×Dic3 [×2], C3×C12, S32 [×8], S3×C6 [×4], S3×C6 [×4], C2×C3⋊S3 [×2], C2×C3⋊S3 [×5], C62, D6⋊C4 [×2], D6⋊C4 [×2], C3×C22⋊C4 [×2], C2×D12 [×5], C2×C3⋊D4 [×2], S3×C23 [×2], C3⋊D12 [×4], C6×Dic3 [×2], C12⋊S3 [×2], C6×C12, C2×S32 [×6], S3×C2×C6 [×2], C22×C3⋊S3 [×2], D6⋊D4 [×2], C6.D12, C3×D6⋊C4 [×2], C2×C3⋊D12 [×2], C2×C12⋊S3, C22×S32, D65D12
Quotients: C1, C2 [×7], C22 [×7], S3 [×2], D4 [×6], C23, D6 [×6], C2×D4 [×3], D12 [×4], C22×S3 [×2], C22≀C2, S32, C2×D12 [×2], S3×D4 [×4], C2×S32, D6⋊D4 [×2], S3×D12 [×2], Dic3⋊D6, D65D12

Smallest permutation representation of D65D12
On 48 points
Generators in S48
(1 17 9 13 5 21)(2 18 10 14 6 22)(3 19 11 15 7 23)(4 20 12 16 8 24)(25 40 29 44 33 48)(26 41 30 45 34 37)(27 42 31 46 35 38)(28 43 32 47 36 39)
(1 41)(2 35)(3 43)(4 25)(5 45)(6 27)(7 47)(8 29)(9 37)(10 31)(11 39)(12 33)(13 34)(14 42)(15 36)(16 44)(17 26)(18 46)(19 28)(20 48)(21 30)(22 38)(23 32)(24 40)
(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 3)(4 12)(5 11)(6 10)(7 9)(13 15)(16 24)(17 23)(18 22)(19 21)(25 44)(26 43)(27 42)(28 41)(29 40)(30 39)(31 38)(32 37)(33 48)(34 47)(35 46)(36 45)

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

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

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

42 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I2J3A3B3C4A4B4C6A···6F6G6H6I6J6K6L6M12A···12H12I12J12K12L
order122222222223334446···6666666612···1212121212
size11116666181836224412122···2444121212124···412121212

42 irreducible representations

dim111111222222244444
type++++++++++++++++++
imageC1C2C2C2C2C2S3D4D4D6D6D6D12S32S3×D4C2×S32S3×D12Dic3⋊D6
kernelD65D12C6.D12C3×D6⋊C4C2×C3⋊D12C2×C12⋊S3C22×S32D6⋊C4S3×C6C2×C3⋊S3C2×Dic3C2×C12C22×S3D6C2×C4C6C22C2C2
# reps112211242222814142

Matrix representation of D65D12 in GL8(ℤ)

-10000000
0-1000000
00100000
00010000
00000100
0000-1-100
00000010
00000001
,
-1-2000000
01000000
00100000
00010000
00000-100
0000-1000
00000010
00000001
,
-1-2000000
11000000
000-10000
00100000
00001000
00000100
000000-11
000000-10
,
-10000000
11000000
00100000
000-10000
00001000
0000-1-100
000000-10
000000-11

G:=sub<GL(8,Integers())| [-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[-1,0,0,0,0,0,0,0,-2,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[-1,1,0,0,0,0,0,0,-2,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,-1,0,0,0,0,0,0,1,0],[-1,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,-1,0,0,0,0,0,0,0,1] >;

D65D12 in GAP, Magma, Sage, TeX

D_6\rtimes_5D_{12}
% in TeX

G:=Group("D6:5D12");
// GroupNames label

G:=SmallGroup(288,571);
// by ID

G=gap.SmallGroup(288,571);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,422,135,142,1356,9414]);
// Polycyclic

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

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