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G = C2×D6.4D6order 288 = 25·32

Direct product of C2 and D6.4D6

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — C2×D6.4D6
 Chief series C1 — C3 — C32 — C3×C6 — S3×C6 — S3×Dic3 — C2×S3×Dic3 — C2×D6.4D6
 Lower central C32 — C3×C6 — C2×D6.4D6
 Upper central C1 — C22 — C23

Generators and relations for C2×D6.4D6
G = < a,b,c,d,e | a2=b6=c2=1, d6=e2=b3, ab=ba, ac=ca, ad=da, ae=ea, cbc=dbd-1=ebe-1=b-1, dcd-1=bc, ece-1=b4c, ede-1=d5 >

Subgroups: 1106 in 355 conjugacy classes, 116 normal (14 characteristic)
C1, C2, C2, C2, C3, C3, C4, C22, C22, C22, S3, C6, C6, C2×C4, D4, Q8, C23, C23, C32, Dic3, Dic3, C12, D6, D6, C2×C6, C2×C6, C22×C4, C2×D4, C2×Q8, C4○D4, C3×S3, C3×C6, C3×C6, C3×C6, Dic6, C4×S3, C2×Dic3, C2×Dic3, C3⋊D4, C3⋊D4, C2×C12, C3×D4, C22×S3, C22×C6, C22×C6, C2×C4○D4, C3×Dic3, C3⋊Dic3, S3×C6, S3×C6, C62, C62, C62, C2×Dic6, S3×C2×C4, D42S3, C22×Dic3, C2×C3⋊D4, C2×C3⋊D4, C6×D4, S3×Dic3, D6⋊S3, C322Q8, C6×Dic3, C3×C3⋊D4, C2×C3⋊Dic3, C2×C3⋊Dic3, S3×C2×C6, C2×C62, C2×D42S3, C2×S3×Dic3, D6.4D6, C2×D6⋊S3, C2×C322Q8, C6×C3⋊D4, C22×C3⋊Dic3, C2×D6.4D6
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, C24, C22×S3, C2×C4○D4, S32, D42S3, S3×C23, C2×S32, C2×D42S3, D6.4D6, C22×S32, C2×D6.4D6

Smallest permutation representation of C2×D6.4D6
On 48 points
Generators in S48
(1 25)(2 26)(3 27)(4 28)(5 29)(6 30)(7 31)(8 32)(9 33)(10 34)(11 35)(12 36)(13 47)(14 48)(15 37)(16 38)(17 39)(18 40)(19 41)(20 42)(21 43)(22 44)(23 45)(24 46)
(1 11 9 7 5 3)(2 4 6 8 10 12)(13 23 21 19 17 15)(14 16 18 20 22 24)(25 35 33 31 29 27)(26 28 30 32 34 36)(37 47 45 43 41 39)(38 40 42 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 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 7 41)(2 40 8 46)(3 45 9 39)(4 38 10 44)(5 43 11 37)(6 48 12 42)(13 31 19 25)(14 36 20 30)(15 29 21 35)(16 34 22 28)(17 27 23 33)(18 32 24 26)

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

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

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

48 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 3A 3B 3C 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 6A ··· 6F 6G ··· 6Q 6R 6S 6T 6U 12A 12B 12C 12D order 1 2 2 2 2 2 2 2 2 2 3 3 3 4 4 4 4 4 4 4 4 4 4 6 ··· 6 6 ··· 6 6 6 6 6 12 12 12 12 size 1 1 1 1 2 2 6 6 6 6 2 2 4 6 6 6 6 9 9 9 9 18 18 2 ··· 2 4 ··· 4 12 12 12 12 12 12 12 12

48 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + - + - image C1 C2 C2 C2 C2 C2 C2 S3 D6 D6 D6 D6 C4○D4 S32 D4⋊2S3 C2×S32 D6.4D6 kernel C2×D6.4D6 C2×S3×Dic3 D6.4D6 C2×D6⋊S3 C2×C32⋊2Q8 C6×C3⋊D4 C22×C3⋊Dic3 C2×C3⋊D4 C2×Dic3 C3⋊D4 C22×S3 C22×C6 C3×C6 C23 C6 C22 C2 # reps 1 2 8 1 1 2 1 2 2 8 2 2 4 1 4 3 4

Matrix representation of C2×D6.4D6 in GL6(𝔽13)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 12 0 0 0 0 0 0 12
,
 12 0 0 0 0 0 0 12 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 12 0 0 0 0 1 12
,
 0 12 0 0 0 0 12 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 12 1 0 0 0 0 0 1
,
 0 1 0 0 0 0 12 0 0 0 0 0 0 0 0 12 0 0 0 0 1 12 0 0 0 0 0 0 0 1 0 0 0 0 1 0
,
 8 0 0 0 0 0 0 8 0 0 0 0 0 0 1 12 0 0 0 0 0 12 0 0 0 0 0 0 0 1 0 0 0 0 1 0

G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,12,12],[0,12,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,1,1],[0,12,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,12,12,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[8,0,0,0,0,0,0,8,0,0,0,0,0,0,1,0,0,0,0,0,12,12,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;

C2×D6.4D6 in GAP, Magma, Sage, TeX

C_2\times D_6._4D_6
% in TeX

G:=Group("C2xD6.4D6");
// GroupNames label

G:=SmallGroup(288,971);
// by ID

G=gap.SmallGroup(288,971);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,120,675,346,1356,9414]);
// Polycyclic

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

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