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## G = C3×Q8⋊3Dic3order 288 = 25·32

### Direct product of C3 and Q8⋊3Dic3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — C3×Q8⋊3Dic3
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C6×C12 — C3×C4.Dic3 — C3×Q8⋊3Dic3
 Lower central C3 — C6 — C12 — C3×Q8⋊3Dic3
 Upper central C1 — C12 — C2×C12 — C3×C4○D4

Generators and relations for C3×Q83Dic3
G = < a,b,c,d,e | a3=b4=d6=1, c2=b2, e2=d3, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=b-1, bd=db, be=eb, dcd-1=b2c, ece-1=b-1c, ede-1=d-1 >

Subgroups: 226 in 108 conjugacy classes, 42 normal (all characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C22, C6, C6, C8, C2×C4, C2×C4, D4, D4, Q8, C32, Dic3, C12, C12, C2×C6, C2×C6, C42, M4(2), C4○D4, C3×C6, C3×C6, C3⋊C8, C24, C2×Dic3, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C3×Q8, C4≀C2, C3×Dic3, C3×C12, C3×C12, C62, C62, C4.Dic3, C4×Dic3, C4×C12, C3×M4(2), C3×C4○D4, C3×C4○D4, C3×C3⋊C8, C6×Dic3, C6×C12, C6×C12, D4×C32, D4×C32, Q8×C32, Q83Dic3, C3×C4≀C2, C3×C4.Dic3, Dic3×C12, C32×C4○D4, C3×Q83Dic3
Quotients: C1, C2, C3, C4, C22, S3, C6, C2×C4, D4, Dic3, C12, D6, C2×C6, C22⋊C4, C3×S3, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C4≀C2, C3×Dic3, S3×C6, C6.D4, C3×C22⋊C4, C6×Dic3, C3×C3⋊D4, Q83Dic3, C3×C4≀C2, C3×C6.D4, C3×Q83Dic3

Smallest permutation representation of C3×Q83Dic3
On 48 points
Generators in S48
(1 3 5)(2 4 6)(7 9 11)(8 10 12)(13 15 17)(14 16 18)(19 23 21)(20 24 22)(25 29 27)(26 30 28)(31 33 35)(32 34 36)(37 41 39)(38 42 40)(43 47 45)(44 48 46)
(1 11 36 13)(2 12 31 14)(3 7 32 15)(4 8 33 16)(5 9 34 17)(6 10 35 18)(19 28 38 47)(20 29 39 48)(21 30 40 43)(22 25 41 44)(23 26 42 45)(24 27 37 46)
(1 16 36 8)(2 9 31 17)(3 18 32 10)(4 11 33 13)(5 14 34 12)(6 7 35 15)(19 41 38 22)(20 23 39 42)(21 37 40 24)(25 47 44 28)(26 29 45 48)(27 43 46 30)
(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 28 4 25)(2 27 5 30)(3 26 6 29)(7 42 10 39)(8 41 11 38)(9 40 12 37)(13 19 16 22)(14 24 17 21)(15 23 18 20)(31 46 34 43)(32 45 35 48)(33 44 36 47)

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

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

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

72 conjugacy classes

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

72 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 2 2 2 2 2 2 4 4 type + + + + + + + + - - image C1 C2 C2 C2 C3 C4 C4 C6 C6 C6 C12 C12 S3 D4 D4 D6 Dic3 Dic3 C3×S3 C3⋊D4 C3×D4 C3⋊D4 C3×D4 C4≀C2 S3×C6 C3×Dic3 C3×Dic3 C3×C3⋊D4 C3×C3⋊D4 C3×C4≀C2 Q8⋊3Dic3 C3×Q8⋊3Dic3 kernel C3×Q8⋊3Dic3 C3×C4.Dic3 Dic3×C12 C32×C4○D4 Q8⋊3Dic3 D4×C32 Q8×C32 C4.Dic3 C4×Dic3 C3×C4○D4 C3×D4 C3×Q8 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 2 2 2 2 2 2 4 4 1 1 1 1 1 1 2 2 2 2 2 4 2 2 2 4 4 8 2 4

Matrix representation of C3×Q83Dic3 in GL4(𝔽73) generated by

 1 0 0 0 0 1 0 0 0 0 64 0 0 0 0 64
,
 0 1 0 0 72 0 0 0 0 0 72 0 0 0 0 72
,
 46 0 0 0 0 27 0 0 0 0 1 0 0 0 0 72
,
 0 46 0 0 27 0 0 0 0 0 65 0 0 0 0 9
,
 14 59 0 0 14 14 0 0 0 0 0 1 0 0 72 0
G:=sub<GL(4,GF(73))| [1,0,0,0,0,1,0,0,0,0,64,0,0,0,0,64],[0,72,0,0,1,0,0,0,0,0,72,0,0,0,0,72],[46,0,0,0,0,27,0,0,0,0,1,0,0,0,0,72],[0,27,0,0,46,0,0,0,0,0,65,0,0,0,0,9],[14,14,0,0,59,14,0,0,0,0,0,72,0,0,1,0] >;

C3×Q83Dic3 in GAP, Magma, Sage, TeX

C_3\times Q_8\rtimes_3{\rm Dic}_3
% in TeX

G:=Group("C3xQ8:3Dic3");
// GroupNames label

G:=SmallGroup(288,271);
// by ID

G=gap.SmallGroup(288,271);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-3,84,365,136,2524,1271,102,9414]);
// Polycyclic

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

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