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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 [×2], C2 [×6], C3 [×2], C3, C4 [×8], C22, C22 [×2], C22 [×10], S3 [×4], C6 [×6], C6 [×15], C2×C4 [×16], D4 [×12], Q8 [×4], C23, C23 [×2], C32, Dic3 [×4], Dic3 [×12], C12 [×4], D6 [×4], D6 [×4], C2×C6 [×6], C2×C6 [×19], C22×C4 [×3], C2×D4 [×3], C2×Q8, C4○D4 [×8], C3×S3 [×4], C3×C6, C3×C6 [×2], C3×C6 [×2], Dic6 [×8], C4×S3 [×8], C2×Dic3 [×2], C2×Dic3 [×26], C3⋊D4 [×8], C3⋊D4 [×8], C2×C12 [×2], C3×D4 [×8], C22×S3 [×2], C22×C6 [×2], C22×C6 [×3], C2×C4○D4, C3×Dic3 [×4], C3⋊Dic3 [×4], S3×C6 [×4], S3×C6 [×4], C62, C62 [×2], C62 [×2], C2×Dic6 [×2], S3×C2×C4 [×2], D42S3 [×16], C22×Dic3 [×5], C2×C3⋊D4 [×2], C2×C3⋊D4 [×2], C6×D4 [×2], S3×Dic3 [×8], D6⋊S3 [×4], C322Q8 [×4], C6×Dic3 [×2], C3×C3⋊D4 [×8], C2×C3⋊Dic3 [×2], C2×C3⋊Dic3 [×4], S3×C2×C6 [×2], C2×C62, C2×D42S3 [×2], C2×S3×Dic3 [×2], D6.4D6 [×8], C2×D6⋊S3, C2×C322Q8, C6×C3⋊D4 [×2], C22×C3⋊Dic3, C2×D6.4D6
Quotients: C1, C2 [×15], C22 [×35], S3 [×2], C23 [×15], D6 [×14], C4○D4 [×2], C24, C22×S3 [×14], C2×C4○D4, S32, D42S3 [×4], S3×C23 [×2], C2×S32 [×3], C2×D42S3 [×2], D6.4D6 [×2], C22×S32, C2×D6.4D6

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

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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