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G = C3×C12.C8order 288 = 25·32

Direct product of C3 and C12.C8

direct product, metacyclic, supersoluble, monomial

Aliases: C3×C12.C8, C24.6C12, C12.1C24, C62.7C8, C24.100D6, C329M5(2), C24.14Dic3, C3⋊C165C6, C12.9(C3⋊C8), C8.22(S3×C6), (C2×C6).6C24, (C3×C12).8C8, C6.9(C2×C24), (C2×C24).22C6, (C6×C24).24C2, (C2×C24).35S3, (C3×C24).13C4, C24.35(C2×C6), (C6×C12).26C4, C32(C3×M5(2)), C8.2(C3×Dic3), C12.46(C2×C12), (C2×C12).13C12, C4.10(C6×Dic3), (C3×C24).67C22, C12.67(C2×Dic3), (C2×C12).28Dic3, C4.(C3×C3⋊C8), C2.4(C6×C3⋊C8), C22.(C3×C3⋊C8), C6.20(C2×C3⋊C8), (C3×C3⋊C16)⋊12C2, (C2×C6).2(C3⋊C8), (C2×C8).7(C3×S3), (C3×C6).40(C2×C8), (C2×C4).5(C3×Dic3), (C3×C12).131(C2×C4), SmallGroup(288,246)

Series: Derived Chief Lower central Upper central

C1C6 — C3×C12.C8
C1C3C6C12C24C3×C24C3×C3⋊C16 — C3×C12.C8
C3C6 — C3×C12.C8
C1C24C2×C24

Generators and relations for C3×C12.C8
 G = < a,b,c | a3=b24=1, c4=b18, ab=ba, ac=ca, cbc-1=b5 >

Subgroups: 90 in 63 conjugacy classes, 42 normal (38 characteristic)
C1, C2, C2, C3 [×2], C3, C4 [×2], C22, C6 [×2], C6 [×5], C8 [×2], C2×C4, C32, C12 [×4], C12 [×2], C2×C6 [×2], C2×C6, C16 [×2], C2×C8, C3×C6, C3×C6, C24 [×4], C24 [×2], C2×C12 [×2], C2×C12, M5(2), C3×C12 [×2], C62, C3⋊C16 [×2], C48 [×2], C2×C24 [×2], C2×C24, C3×C24 [×2], C6×C12, C12.C8, C3×M5(2), C3×C3⋊C16 [×2], C6×C24, C3×C12.C8
Quotients: C1, C2 [×3], C3, C4 [×2], C22, S3, C6 [×3], C8 [×2], C2×C4, Dic3 [×2], C12 [×2], D6, C2×C6, C2×C8, C3×S3, C3⋊C8 [×2], C24 [×2], C2×Dic3, C2×C12, M5(2), C3×Dic3 [×2], S3×C6, C2×C3⋊C8, C2×C24, C3×C3⋊C8 [×2], C6×Dic3, C12.C8, C3×M5(2), C6×C3⋊C8, C3×C12.C8

Smallest permutation representation of C3×C12.C8
On 48 points
Generators in S48
(1 17 9)(2 18 10)(3 19 11)(4 20 12)(5 21 13)(6 22 14)(7 23 15)(8 24 16)(25 33 41)(26 34 42)(27 35 43)(28 36 44)(29 37 45)(30 38 46)(31 39 47)(32 40 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 25 10 46 19 43 4 40 13 37 22 34 7 31 16 28)(2 30 11 27 20 48 5 45 14 42 23 39 8 36 17 33)(3 35 12 32 21 29 6 26 15 47 24 44 9 41 18 38)

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

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

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

108 conjugacy classes

class 1 2A2B3A3B3C3D3E4A4B4C6A6B6C···6M8A8B8C8D8E8F12A12B12C12D12E···12R16A···16H24A···24H24I···24AJ48A···48P
order12233333444666···68888881212121212···1216···1624···2424···2448···48
size11211222112112···211112211112···26···61···12···26···6

108 irreducible representations

dim111111111111112222222222222222
type++++-+-
imageC1C2C2C3C4C4C6C6C8C8C12C12C24C24S3Dic3D6Dic3C3×S3C3⋊C8C3⋊C8M5(2)C3×Dic3S3×C6C3×Dic3C3×C3⋊C8C3×C3⋊C8C12.C8C3×M5(2)C3×C12.C8
kernelC3×C12.C8C3×C3⋊C16C6×C24C12.C8C3×C24C6×C12C3⋊C16C2×C24C3×C12C62C24C2×C12C12C2×C6C2×C24C24C24C2×C12C2×C8C12C2×C6C32C8C8C2×C4C4C22C3C3C1
# reps1212224244448811112224222448816

Matrix representation of C3×C12.C8 in GL2(𝔽97) generated by

610
061
,
430
093
,
01
500
G:=sub<GL(2,GF(97))| [61,0,0,61],[43,0,0,93],[0,50,1,0] >;

C3×C12.C8 in GAP, Magma, Sage, TeX

C_3\times C_{12}.C_8
% in TeX

G:=Group("C3xC12.C8");
// GroupNames label

G:=SmallGroup(288,246);
// by ID

G=gap.SmallGroup(288,246);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-3,84,701,80,102,9414]);
// Polycyclic

G:=Group<a,b,c|a^3=b^24=1,c^4=b^18,a*b=b*a,a*c=c*a,c*b*c^-1=b^5>;
// generators/relations

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