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G = D1220D6order 288 = 25·32

4th semidirect product of D12 and D6 acting via D6/C6=C2

metabelian, supersoluble, monomial

Aliases: D1220D6, Dic618D6, C62.42D4, C4○D123S3, (C2×D12)⋊8S3, (C6×D12)⋊2C2, (C3×C12).61D4, C322D85C2, C34(D4⋊D6), (C2×C12).112D6, C326(C8⋊C22), Dic6⋊S35C2, C34(D126C22), C12.58D69C2, (C3×D12)⋊20C22, C12.43(C3⋊D4), (C3×C12).59C23, (C6×C12).72C22, C12.79(C22×S3), C324C82C22, C4.13(D6⋊S3), (C3×Dic6)⋊18C22, C22.4(D6⋊S3), (C2×C4).5S32, C4.74(C2×S32), (C3×C4○D12)⋊2C2, (C3×C6).63(C2×D4), C6.71(C2×C3⋊D4), C2.6(C2×D6⋊S3), (C2×C6).55(C3⋊D4), SmallGroup(288,471)

Series: Derived Chief Lower central Upper central

C1C3×C12 — D1220D6
C1C3C32C3×C6C3×C12C3×D12C322D8 — D1220D6
C32C3×C6C3×C12 — D1220D6
C1C2C2×C4

Generators and relations for D1220D6
 G = < a,b,c,d | a12=b2=c6=d2=1, bab=a-1, ac=ca, dad=a7, cbc-1=a6b, dbd=a3b, dcd=c-1 >

Subgroups: 530 in 146 conjugacy classes, 44 normal (34 characteristic)
C1, C2, C2 [×4], C3 [×2], C3, C4 [×2], C4, C22, C22 [×5], S3 [×3], C6 [×2], C6 [×7], C8 [×2], C2×C4, C2×C4, D4 [×5], Q8, C23, C32, Dic3, C12 [×4], C12 [×3], D6 [×5], C2×C6 [×2], C2×C6 [×6], M4(2), D8 [×2], SD16 [×2], C2×D4, C4○D4, C3×S3 [×3], C3×C6, C3×C6, C3⋊C8 [×6], Dic6, C4×S3, D12, D12 [×2], D12, C3⋊D4, C2×C12 [×2], C2×C12 [×2], C3×D4 [×5], C3×Q8, C22×S3, C22×C6, C8⋊C22, C3×Dic3, C3×C12 [×2], S3×C6 [×5], C62, C4.Dic3 [×3], D4⋊S3 [×4], D4.S3 [×2], Q82S3 [×2], C2×D12, C4○D12, C6×D4, C3×C4○D4, C324C8 [×2], C3×Dic6, S3×C12, C3×D12, C3×D12 [×2], C3×D12, C3×C3⋊D4, C6×C12, S3×C2×C6, D126C22, D4⋊D6, C322D8 [×2], Dic6⋊S3 [×2], C12.58D6, C6×D12, C3×C4○D12, D1220D6
Quotients: C1, C2 [×7], C22 [×7], S3 [×2], D4 [×2], C23, D6 [×6], C2×D4, C3⋊D4 [×4], C22×S3 [×2], C8⋊C22, S32, C2×C3⋊D4 [×2], D6⋊S3 [×2], C2×S32, D126C22, D4⋊D6, C2×D6⋊S3, D1220D6

Smallest permutation representation of D1220D6
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 24)(2 23)(3 22)(4 21)(5 20)(6 19)(7 18)(8 17)(9 16)(10 15)(11 14)(12 13)(25 45)(26 44)(27 43)(28 42)(29 41)(30 40)(31 39)(32 38)(33 37)(34 48)(35 47)(36 46)
(1 9 5)(2 10 6)(3 11 7)(4 12 8)(13 23 21 19 17 15)(14 24 22 20 18 16)(25 29 33)(26 30 34)(27 31 35)(28 32 36)(37 39 41 43 45 47)(38 40 42 44 46 48)
(1 30)(2 25)(3 32)(4 27)(5 34)(6 29)(7 36)(8 31)(9 26)(10 33)(11 28)(12 35)(13 44)(14 39)(15 46)(16 41)(17 48)(18 43)(19 38)(20 45)(21 40)(22 47)(23 42)(24 37)

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

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

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

39 conjugacy classes

class 1 2A2B2C2D2E3A3B3C4A4B4C6A6B6C6D6E6F6G6H6I···6N8A8B12A12B12C···12I12J12K
order122222333444666666666···688121212···121212
size11212121222422122222444412···123636224···41212

39 irreducible representations

dim11111122222222244444444
type+++++++++++++++-+-+
imageC1C2C2C2C2C2S3S3D4D4D6D6D6C3⋊D4C3⋊D4C8⋊C22S32D6⋊S3C2×S32D6⋊S3D126C22D4⋊D6D1220D6
kernelD1220D6C322D8Dic6⋊S3C12.58D6C6×D12C3×C4○D12C2×D12C4○D12C3×C12C62Dic6D12C2×C12C12C2×C6C32C2×C4C4C4C22C3C3C1
# reps12211111111324411111224

Matrix representation of D1220D6 in GL4(𝔽73) generated by

49000
0300
00240
00070
,
0300
49000
00065
0090
,
8000
06500
00640
0009
,
00640
0009
8000
06500
G:=sub<GL(4,GF(73))| [49,0,0,0,0,3,0,0,0,0,24,0,0,0,0,70],[0,49,0,0,3,0,0,0,0,0,0,9,0,0,65,0],[8,0,0,0,0,65,0,0,0,0,64,0,0,0,0,9],[0,0,8,0,0,0,0,65,64,0,0,0,0,9,0,0] >;

D1220D6 in GAP, Magma, Sage, TeX

D_{12}\rtimes_{20}D_6
% in TeX

G:=Group("D12:20D6");
// GroupNames label

G:=SmallGroup(288,471);
// by ID

G=gap.SmallGroup(288,471);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,141,219,675,346,80,1356,9414]);
// Polycyclic

G:=Group<a,b,c,d|a^12=b^2=c^6=d^2=1,b*a*b=a^-1,a*c=c*a,d*a*d=a^7,c*b*c^-1=a^6*b,d*b*d=a^3*b,d*c*d=c^-1>;
// generators/relations

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