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G = D64D12order 288 = 25·32

1st semidirect product of D6 and D12 acting via D12/D6=C2

metabelian, supersoluble, monomial

Aliases: D64D12, C62.92C23, (S3×C6)⋊3D4, D6⋊C418S3, (C2×D12)⋊6S3, (C2×C12)⋊13D6, C6.26(S3×D4), (C6×D12)⋊23C2, D65(C3⋊D4), (C2×Dic3)⋊1D6, C6.28(C2×D12), C2.28(S3×D12), C322C22≀C2, (C22×S3)⋊2D6, C33(D6⋊D4), C31(C232D6), D6⋊Dic314C2, (C6×C12)⋊17C22, (C6×Dic3)⋊1C22, C6.11D1212C2, C2.20(D6⋊D6), (C2×C4)⋊1S32, (C2×C3⋊S3)⋊2D4, (C22×S32)⋊1C2, (S3×C2×C6)⋊1C22, (C3×D6⋊C4)⋊15C2, C2.19(S3×C3⋊D4), C6.41(C2×C3⋊D4), (C2×D6⋊S3)⋊6C2, (C2×C3⋊D12)⋊7C2, C22.127(C2×S32), (C3×C6).114(C2×D4), (C2×C3⋊Dic3)⋊2C22, (C2×C6).111(C22×S3), (C22×C3⋊S3).27C22, SmallGroup(288,570)

Series: Derived Chief Lower central Upper central

C1C62 — D64D12
C1C3C32C3×C6C62S3×C2×C6C22×S32 — D64D12
C32C62 — D64D12
C1C22C2×C4

Generators and relations for D64D12
 G = < a,b,c,d | a6=b2=c12=d2=1, bab=a-1, ac=ca, ad=da, cbc-1=dbd=a3b, dcd=c-1 >

Subgroups: 1370 in 277 conjugacy classes, 54 normal (44 characteristic)
C1, C2 [×3], C2 [×7], C3 [×2], C3, C4 [×3], C22, C22 [×23], S3 [×11], C6 [×6], C6 [×8], C2×C4, C2×C4 [×2], D4 [×6], C23 [×10], C32, Dic3 [×4], C12 [×4], D6 [×4], D6 [×35], C2×C6 [×2], C2×C6 [×12], C22⋊C4 [×3], C2×D4 [×3], C24, C3×S3 [×5], C3⋊S3 [×2], C3×C6 [×3], D12 [×4], C2×Dic3, C2×Dic3 [×3], C3⋊D4 [×6], C2×C12 [×2], C2×C12 [×2], C3×D4 [×2], C22×S3 [×3], C22×S3 [×15], C22×C6 [×3], C22≀C2, C3×Dic3, C3⋊Dic3, C3×C12, S32 [×8], S3×C6 [×4], S3×C6 [×7], C2×C3⋊S3 [×2], C2×C3⋊S3 [×2], C62, D6⋊C4, D6⋊C4 [×4], C6.D4, C3×C22⋊C4, C2×D12, C2×D12, C2×C3⋊D4 [×3], C6×D4, S3×C23 [×2], D6⋊S3 [×2], C3⋊D12 [×2], C3×D12 [×2], C6×Dic3, C2×C3⋊Dic3, C6×C12, C2×S32 [×6], S3×C2×C6 [×3], C22×C3⋊S3, D6⋊D4, C232D6, D6⋊Dic3, C3×D6⋊C4, C6.11D12, C2×D6⋊S3, C2×C3⋊D12, C6×D12, C22×S32, D64D12
Quotients: C1, C2 [×7], C22 [×7], S3 [×2], D4 [×6], C23, D6 [×6], C2×D4 [×3], D12 [×2], C3⋊D4 [×2], C22×S3 [×2], C22≀C2, S32, C2×D12, S3×D4 [×4], C2×C3⋊D4, C2×S32, D6⋊D4, C232D6, S3×D12, D6⋊D6, S3×C3⋊D4, D64D12

Smallest permutation representation of D64D12
On 48 points
Generators in S48
(1 15 5 19 9 23)(2 16 6 20 10 24)(3 17 7 21 11 13)(4 18 8 22 12 14)(25 47 33 43 29 39)(26 48 34 44 30 40)(27 37 35 45 31 41)(28 38 36 46 32 42)
(1 33)(2 40)(3 35)(4 42)(5 25)(6 44)(7 27)(8 46)(9 29)(10 48)(11 31)(12 38)(13 45)(14 28)(15 47)(16 30)(17 37)(18 32)(19 39)(20 34)(21 41)(22 36)(23 43)(24 26)
(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 30)(14 29)(15 28)(16 27)(17 26)(18 25)(19 36)(20 35)(21 34)(22 33)(23 32)(24 31)

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

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

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

42 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I2J3A3B3C4A4B4C6A···6F6G6H6I6J···6O12A···12H12I12J
order122222222223334446···66666···612···121212
size11116666121818224412362···244412···124···41212

42 irreducible representations

dim11111111222222222444444
type++++++++++++++++++++
imageC1C2C2C2C2C2C2C2S3S3D4D4D6D6D6D12C3⋊D4S32S3×D4C2×S32S3×D12D6⋊D6S3×C3⋊D4
kernelD64D12D6⋊Dic3C3×D6⋊C4C6.11D12C2×D6⋊S3C2×C3⋊D12C6×D12C22×S32D6⋊C4C2×D12S3×C6C2×C3⋊S3C2×Dic3C2×C12C22×S3D6D6C2×C4C6C22C2C2C2
# reps11111111114212344141222

Matrix representation of D64D12 in GL8(𝔽13)

120000000
012000000
00100000
00010000
000012100
000012000
000000120
000000012
,
012000000
120000000
00100000
00010000
000012000
000012100
000000120
000000121
,
80000000
05000000
001120000
00100000
000012000
000001200
000000111
000000012
,
05000000
80000000
00100000
001120000
000012000
000001200
000000111
000000012

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

D64D12 in GAP, Magma, Sage, TeX

D_6\rtimes_4D_{12}
% in TeX

G:=Group("D6:4D12");
// GroupNames label

G:=SmallGroup(288,570);
// by ID

G=gap.SmallGroup(288,570);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,422,135,100,1356,9414]);
// Polycyclic

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

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