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## G = C3×Q8.15D6order 288 = 25·32

### Direct product of C3 and Q8.15D6

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C3×Q8.15D6
 Chief series C1 — C3 — C6 — C3×C6 — S3×C6 — S3×C12 — C3×S3×Q8 — C3×Q8.15D6
 Lower central C3 — C6 — C3×Q8.15D6
 Upper central C1 — C6 — C6×Q8

Generators and relations for C3×Q8.15D6
G = < a,b,c,d,e | a3=b4=1, c2=d6=e2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=ebe-1=b-1, bd=db, dcd-1=ece-1=b2c, ede-1=d5 >

Subgroups: 586 in 311 conjugacy classes, 170 normal (18 characteristic)
C1, C2, C2 [×5], C3 [×2], C3, C4 [×6], C4 [×4], C22, C22 [×4], S3 [×4], C6 [×2], C6 [×9], C2×C4 [×3], C2×C4 [×12], D4 [×10], Q8 [×4], Q8 [×6], C32, Dic3 [×4], C12 [×12], C12 [×10], D6 [×4], C2×C6 [×2], C2×C6 [×5], C2×Q8, C2×Q8 [×4], C4○D4 [×10], C3×S3 [×4], C3×C6, C3×C6, Dic6 [×6], C4×S3 [×12], D12 [×6], C3⋊D4 [×4], C2×C12 [×6], C2×C12 [×15], C3×D4 [×10], C3×Q8 [×8], C3×Q8 [×10], 2- 1+4, C3×Dic3 [×4], C3×C12 [×6], S3×C6 [×4], C62, C4○D12 [×6], S3×Q8 [×4], Q83S3 [×4], C6×Q8 [×2], C6×Q8 [×5], C3×C4○D4 [×10], C3×Dic6 [×6], S3×C12 [×12], C3×D12 [×6], C3×C3⋊D4 [×4], C6×C12 [×3], Q8×C32 [×4], Q8.15D6, C3×2- 1+4, C3×C4○D12 [×6], C3×S3×Q8 [×4], C3×Q83S3 [×4], Q8×C3×C6, C3×Q8.15D6
Quotients: C1, C2 [×15], C3, C22 [×35], S3, C6 [×15], C23 [×15], D6 [×7], C2×C6 [×35], C24, C3×S3, C22×S3 [×7], C22×C6 [×15], 2- 1+4, S3×C6 [×7], S3×C23, C23×C6, S3×C2×C6 [×7], Q8.15D6, C3×2- 1+4, S3×C22×C6, C3×Q8.15D6

Smallest permutation representation of C3×Q8.15D6
On 48 points
Generators in S48
(1 5 9)(2 6 10)(3 7 11)(4 8 12)(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 45 41)(38 46 42)(39 47 43)(40 48 44)
(1 10 7 4)(2 11 8 5)(3 12 9 6)(13 22 19 16)(14 23 20 17)(15 24 21 18)(25 28 31 34)(26 29 32 35)(27 30 33 36)(37 40 43 46)(38 41 44 47)(39 42 45 48)
(1 13 7 19)(2 20 8 14)(3 15 9 21)(4 22 10 16)(5 17 11 23)(6 24 12 18)(25 37 31 43)(26 44 32 38)(27 39 33 45)(28 46 34 40)(29 41 35 47)(30 48 36 42)
(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 33 7 27)(2 26 8 32)(3 31 9 25)(4 36 10 30)(5 29 11 35)(6 34 12 28)(13 39 19 45)(14 44 20 38)(15 37 21 43)(16 42 22 48)(17 47 23 41)(18 40 24 46)

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

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

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

81 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 3A 3B 3C 3D 3E 4A ··· 4F 4G 4H 4I 4J 6A 6B 6C ··· 6M 6N ··· 6U 12A ··· 12L 12M ··· 12AD 12AE ··· 12AL order 1 2 2 2 2 2 2 3 3 3 3 3 4 ··· 4 4 4 4 4 6 6 6 ··· 6 6 ··· 6 12 ··· 12 12 ··· 12 12 ··· 12 size 1 1 2 6 6 6 6 1 1 2 2 2 2 ··· 2 6 6 6 6 1 1 2 ··· 2 6 ··· 6 2 ··· 2 4 ··· 4 6 ··· 6

81 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + - image C1 C2 C2 C2 C2 C3 C6 C6 C6 C6 S3 D6 D6 C3×S3 S3×C6 S3×C6 2- 1+4 Q8.15D6 C3×2- 1+4 C3×Q8.15D6 kernel C3×Q8.15D6 C3×C4○D12 C3×S3×Q8 C3×Q8⋊3S3 Q8×C3×C6 Q8.15D6 C4○D12 S3×Q8 Q8⋊3S3 C6×Q8 C6×Q8 C2×C12 C3×Q8 C2×Q8 C2×C4 Q8 C32 C3 C3 C1 # reps 1 6 4 4 1 2 12 8 8 2 1 3 4 2 6 8 1 2 2 4

Matrix representation of C3×Q8.15D6 in GL4(𝔽7) generated by

 2 0 0 0 0 2 0 0 0 0 2 0 0 0 0 2
,
 4 3 4 1 0 1 6 0 0 2 6 0 4 5 5 3
,
 5 1 1 1 0 2 5 2 1 3 1 1 1 4 3 6
,
 0 3 2 3 4 1 4 5 3 4 3 0 2 5 6 3
,
 4 5 0 1 0 6 6 1 4 1 6 4 4 6 5 5
G:=sub<GL(4,GF(7))| [2,0,0,0,0,2,0,0,0,0,2,0,0,0,0,2],[4,0,0,4,3,1,2,5,4,6,6,5,1,0,0,3],[5,0,1,1,1,2,3,4,1,5,1,3,1,2,1,6],[0,4,3,2,3,1,4,5,2,4,3,6,3,5,0,3],[4,0,4,4,5,6,1,6,0,6,6,5,1,1,4,5] >;

C3×Q8.15D6 in GAP, Magma, Sage, TeX

C_3\times Q_8._{15}D_6
% in TeX

G:=Group("C3xQ8.15D6");
// GroupNames label

G:=SmallGroup(288,997);
// by ID

G=gap.SmallGroup(288,997);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-2,-3,344,555,268,1571,9414]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^4=1,c^2=d^6=e^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=e*b*e^-1=b^-1,b*d=d*b,d*c*d^-1=e*c*e^-1=b^2*c,e*d*e^-1=d^5>;
// generators/relations

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