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## G = C3×Dic3⋊4D4order 288 = 25·32

### Direct product of C3 and Dic3⋊4D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C3×Dic3⋊4D4
 Chief series C1 — C3 — C6 — C2×C6 — C62 — S3×C2×C6 — C6×C3⋊D4 — C3×Dic3⋊4D4
 Lower central C3 — C6 — C3×Dic3⋊4D4
 Upper central C1 — C2×C6 — C3×C22⋊C4

Generators and relations for C3×Dic34D4
G = < a,b,c,d,e | a3=b6=d4=e2=1, c2=b3, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=dbd-1=b-1, be=eb, cd=dc, ce=ec, ede=d-1 >

Subgroups: 450 in 205 conjugacy classes, 86 normal (58 characteristic)
C1, C2 [×3], C2 [×4], C3 [×2], C3, C4 [×7], C22, C22 [×2], C22 [×6], S3 [×2], C6 [×6], C6 [×11], C2×C4 [×2], C2×C4 [×7], D4 [×4], C23, C23, C32, Dic3 [×4], Dic3, C12 [×11], D6 [×2], D6 [×2], C2×C6 [×2], C2×C6 [×4], C2×C6 [×13], C42, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4 [×2], C2×D4, C3×S3 [×2], C3×C6 [×3], C3×C6 [×2], C4×S3 [×2], C2×Dic3 [×3], C2×Dic3 [×2], C3⋊D4 [×4], C2×C12 [×4], C2×C12 [×9], C3×D4 [×4], C22×S3, C22×C6 [×2], C22×C6 [×2], C4×D4, C3×Dic3 [×4], C3×Dic3, C3×C12 [×2], S3×C6 [×2], S3×C6 [×2], C62, C62 [×2], C62 [×2], C4×Dic3, Dic3⋊C4, D6⋊C4, C4×C12, C3×C22⋊C4 [×2], C3×C22⋊C4 [×2], C3×C4⋊C4, S3×C2×C4, C22×Dic3, C2×C3⋊D4, C22×C12 [×2], C6×D4, S3×C12 [×2], C6×Dic3 [×3], C6×Dic3 [×2], C3×C3⋊D4 [×4], C6×C12 [×2], S3×C2×C6, C2×C62, Dic34D4, D4×C12, Dic3×C12, C3×Dic3⋊C4, C3×D6⋊C4, C32×C22⋊C4, S3×C2×C12, Dic3×C2×C6, C6×C3⋊D4, C3×Dic34D4
Quotients: C1, C2 [×7], C3, C4 [×4], C22 [×7], S3, C6 [×7], C2×C4 [×6], D4 [×2], C23, C12 [×4], D6 [×3], C2×C6 [×7], C22×C4, C2×D4, C4○D4, C3×S3, C4×S3 [×2], C2×C12 [×6], C3×D4 [×2], C22×S3, C22×C6, C4×D4, S3×C6 [×3], S3×C2×C4, S3×D4, D42S3, C22×C12, C6×D4, C3×C4○D4, S3×C12 [×2], S3×C2×C6, Dic34D4, D4×C12, S3×C2×C12, C3×S3×D4, C3×D42S3, C3×Dic34D4

Smallest permutation representation of C3×Dic34D4
On 48 points
Generators in S48
(1 5 3)(2 6 4)(7 9 11)(8 10 12)(13 17 15)(14 18 16)(19 23 21)(20 24 22)(25 29 27)(26 30 28)(31 33 35)(32 34 36)(37 39 41)(38 40 42)(43 45 47)(44 46 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 47 4 44)(2 46 5 43)(3 45 6 48)(7 17 10 14)(8 16 11 13)(9 15 12 18)(19 35 22 32)(20 34 23 31)(21 33 24 36)(25 39 28 42)(26 38 29 41)(27 37 30 40)
(1 32 17 40)(2 31 18 39)(3 36 13 38)(4 35 14 37)(5 34 15 42)(6 33 16 41)(7 30 44 22)(8 29 45 21)(9 28 46 20)(10 27 47 19)(11 26 48 24)(12 25 43 23)
(19 27)(20 28)(21 29)(22 30)(23 25)(24 26)(31 39)(32 40)(33 41)(34 42)(35 37)(36 38)

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

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

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

90 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 3C 3D 3E 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 6A ··· 6F 6G ··· 6S 6T ··· 6Y 6Z 6AA 6AB 6AC 12A ··· 12H 12I ··· 12P 12Q ··· 12AB 12AC ··· 12AJ order 1 2 2 2 2 2 2 2 3 3 3 3 3 4 4 4 4 4 4 4 4 4 4 4 4 6 ··· 6 6 ··· 6 6 ··· 6 6 6 6 6 12 ··· 12 12 ··· 12 12 ··· 12 12 ··· 12 size 1 1 1 1 2 2 6 6 1 1 2 2 2 2 2 2 2 3 3 3 3 6 6 6 6 1 ··· 1 2 ··· 2 4 ··· 4 6 6 6 6 2 ··· 2 3 ··· 3 4 ··· 4 6 ··· 6

90 irreducible representations

 dim 1 1 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 4 4 4 4 type + + + + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C2 C2 C3 C4 C6 C6 C6 C6 C6 C6 C6 C12 S3 D4 D6 D6 C4○D4 C3×S3 C3×D4 C4×S3 S3×C6 S3×C6 C3×C4○D4 S3×C12 S3×D4 D4⋊2S3 C3×S3×D4 C3×D4⋊2S3 kernel C3×Dic3⋊4D4 Dic3×C12 C3×Dic3⋊C4 C3×D6⋊C4 C32×C22⋊C4 S3×C2×C12 Dic3×C2×C6 C6×C3⋊D4 Dic3⋊4D4 C3×C3⋊D4 C4×Dic3 Dic3⋊C4 D6⋊C4 C3×C22⋊C4 S3×C2×C4 C22×Dic3 C2×C3⋊D4 C3⋊D4 C3×C22⋊C4 C3×Dic3 C2×C12 C22×C6 C3×C6 C22⋊C4 Dic3 C2×C6 C2×C4 C23 C6 C22 C6 C6 C2 C2 # reps 1 1 1 1 1 1 1 1 2 8 2 2 2 2 2 2 2 16 1 2 2 1 2 2 4 4 4 2 4 8 1 1 2 2

Matrix representation of C3×Dic34D4 in GL4(𝔽13) generated by

 9 0 0 0 0 9 0 0 0 0 1 0 0 0 0 1
,
 4 0 0 0 10 10 0 0 0 0 1 0 0 0 0 1
,
 5 3 0 0 0 8 0 0 0 0 1 0 0 0 0 1
,
 12 2 0 0 0 1 0 0 0 0 5 2 0 0 0 8
,
 1 0 0 0 0 1 0 0 0 0 1 0 0 0 8 12
G:=sub<GL(4,GF(13))| [9,0,0,0,0,9,0,0,0,0,1,0,0,0,0,1],[4,10,0,0,0,10,0,0,0,0,1,0,0,0,0,1],[5,0,0,0,3,8,0,0,0,0,1,0,0,0,0,1],[12,0,0,0,2,1,0,0,0,0,5,0,0,0,2,8],[1,0,0,0,0,1,0,0,0,0,1,8,0,0,0,12] >;

C3×Dic34D4 in GAP, Magma, Sage, TeX

C_3\times {\rm Dic}_3\rtimes_4D_4
% in TeX

G:=Group("C3xDic3:4D4");
// GroupNames label

G:=SmallGroup(288,652);
// by ID

G=gap.SmallGroup(288,652);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,701,555,142,9414]);
// Polycyclic

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

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