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G = C3×C23.16D6order 288 = 25·32

Direct product of C3 and C23.16D6

direct product, metabelian, supersoluble, monomial

Aliases: C3×C23.16D6, C62.168C23, Dic3⋊C47C6, (C4×Dic3)⋊9C6, (C6×Dic3)⋊8C4, (C2×Dic3)⋊3C12, (C2×C12).263D6, C62.58(C2×C4), C6.5(C22×C12), C22.6(S3×C12), C23.21(S3×C6), (Dic3×C12)⋊27C2, Dic3.8(C2×C12), C6.D4.1C6, (C22×C6).102D6, (C6×C12).241C22, (C2×C62).44C22, C6.109(D42S3), (C22×Dic3).3C6, C3213(C42⋊C2), (C6×Dic3).166C22, C2.7(S3×C2×C12), C6.104(S3×C2×C4), (C2×C6).61(C4×S3), (C2×C4).24(S3×C6), (C2×C6).4(C2×C12), C6.19(C3×C4○D4), C22.12(S3×C2×C6), (C2×C12).50(C2×C6), C32(C3×C42⋊C2), C2.1(C3×D42S3), C22⋊C4.3(C3×S3), (C3×C22⋊C4).4C6, (Dic3×C2×C6).10C2, (C3×Dic3⋊C4)⋊26C2, (C3×C22⋊C4).17S3, (C2×C6).23(C22×C6), (C3×C6).76(C22×C4), (C22×C6).18(C2×C6), (C3×C6).125(C4○D4), (C2×C6).301(C22×S3), (C3×Dic3).29(C2×C4), (C2×Dic3).18(C2×C6), (C3×C6.D4).1C2, (C32×C22⋊C4).5C2, SmallGroup(288,648)

Series: Derived Chief Lower central Upper central

C1C6 — C3×C23.16D6
C1C3C6C2×C6C62C6×Dic3Dic3×C2×C6 — C3×C23.16D6
C3C6 — C3×C23.16D6
C1C2×C6C3×C22⋊C4

Generators and relations for C3×C23.16D6
 G = < a,b,c,d,e,f | a3=b2=c2=d2=1, e6=c, f2=dc=cd, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, ebe-1=fbf-1=bd=db, ce=ec, cf=fc, de=ed, df=fd, fef-1=e5 >

Subgroups: 322 in 169 conjugacy classes, 82 normal (30 characteristic)
C1, C2, C2 [×2], C2 [×2], C3 [×2], C3, C4 [×8], C22, C22 [×2], C22 [×2], C6 [×2], C6 [×4], C6 [×9], C2×C4 [×2], C2×C4 [×8], C23, C32, Dic3 [×4], Dic3 [×2], C12 [×12], C2×C6 [×2], C2×C6 [×4], C2×C6 [×9], C42 [×2], C22⋊C4, C22⋊C4, C4⋊C4 [×2], C22×C4, C3×C6, C3×C6 [×2], C3×C6 [×2], C2×Dic3 [×2], C2×Dic3 [×6], C2×C12 [×4], C2×C12 [×10], C22×C6 [×2], C22×C6, C42⋊C2, C3×Dic3 [×4], C3×Dic3 [×2], C3×C12 [×2], C62, C62 [×2], C62 [×2], C4×Dic3 [×2], Dic3⋊C4 [×2], C6.D4, C4×C12 [×2], C3×C22⋊C4 [×2], C3×C22⋊C4 [×2], C3×C4⋊C4 [×2], C22×Dic3, C22×C12, C6×Dic3 [×2], C6×Dic3 [×6], C6×C12 [×2], C2×C62, C23.16D6, C3×C42⋊C2, Dic3×C12 [×2], C3×Dic3⋊C4 [×2], C3×C6.D4, C32×C22⋊C4, Dic3×C2×C6, C3×C23.16D6
Quotients: C1, C2 [×7], C3, C4 [×4], C22 [×7], S3, C6 [×7], C2×C4 [×6], C23, C12 [×4], D6 [×3], C2×C6 [×7], C22×C4, C4○D4 [×2], C3×S3, C4×S3 [×2], C2×C12 [×6], C22×S3, C22×C6, C42⋊C2, S3×C6 [×3], S3×C2×C4, D42S3 [×2], C22×C12, C3×C4○D4 [×2], S3×C12 [×2], S3×C2×C6, C23.16D6, C3×C42⋊C2, S3×C2×C12, C3×D42S3 [×2], C3×C23.16D6

