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## G = C3×Q8.13D6order 288 = 25·32

### Direct product of C3 and Q8.13D6

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — C3×Q8.13D6
 Chief series C1 — C3 — C6 — C12 — C3×C12 — C3×D12 — C3×C4○D12 — C3×Q8.13D6
 Lower central C3 — C6 — C12 — C3×Q8.13D6
 Upper central C1 — C12 — C2×C12 — C3×C4○D4

Generators and relations for C3×Q8.13D6
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, cd=dc, ece-1=b-1c, ede-1=d5 >

Subgroups: 322 in 144 conjugacy classes, 58 normal (all characteristic)
C1, C2, C2 [×3], C3 [×2], C3, C4 [×2], C4 [×2], C22, C22 [×2], S3, C6 [×2], C6 [×9], C8 [×2], C2×C4, C2×C4 [×2], D4, D4 [×3], Q8, Q8, C32, Dic3, C12 [×4], C12 [×7], D6, C2×C6 [×2], C2×C6 [×6], C2×C8, D8, SD16 [×2], Q16, C4○D4, C4○D4, C3×S3, C3×C6, C3×C6 [×2], C3⋊C8 [×2], C24 [×2], Dic6, C4×S3, D12, C3⋊D4, C2×C12 [×2], C2×C12 [×6], C3×D4 [×2], C3×D4 [×7], C3×Q8 [×2], C3×Q8 [×2], C4○D8, C3×Dic3, C3×C12 [×2], C3×C12, S3×C6, C62, C62, C2×C3⋊C8, D4⋊S3, D4.S3, Q82S3, C3⋊Q16, C2×C24, C3×D8, C3×SD16 [×2], C3×Q16, C4○D12, C3×C4○D4 [×2], C3×C4○D4 [×2], C3×C3⋊C8 [×2], C3×Dic6, S3×C12, C3×D12, C3×C3⋊D4, C6×C12, C6×C12, D4×C32, D4×C32, Q8×C32, Q8.13D6, C3×C4○D8, C6×C3⋊C8, C3×D4⋊S3, C3×D4.S3, C3×Q82S3, C3×C3⋊Q16, C3×C4○D12, C32×C4○D4, C3×Q8.13D6
Quotients: C1, C2 [×7], C3, C22 [×7], S3, C6 [×7], D4 [×2], C23, D6 [×3], C2×C6 [×7], C2×D4, C3×S3, C3⋊D4 [×2], C3×D4 [×2], C22×S3, C22×C6, C4○D8, S3×C6 [×3], C2×C3⋊D4, C6×D4, C3×C3⋊D4 [×2], S3×C2×C6, Q8.13D6, C3×C4○D8, C6×C3⋊D4, C3×Q8.13D6

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

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

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

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

72 conjugacy classes

 class 1 2A 2B 2C 2D 3A 3B 3C 3D 3E 4A 4B 4C 4D 4E 6A 6B 6C ··· 6G 6H ··· 6R 6S 6T 8A 8B 8C 8D 12A 12B 12C 12D 12E ··· 12L 12M ··· 12W 12X 12Y 24A ··· 24H order 1 2 2 2 2 3 3 3 3 3 4 4 4 4 4 6 6 6 ··· 6 6 ··· 6 6 6 8 8 8 8 12 12 12 12 12 ··· 12 12 ··· 12 12 12 24 ··· 24 size 1 1 2 4 12 1 1 2 2 2 1 1 2 4 12 1 1 2 ··· 2 4 ··· 4 12 12 6 6 6 6 1 1 1 1 2 ··· 2 4 ··· 4 12 12 6 ··· 6

72 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C3 C6 C6 C6 C6 C6 C6 C6 S3 D4 D4 D6 D6 D6 C3×S3 C3⋊D4 C3×D4 C3⋊D4 C3×D4 C4○D8 S3×C6 S3×C6 S3×C6 C3×C3⋊D4 C3×C3⋊D4 C3×C4○D8 Q8.13D6 C3×Q8.13D6 kernel C3×Q8.13D6 C6×C3⋊C8 C3×D4⋊S3 C3×D4.S3 C3×Q8⋊2S3 C3×C3⋊Q16 C3×C4○D12 C32×C4○D4 Q8.13D6 C2×C3⋊C8 D4⋊S3 D4.S3 Q8⋊2S3 C3⋊Q16 C4○D12 C3×C4○D4 C3×C4○D4 C3×C12 C62 C2×C12 C3×D4 C3×Q8 C4○D4 C12 C12 C2×C6 C2×C6 C32 C2×C4 D4 Q8 C4 C22 C3 C3 C1 # reps 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 1 1 1 1 1 1 2 2 2 2 2 4 2 2 2 4 4 8 2 4

Matrix representation of C3×Q8.13D6 in GL4(𝔽73) generated by

 64 0 0 0 0 64 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 1 0 0 0 0 27 0 0 0 0 46
,
 1 0 0 0 0 1 0 0 0 0 0 1 0 0 72 0
,
 8 0 0 0 0 64 0 0 0 0 27 0 0 0 0 27
,
 0 64 0 0 8 0 0 0 0 0 0 63 0 0 22 0
G:=sub<GL(4,GF(73))| [64,0,0,0,0,64,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,27,0,0,0,0,46],[1,0,0,0,0,1,0,0,0,0,0,72,0,0,1,0],[8,0,0,0,0,64,0,0,0,0,27,0,0,0,0,27],[0,8,0,0,64,0,0,0,0,0,0,22,0,0,63,0] >;

C3×Q8.13D6 in GAP, Magma, Sage, TeX

C_3\times Q_8._{13}D_6
% in TeX

G:=Group("C3xQ8.13D6");
// GroupNames label

G:=SmallGroup(288,721);
// by ID

G=gap.SmallGroup(288,721);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,344,590,2524,648,102,9414]);
// 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,c*d=d*c,e*c*e^-1=b^-1*c,e*d*e^-1=d^5>;
// generators/relations

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