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G = C4×C3⋊D12order 288 = 25·32

Direct product of C4 and C3⋊D12

direct product, metabelian, supersoluble, monomial

Aliases: C4×C3⋊D12, C128D12, C62.73C23, D66(C4×S3), C34(C4×D12), (C3×C12)⋊16D4, C3210(C4×D4), C127(C3⋊D4), Dic34(C4×S3), C6.79(C2×D12), D6⋊Dic339C2, (Dic3×C12)⋊3C2, (C4×Dic3)⋊16S3, (C2×C12).308D6, C6.31(C4○D12), (C2×Dic3).98D6, (C22×S3).65D6, Dic3⋊Dic339C2, C6.D1224C2, (C6×C12).234C22, C2.4(D6.D6), (C6×Dic3).113C22, (S3×C2×C4)⋊10S3, C2.21(C4×S32), C31(C4×C3⋊D4), C6.20(S3×C2×C4), (S3×C2×C12)⋊18C2, (C2×C4).141S32, (S3×C6)⋊12(C2×C4), C22.41(C2×S32), C6.15(C2×C3⋊D4), (C3×Dic3)⋊8(C2×C4), C2.1(C2×C3⋊D12), (C3×C6).100(C2×D4), (S3×C2×C6).78C22, (C3×C6).43(C4○D4), (C2×C6).92(C22×S3), (C3×C6).19(C22×C4), (C2×C3⋊D12).10C2, (C22×C3⋊S3).72C22, (C2×C3⋊Dic3).135C22, (C2×C4×C3⋊S3)⋊13C2, (C2×C3⋊S3)⋊8(C2×C4), SmallGroup(288,551)

Series: Derived Chief Lower central Upper central

C1C3×C6 — C4×C3⋊D12
C1C3C32C3×C6C62S3×C2×C6C2×C3⋊D12 — C4×C3⋊D12
C32C3×C6 — C4×C3⋊D12
C1C2×C4

Generators and relations for C4×C3⋊D12
 G = < a,b,c,d | a4=b3=c12=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c-1 >

Subgroups: 818 in 215 conjugacy classes, 66 normal (44 characteristic)
C1, C2 [×3], C2 [×4], C3 [×2], C3, C4 [×2], C4 [×5], C22, C22 [×8], S3 [×10], C6 [×6], C6 [×5], C2×C4, C2×C4 [×8], D4 [×4], C23 [×2], C32, Dic3 [×2], Dic3 [×6], C12 [×4], C12 [×6], D6 [×2], D6 [×16], C2×C6 [×2], C2×C6 [×5], C42, C22⋊C4 [×2], C4⋊C4, C22×C4 [×2], C2×D4, C3×S3 [×2], C3⋊S3 [×2], C3×C6 [×3], C4×S3 [×10], D12 [×4], C2×Dic3 [×3], C2×Dic3 [×3], C3⋊D4 [×4], C2×C12 [×2], C2×C12 [×6], C22×S3, C22×S3 [×3], C22×C6, C4×D4, C3×Dic3 [×2], C3×Dic3 [×2], C3⋊Dic3, C3×C12 [×2], S3×C6 [×2], S3×C6 [×2], C2×C3⋊S3 [×2], C2×C3⋊S3 [×2], C62, C4×Dic3, Dic3⋊C4, C4⋊Dic3, D6⋊C4 [×3], C6.D4, C4×C12, S3×C2×C4, S3×C2×C4 [×3], C2×D12, C2×C3⋊D4, C22×C12, C3⋊D12 [×4], S3×C12 [×2], C6×Dic3 [×3], C4×C3⋊S3 [×2], C2×C3⋊Dic3, C6×C12, S3×C2×C6, C22×C3⋊S3, C4×D12, C4×C3⋊D4, D6⋊Dic3, C6.D12, Dic3⋊Dic3, Dic3×C12, C2×C3⋊D12, S3×C2×C12, C2×C4×C3⋊S3, C4×C3⋊D12
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3 [×2], C2×C4 [×6], D4 [×2], C23, D6 [×6], C22×C4, C2×D4, C4○D4, C4×S3 [×4], D12 [×2], C3⋊D4 [×2], C22×S3 [×2], C4×D4, S32, S3×C2×C4 [×2], C2×D12, C4○D12 [×2], C2×C3⋊D4, C3⋊D12 [×2], C2×S32, C4×D12, C4×C3⋊D4, D6.D6, C4×S32, C2×C3⋊D12, C4×C3⋊D12

Smallest permutation representation of C4×C3⋊D12
On 48 points
Generators in S48
(1 29 24 48)(2 30 13 37)(3 31 14 38)(4 32 15 39)(5 33 16 40)(6 34 17 41)(7 35 18 42)(8 36 19 43)(9 25 20 44)(10 26 21 45)(11 27 22 46)(12 28 23 47)
(1 5 9)(2 10 6)(3 7 11)(4 12 8)(13 21 17)(14 18 22)(15 23 19)(16 20 24)(25 29 33)(26 34 30)(27 31 35)(28 36 32)(37 45 41)(38 42 46)(39 47 43)(40 44 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 3)(4 12)(5 11)(6 10)(7 9)(14 24)(15 23)(16 22)(17 21)(18 20)(25 35)(26 34)(27 33)(28 32)(29 31)(38 48)(39 47)(40 46)(41 45)(42 44)

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

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

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

60 conjugacy classes

class 1 2A2B2C2D2E2F2G3A3B3C4A4B4C4D4E···4J4K4L6A···6F6G6H6I6J6K6L6M12A···12H12I12J12K12L12M···12X
order1222222233344444···4446···6666666612···121212121212···12
size111166181822411116···618182···244466662···244446···6

60 irreducible representations

dim11111111122222222222244444
type++++++++++++++++++
imageC1C2C2C2C2C2C2C2C4S3S3D4D6D6D6C4○D4C4×S3D12C3⋊D4C4×S3C4○D12S32C3⋊D12C2×S32D6.D6C4×S32
kernelC4×C3⋊D12D6⋊Dic3C6.D12Dic3⋊Dic3Dic3×C12C2×C3⋊D12S3×C2×C12C2×C4×C3⋊S3C3⋊D12C4×Dic3S3×C2×C4C3×C12C2×Dic3C2×C12C22×S3C3×C6Dic3C12C12D6C6C2×C4C4C22C2C2
# reps11111111811232124444812122

Matrix representation of C4×C3⋊D12 in GL6(𝔽13)

1200000
0120000
0012000
0001200
000050
000005
,
100000
010000
001000
000100
00001212
000010
,
12120000
100000
000100
0012000
0000120
000011
,
12120000
010000
0012000
000100
000010
00001212

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

C4×C3⋊D12 in GAP, Magma, Sage, TeX

C_4\times C_3\rtimes D_{12}
% in TeX

G:=Group("C4xC3:D12");
// GroupNames label

G:=SmallGroup(288,551);
// by ID

G=gap.SmallGroup(288,551);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,253,120,58,1356,9414]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^3=c^12=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations

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