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## G = C3×C23.26D6order 288 = 25·32

### Direct product of C3 and C23.26D6

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C3×C23.26D6
 Chief series C1 — C3 — C6 — C2×C6 — C62 — C6×Dic3 — Dic3×C12 — C3×C23.26D6
 Lower central C3 — C6 — C3×C23.26D6
 Upper central C1 — C2×C12 — C22×C12

Generators and relations for C3×C23.26D6
G = < a,b,c,d,e,f | a3=b2=c2=d2=1, e6=d, f2=dc=cd, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, fbf-1=bd=db, be=eb, ce=ec, cf=fc, de=ed, df=fd, fef-1=e5 >

Subgroups: 314 in 179 conjugacy classes, 98 normal (30 characteristic)
C1, C2, C2 [×2], C2 [×2], C3 [×2], C3, C4 [×4], C4 [×4], C22, C22 [×2], C22 [×2], C6 [×2], C6 [×4], C6 [×11], C2×C4 [×2], C2×C4 [×4], C2×C4 [×4], C23, C32, Dic3 [×4], C12 [×8], C12 [×8], C2×C6 [×2], C2×C6 [×4], C2×C6 [×11], C42 [×2], C22⋊C4 [×2], C4⋊C4 [×2], C22×C4, C3×C6, C3×C6 [×2], C3×C6 [×2], C2×Dic3 [×4], C2×C12 [×4], C2×C12 [×8], C2×C12 [×10], C22×C6 [×2], C22×C6, C42⋊C2, C3×Dic3 [×4], C3×C12 [×4], C62, C62 [×2], C62 [×2], C4×Dic3 [×2], C4⋊Dic3 [×2], C6.D4 [×2], C4×C12 [×2], C3×C22⋊C4 [×2], C3×C4⋊C4 [×2], C22×C12 [×2], C22×C12, C6×Dic3 [×4], C6×C12 [×2], C6×C12 [×4], C2×C62, C23.26D6, C3×C42⋊C2, Dic3×C12 [×2], C3×C4⋊Dic3 [×2], C3×C6.D4 [×2], C2×C6×C12, C3×C23.26D6
Quotients: C1, C2 [×7], C3, C4 [×4], C22 [×7], S3, C6 [×7], C2×C4 [×6], C23, Dic3 [×4], C12 [×4], D6 [×3], C2×C6 [×7], C22×C4, C4○D4 [×2], C3×S3, C2×Dic3 [×6], C2×C12 [×6], C22×S3, C22×C6, C42⋊C2, C3×Dic3 [×4], S3×C6 [×3], C4○D12 [×2], C22×Dic3, C22×C12, C3×C4○D4 [×2], C6×Dic3 [×6], S3×C2×C6, C23.26D6, C3×C42⋊C2, C3×C4○D12 [×2], Dic3×C2×C6, C3×C23.26D6

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

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

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

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

108 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3A 3B 3C 3D 3E 4A 4B 4C 4D 4E 4F 4G ··· 4N 6A ··· 6F 6G ··· 6AE 12A ··· 12H 12I ··· 12AJ 12AK ··· 12AZ order 1 2 2 2 2 2 3 3 3 3 3 4 4 4 4 4 4 4 ··· 4 6 ··· 6 6 ··· 6 12 ··· 12 12 ··· 12 12 ··· 12 size 1 1 1 1 2 2 1 1 2 2 2 1 1 1 1 2 2 6 ··· 6 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2 6 ··· 6

108 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + + - + + image C1 C2 C2 C2 C2 C3 C4 C6 C6 C6 C6 C12 S3 Dic3 D6 D6 C4○D4 C3×S3 C3×Dic3 S3×C6 S3×C6 C4○D12 C3×C4○D4 C3×C4○D12 kernel C3×C23.26D6 Dic3×C12 C3×C4⋊Dic3 C3×C6.D4 C2×C6×C12 C23.26D6 C6×C12 C4×Dic3 C4⋊Dic3 C6.D4 C22×C12 C2×C12 C22×C12 C2×C12 C2×C12 C22×C6 C3×C6 C22×C4 C2×C4 C2×C4 C23 C6 C6 C2 # reps 1 2 2 2 1 2 8 4 4 4 2 16 1 4 2 1 4 2 8 4 2 8 8 16

Matrix representation of C3×C23.26D6 in GL4(𝔽13) generated by

 3 0 0 0 0 3 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 1 0 0 0 0 1 0 0 0 11 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
,
 4 0 0 0 0 10 0 0 0 0 8 0 0 0 0 8
,
 0 10 0 0 9 0 0 0 0 0 12 12 0 0 2 1
G:=sub<GL(4,GF(13))| [3,0,0,0,0,3,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,1,11,0,0,0,12],[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],[4,0,0,0,0,10,0,0,0,0,8,0,0,0,0,8],[0,9,0,0,10,0,0,0,0,0,12,2,0,0,12,1] >;

C3×C23.26D6 in GAP, Magma, Sage, TeX

C_3\times C_2^3._{26}D_6
% in TeX

G:=Group("C3xC2^3.26D6");
// GroupNames label

G:=SmallGroup(288,697);
// by ID

G=gap.SmallGroup(288,697);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,168,344,1094,9414]);
// Polycyclic

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