Copied to
clipboard

## G = C3×C23.23D6order 288 = 25·32

### Direct product of C3 and C23.23D6

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C3×C23.23D6
 Chief series C1 — C3 — C6 — C2×C6 — C62 — C6×Dic3 — Dic3×C2×C6 — C3×C23.23D6
 Lower central C3 — C2×C6 — C3×C23.23D6
 Upper central C1 — C2×C6 — C6×D4

Generators and relations for C3×C23.23D6
G = < a,b,c,d,e,f | a3=b2=c2=d2=e6=1, 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=ce-1 >

Subgroups: 394 in 183 conjugacy classes, 66 normal (34 characteristic)
C1, C2, C2 [×2], C2 [×3], C3 [×2], C3, C4 [×5], C22, C22 [×2], C22 [×5], C6 [×2], C6 [×4], C6 [×13], C2×C4, C2×C4 [×6], D4 [×2], C23 [×2], C32, Dic3 [×4], C12 [×8], C2×C6 [×2], C2×C6 [×4], C2×C6 [×19], C22⋊C4 [×3], C4⋊C4 [×2], C22×C4, C2×D4, C3×C6, C3×C6 [×2], C3×C6 [×3], C2×Dic3 [×4], C2×Dic3 [×2], C2×C12 [×2], C2×C12 [×7], C3×D4 [×8], C22×C6 [×4], C22×C6 [×2], C22.D4, C3×Dic3 [×4], C3×C12, C62, C62 [×2], C62 [×5], Dic3⋊C4 [×2], C6.D4, C6.D4 [×2], C3×C22⋊C4 [×3], C3×C4⋊C4 [×2], C22×Dic3, C22×C12, C6×D4 [×2], C6×D4, C6×Dic3 [×4], C6×Dic3 [×2], C6×C12, D4×C32 [×2], C2×C62 [×2], C23.23D6, C3×C22.D4, C3×Dic3⋊C4 [×2], C3×C6.D4, C3×C6.D4 [×2], Dic3×C2×C6, D4×C3×C6, C3×C23.23D6
Quotients: C1, C2 [×7], C3, C22 [×7], S3, C6 [×7], D4 [×2], C23, D6 [×3], C2×C6 [×7], C2×D4, C4○D4 [×2], C3×S3, C3⋊D4 [×2], C3×D4 [×2], C22×S3, C22×C6, C22.D4, S3×C6 [×3], D42S3 [×2], C2×C3⋊D4, C6×D4, C3×C4○D4 [×2], C3×C3⋊D4 [×2], S3×C2×C6, C23.23D6, C3×C22.D4, C3×D42S3 [×2], C6×C3⋊D4, C3×C23.23D6

Smallest permutation representation of C3×C23.23D6
On 48 points
Generators in S48
(1 2 3)(4 5 6)(7 8 9)(10 11 12)(13 17 15)(14 18 16)(19 23 21)(20 24 22)(25 27 29)(26 28 30)(31 33 35)(32 34 36)(37 39 41)(38 40 42)(43 45 47)(44 46 48)
(1 18)(2 16)(3 14)(4 17)(5 15)(6 13)(7 23)(8 21)(9 19)(10 24)(11 22)(12 20)(25 47)(26 40)(27 43)(28 42)(29 45)(30 38)(31 41)(32 44)(33 37)(34 46)(35 39)(36 48)
(1 7)(2 8)(3 9)(4 11)(5 12)(6 10)(13 24)(14 19)(15 20)(16 21)(17 22)(18 23)(25 28)(26 29)(27 30)(31 34)(32 35)(33 36)(37 48)(38 43)(39 44)(40 45)(41 46)(42 47)
(1 5)(2 6)(3 4)(7 12)(8 10)(9 11)(13 16)(14 17)(15 18)(19 22)(20 23)(21 24)(25 35)(26 36)(27 31)(28 32)(29 33)(30 34)(37 45)(38 46)(39 47)(40 48)(41 43)(42 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 36 12 29)(2 32 10 25)(3 34 11 27)(4 30 9 31)(5 26 7 33)(6 28 8 35)(13 44 21 47)(14 38 22 41)(15 48 23 45)(16 42 24 39)(17 46 19 43)(18 40 20 37)

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

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

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

72 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 3A 3B 3C 3D 3E 4A 4B 4C 4D 4E 4F 4G 6A ··· 6F 6G ··· 6S 6T ··· 6AG 12A ··· 12H 12I ··· 12P 12Q 12R 12S 12T order 1 2 2 2 2 2 2 3 3 3 3 3 4 4 4 4 4 4 4 6 ··· 6 6 ··· 6 6 ··· 6 12 ··· 12 12 ··· 12 12 12 12 12 size 1 1 1 1 2 2 4 1 1 2 2 2 4 6 6 6 6 12 12 1 ··· 1 2 ··· 2 4 ··· 4 4 ··· 4 6 ··· 6 12 12 12 12

72 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + - image C1 C2 C2 C2 C2 C3 C6 C6 C6 C6 S3 D4 D6 D6 C4○D4 C3×S3 C3⋊D4 C3×D4 S3×C6 S3×C6 C3×C4○D4 C3×C3⋊D4 D4⋊2S3 C3×D4⋊2S3 kernel C3×C23.23D6 C3×Dic3⋊C4 C3×C6.D4 Dic3×C2×C6 D4×C3×C6 C23.23D6 Dic3⋊C4 C6.D4 C22×Dic3 C6×D4 C6×D4 C62 C2×C12 C22×C6 C3×C6 C2×D4 C2×C6 C2×C6 C2×C4 C23 C6 C22 C6 C2 # reps 1 2 3 1 1 2 4 6 2 2 1 2 1 2 4 2 4 4 2 4 8 8 2 4

Matrix representation of C3×C23.23D6 in GL4(𝔽13) generated by

 9 0 0 0 0 9 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 1 0 0 0 0 1 5 0 0 0 12
,
 12 0 0 0 0 12 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 1 0 0 0 0 12 0 0 0 0 12
,
 9 0 0 0 0 10 0 0 0 0 1 0 0 0 10 12
,
 0 3 0 0 4 0 0 0 0 0 5 0 0 0 11 8
G:=sub<GL(4,GF(13))| [9,0,0,0,0,9,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,1,0,0,0,5,12],[12,0,0,0,0,12,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,12,0,0,0,0,12],[9,0,0,0,0,10,0,0,0,0,1,10,0,0,0,12],[0,4,0,0,3,0,0,0,0,0,5,11,0,0,0,8] >;

C3×C23.23D6 in GAP, Magma, Sage, TeX

C_3\times C_2^3._{23}D_6
% in TeX

G:=Group("C3xC2^3.23D6");
// GroupNames label

G:=SmallGroup(288,706);
// by ID

G=gap.SmallGroup(288,706);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,336,590,555,9414]);
// Polycyclic

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

׿
×
𝔽