Copied to
clipboard

G = (C3×C12).148D6order 432 = 24·33

80th non-split extension by C3×C12 of D6 acting via D6/C3=C22

metabelian, supersoluble, monomial

Aliases: C12.43(S32), C12⋊S312S3, (C3×C12).148D6, C3319(C4○D4), C339D411C2, C3⋊Dic3.21D6, C35(D125S3), C35(D12⋊S3), C324Q813S3, C35(D6.6D6), C3213(C4○D12), C328(Q83S3), C4.5(C324D6), (C32×C6).66C23, C3214(D42S3), (C32×C12).50C22, (C4×C3⋊S3)⋊9S3, C6.95(C2×S32), (C12×C3⋊S3)⋊8C2, (C2×C3⋊S3).46D6, C339(C2×C4)⋊4C2, (C3×C12⋊S3)⋊12C2, (C6×C3⋊S3).30C22, (C3×C324Q8)⋊13C2, C2.4(C2×C324D6), (C3×C6).116(C22×S3), (C3×C3⋊Dic3).24C22, SmallGroup(432,688)

Series: Derived Chief Lower central Upper central

C1C32×C6 — (C3×C12).148D6
C1C3C32C33C32×C6C6×C3⋊S3C339(C2×C4) — (C3×C12).148D6
C33C32×C6 — (C3×C12).148D6
C1C2C4

Generators and relations for (C3×C12).148D6
 G = < a,b,c,d | a3=b12=1, c6=d2=b6, ab=ba, cac-1=a-1, ad=da, cbc-1=b5, dbd-1=b-1, dcd-1=c5 >

Subgroups: 1064 in 210 conjugacy classes, 47 normal (35 characteristic)
C1, C2, C2 [×3], C3 [×3], C3 [×4], C4, C4 [×3], C22 [×3], S3 [×10], C6 [×3], C6 [×7], C2×C4 [×3], D4 [×3], Q8, C32 [×3], C32 [×4], Dic3 [×9], C12 [×3], C12 [×7], D6 [×9], C2×C6 [×3], C4○D4, C3×S3 [×10], C3⋊S3 [×3], C3×C6 [×3], C3×C6 [×4], Dic6 [×3], C4×S3 [×7], D12 [×5], C2×Dic3 [×2], C3⋊D4 [×4], C2×C12, C3×D4, C3×Q8, C33, C3×Dic3 [×9], C3⋊Dic3, C3⋊Dic3 [×2], C3×C12 [×3], C3×C12 [×4], S3×C6 [×9], C2×C3⋊S3, C2×C3⋊S3 [×2], C4○D12, D42S3, Q83S3, C3×C3⋊S3 [×3], C32×C6, S3×Dic3 [×4], C6.D6 [×2], D6⋊S3 [×2], C3⋊D12 [×4], C3×Dic6 [×3], S3×C12 [×3], C3×D12 [×3], C324Q8, C4×C3⋊S3, C12⋊S3, C3×C3⋊Dic3, C3×C3⋊Dic3 [×2], C32×C12, C6×C3⋊S3, C6×C3⋊S3 [×2], D125S3, D12⋊S3, D6.6D6, C339(C2×C4) [×2], C339D4 [×2], C3×C324Q8, C12×C3⋊S3, C3×C12⋊S3, (C3×C12).148D6
Quotients: C1, C2 [×7], C22 [×7], S3 [×3], C23, D6 [×9], C4○D4, C22×S3 [×3], S32 [×3], C4○D12, D42S3, Q83S3, C2×S32 [×3], C324D6, D125S3, D12⋊S3, D6.6D6, C2×C324D6, (C3×C12).148D6

Smallest permutation representation of (C3×C12).148D6
On 48 points
Generators in S48
(1 5 9)(2 6 10)(3 7 11)(4 8 12)(13 17 21)(14 18 22)(15 19 23)(16 20 24)(25 33 29)(26 34 30)(27 35 31)(28 36 32)(37 45 41)(38 46 42)(39 47 43)(40 48 44)
(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 18 3 16 5 14 7 24 9 22 11 20)(2 23 4 21 6 19 8 17 10 15 12 13)(25 43 35 45 33 47 31 37 29 39 27 41)(26 48 36 38 34 40 32 42 30 44 28 46)
(1 41 7 47)(2 40 8 46)(3 39 9 45)(4 38 10 44)(5 37 11 43)(6 48 12 42)(13 26 19 32)(14 25 20 31)(15 36 21 30)(16 35 22 29)(17 34 23 28)(18 33 24 27)

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

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

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

48 conjugacy classes

class 1 2A2B2C2D3A3B3C3D···3H4A4B4C4D4E6A6B6C6D···6H6I6J6K6L12A12B12C···12N12O12P12Q12R
order122223333···3444446666···66666121212···1212121212
size111818182224···429918182224···418183636224···418183636

48 irreducible representations

dim111111222222224444444444
type+++++++++++++-++-+
imageC1C2C2C2C2C2S3S3S3D6D6D6C4○D4C4○D12S32D42S3Q83S3C2×S32C324D6D125S3D12⋊S3D6.6D6C2×C324D6(C3×C12).148D6
kernel(C3×C12).148D6C339(C2×C4)C339D4C3×C324Q8C12×C3⋊S3C3×C12⋊S3C324Q8C4×C3⋊S3C12⋊S3C3⋊Dic3C3×C12C2×C3⋊S3C33C32C12C32C32C6C4C3C3C3C2C1
# reps122111111333243113222224

Matrix representation of (C3×C12).148D6 in GL6(𝔽13)

100000
010000
00121200
001000
000010
000001
,
010000
1210000
0012000
0001200
000037
0000610
,
010000
100000
001000
00121200
000085
000080
,
0120000
1200000
001000
000100
000005
000050

G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,1,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,12,0,0,0,0,1,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,3,6,0,0,0,0,7,10],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,12,0,0,0,0,0,12,0,0,0,0,0,0,8,8,0,0,0,0,5,0],[0,12,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,5,0,0,0,0,5,0] >;

(C3×C12).148D6 in GAP, Magma, Sage, TeX

(C_3\times C_{12})._{148}D_6
% in TeX

G:=Group("(C3xC12).148D6");
// GroupNames label

G:=SmallGroup(432,688);
// by ID

G=gap.SmallGroup(432,688);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,64,254,58,1124,571,2028,14118]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^12=1,c^6=d^2=b^6,a*b=b*a,c*a*c^-1=a^-1,a*d=d*a,c*b*c^-1=b^5,d*b*d^-1=b^-1,d*c*d^-1=c^5>;
// generators/relations

׿
×
𝔽