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G = C3×C127D4order 288 = 25·32

Direct product of C3 and C127D4

direct product, metabelian, supersoluble, monomial

Aliases: C3×C127D4, C6218D4, C62.198C23, D6⋊C43C6, C127(C3×D4), (C2×C6)⋊7D12, (C2×D12)⋊5C6, (C3×C12)⋊23D4, C4⋊Dic39C6, C6.43(C6×D4), (C6×D12)⋊29C2, C2.17(C6×D12), C223(C3×D12), C1214(C3⋊D4), (C22×C12)⋊15S3, (C22×C12)⋊10C6, (C2×C12).448D6, C6.105(C2×D12), C23.34(S3×C6), C3220(C4⋊D4), (C22×C6).129D6, C6.128(C4○D12), (C6×C12).284C22, (C2×C62).101C22, (C6×Dic3).99C22, (C2×C6×C12)⋊12C2, (C2×C6)⋊8(C3×D4), C43(C3×C3⋊D4), C33(C3×C4⋊D4), (C2×C3⋊D4)⋊3C6, C2.7(C6×C3⋊D4), (C3×D6⋊C4)⋊34C2, (C6×C3⋊D4)⋊17C2, (C2×C4).85(S3×C6), (C22×C4)⋊8(C3×S3), C6.18(C3×C4○D4), C22.56(S3×C2×C6), (C2×C12).93(C2×C6), (C3×C4⋊Dic3)⋊33C2, C2.19(C3×C4○D12), (C3×C6).187(C2×D4), C6.144(C2×C3⋊D4), (S3×C2×C6).60C22, (C22×C6).65(C2×C6), (C2×C6).53(C22×C6), (C3×C6).106(C4○D4), (C22×S3).10(C2×C6), (C2×C6).331(C22×S3), (C2×Dic3).11(C2×C6), SmallGroup(288,701)

Series: Derived Chief Lower central Upper central

C1C2×C6 — C3×C127D4
C1C3C6C2×C6C62S3×C2×C6C6×D12 — C3×C127D4
C3C2×C6 — C3×C127D4
C1C2×C6C22×C12

Generators and relations for C3×C127D4
 G = < a,b,c,d | a3=b12=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c-1 >

Subgroups: 538 in 215 conjugacy classes, 74 normal (42 characteristic)
C1, C2 [×3], C2 [×4], C3 [×2], C3, C4 [×2], C4 [×3], C22, C22 [×2], C22 [×8], S3 [×2], C6 [×6], C6 [×13], C2×C4 [×2], C2×C4 [×4], D4 [×6], C23, C23 [×2], C32, Dic3 [×2], C12 [×4], C12 [×8], D6 [×6], C2×C6 [×2], C2×C6 [×4], C2×C6 [×17], C22⋊C4 [×2], C4⋊C4, C22×C4, C2×D4 [×3], C3×S3 [×2], C3×C6 [×3], C3×C6 [×2], D12 [×2], C2×Dic3 [×2], C3⋊D4 [×4], C2×C12 [×4], C2×C12 [×12], C3×D4 [×6], C22×S3 [×2], C22×C6 [×2], C22×C6 [×3], C4⋊D4, C3×Dic3 [×2], C3×C12 [×2], C3×C12, S3×C6 [×6], C62, C62 [×2], C62 [×2], C4⋊Dic3, D6⋊C4 [×2], C3×C22⋊C4 [×2], C3×C4⋊C4, C2×D12, C2×C3⋊D4 [×2], C22×C12 [×2], C22×C12, C6×D4 [×3], C3×D12 [×2], C6×Dic3 [×2], C3×C3⋊D4 [×4], C6×C12 [×2], C6×C12 [×2], S3×C2×C6 [×2], C2×C62, C127D4, C3×C4⋊D4, C3×C4⋊Dic3, C3×D6⋊C4 [×2], C6×D12, C6×C3⋊D4 [×2], C2×C6×C12, C3×C127D4
Quotients: C1, C2 [×7], C3, C22 [×7], S3, C6 [×7], D4 [×4], C23, D6 [×3], C2×C6 [×7], C2×D4 [×2], C4○D4, C3×S3, D12 [×2], C3⋊D4 [×2], C3×D4 [×4], C22×S3, C22×C6, C4⋊D4, S3×C6 [×3], C2×D12, C4○D12, C2×C3⋊D4, C6×D4 [×2], C3×C4○D4, C3×D12 [×2], C3×C3⋊D4 [×2], S3×C2×C6, C127D4, C3×C4⋊D4, C6×D12, C3×C4○D12, C6×C3⋊D4, C3×C127D4

Smallest permutation representation of C3×C127D4
On 48 points
Generators in S48
(1 9 5)(2 10 6)(3 11 7)(4 12 8)(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 41 45)(38 42 46)(39 43 47)(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 16 27 40)(2 15 28 39)(3 14 29 38)(4 13 30 37)(5 24 31 48)(6 23 32 47)(7 22 33 46)(8 21 34 45)(9 20 35 44)(10 19 36 43)(11 18 25 42)(12 17 26 41)
(1 46)(2 45)(3 44)(4 43)(5 42)(6 41)(7 40)(8 39)(9 38)(10 37)(11 48)(12 47)(13 36)(14 35)(15 34)(16 33)(17 32)(18 31)(19 30)(20 29)(21 28)(22 27)(23 26)(24 25)

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

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

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

90 conjugacy classes

class 1 2A2B2C2D2E2F2G3A3B3C3D3E4A4B4C4D4E4F6A···6F6G···6AE6AF6AG6AH6AI12A···12AF12AG12AH12AI12AJ
order12222222333334444446···66···6666612···1212121212
size111122121211222222212121···12···2121212122···212121212

90 irreducible representations

dim111111111111222222222222222222
type++++++++++++
imageC1C2C2C2C2C2C3C6C6C6C6C6S3D4D4D6D6C4○D4C3×S3C3⋊D4C3×D4D12C3×D4S3×C6S3×C6C4○D12C3×C4○D4C3×C3⋊D4C3×D12C3×C4○D12
kernelC3×C127D4C3×C4⋊Dic3C3×D6⋊C4C6×D12C6×C3⋊D4C2×C6×C12C127D4C4⋊Dic3D6⋊C4C2×D12C2×C3⋊D4C22×C12C22×C12C3×C12C62C2×C12C22×C6C3×C6C22×C4C12C12C2×C6C2×C6C2×C4C23C6C6C4C22C2
# reps112121224242122212244444244888

Matrix representation of C3×C127D4 in GL4(𝔽13) generated by

3000
0300
0030
0003
,
4400
01000
0070
00112
,
121000
5100
00711
00116
,
1000
81200
0062
0027
G:=sub<GL(4,GF(13))| [3,0,0,0,0,3,0,0,0,0,3,0,0,0,0,3],[4,0,0,0,4,10,0,0,0,0,7,11,0,0,0,2],[12,5,0,0,10,1,0,0,0,0,7,11,0,0,11,6],[1,8,0,0,0,12,0,0,0,0,6,2,0,0,2,7] >;

C3×C127D4 in GAP, Magma, Sage, TeX

C_3\times C_{12}\rtimes_7D_4
% in TeX

G:=Group("C3xC12:7D4");
// GroupNames label

G:=SmallGroup(288,701);
// by ID

G=gap.SmallGroup(288,701);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,701,344,590,9414]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^12=c^4=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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