Smallest permutation representation of C3×C23.16D6
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 29 33)(26 30 34)(27 31 35)(28 32 36)(37 45 41)(38 46 42)(39 47 43)(40 48 44)
(1 7)(2 48)(3 9)(4 38)(5 11)(6 40)(8 42)(10 44)(12 46)(13 19)(14 32)(15 21)(16 34)(17 23)(18 36)(20 26)(22 28)(24 30)(25 31)(27 33)(29 35)(37 43)(39 45)(41 47)
(1 7)(2 8)(3 9)(4 10)(5 11)(6 12)(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)(25 31)(26 32)(27 33)(28 34)(29 35)(30 36)(37 43)(38 44)(39 45)(40 46)(41 47)(42 48)
(1 41)(2 42)(3 43)(4 44)(5 45)(6 46)(7 47)(8 48)(9 37)(10 38)(11 39)(12 40)(13 25)(14 26)(15 27)(16 28)(17 29)(18 30)(19 31)(20 32)(21 33)(22 34)(23 35)(24 36)
(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 30 47 24)(2 35 48 17)(3 28 37 22)(4 33 38 15)(5 26 39 20)(6 31 40 13)(7 36 41 18)(8 29 42 23)(9 34 43 16)(10 27 44 21)(11 32 45 14)(12 25 46 19)

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

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

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

90 conjugacy classes

class 1 2A2B2C2D2E3A3B3C3D3E4A4B4C4D4E4F4G4H4I···4N6A···6F6G···6S6T···6Y12A···12H12I···12P12Q···12AB12AC···12AN
order12222233333444444444···46···66···66···612···1212···1212···1212···12
size11112211222222233336···61···12···24···42···23···34···46···6

90 irreducible representations

dim11111111111111222222222244
type+++++++++-
imageC1C2C2C2C2C2C3C4C6C6C6C6C6C12S3D6D6C4○D4C3×S3C4×S3S3×C6S3×C6C3×C4○D4S3×C12D42S3C3×D42S3
kernelC3×C23.16D6Dic3×C12C3×Dic3⋊C4C3×C6.D4C32×C22⋊C4Dic3×C2×C6C23.16D6C6×Dic3C4×Dic3Dic3⋊C4C6.D4C3×C22⋊C4C22×Dic3C2×Dic3C3×C22⋊C4C2×C12C22×C6C3×C6C22⋊C4C2×C6C2×C4C23C6C22C6C2
# reps122111284422216121424428824

Matrix representation of C3×C23.16D6 in GL4(𝔽13) generated by

9000
0900
0010
0001
,
12000
01200
00120
0021
,
12000
01200
00120
00012
,
1000
0100
00120
00012
,
2000
0600
0055
0008
,
0600
2000
0011
00012
G:=sub<GL(4,GF(13))| [9,0,0,0,0,9,0,0,0,0,1,0,0,0,0,1],[12,0,0,0,0,12,0,0,0,0,12,2,0,0,0,1],[12,0,0,0,0,12,0,0,0,0,12,0,0,0,0,12],[1,0,0,0,0,1,0,0,0,0,12,0,0,0,0,12],[2,0,0,0,0,6,0,0,0,0,5,0,0,0,5,8],[0,2,0,0,6,0,0,0,0,0,1,0,0,0,1,12] >;

C3×C23.16D6 in GAP, Magma, Sage, TeX

C_3\times C_2^3._{16}D_6
% in TeX

G:=Group("C3xC2^3.16D6");
// GroupNames label

G:=SmallGroup(288,648);
// by ID

G=gap.SmallGroup(288,648);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,336,555,142,9414]);
// Polycyclic

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

׿
×
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