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## G = C2×D4⋊6D6order 192 = 26·3

### Direct product of C2 and D4⋊6D6

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C2×D4⋊6D6
 Chief series C1 — C3 — C6 — D6 — C22×S3 — S3×C23 — C2×S3×D4 — C2×D4⋊6D6
 Lower central C3 — C6 — C2×D4⋊6D6
 Upper central C1 — C22 — C22×D4

Generators and relations for C2×D46D6
G = < a,b,c,d,e | a2=b4=c2=d6=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=dbd-1=b-1, be=eb, dcd-1=ece=b2c, ede=d-1 >

Subgroups: 2056 in 898 conjugacy classes, 447 normal (13 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C22, S3, C6, C6, C6, C2×C4, C2×C4, D4, D4, Q8, C23, C23, C23, Dic3, C12, D6, D6, C2×C6, C2×C6, C2×C6, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C24, C24, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C22×S3, C22×S3, C22×C6, C22×C6, C22×C6, C22×D4, C22×D4, C2×C4○D4, 2+ 1+4, C2×Dic6, S3×C2×C4, C2×D12, C4○D12, S3×D4, D42S3, C22×Dic3, C2×C3⋊D4, C22×C12, C6×D4, S3×C23, C23×C6, C2×2+ 1+4, C2×C4○D12, C2×S3×D4, C2×D42S3, D46D6, C22×C3⋊D4, D4×C2×C6, C2×D46D6
Quotients: C1, C2, C22, S3, C23, D6, C24, C22×S3, 2+ 1+4, C25, S3×C23, C2×2+ 1+4, D46D6, S3×C24, C2×D46D6

Smallest permutation representation of C2×D46D6
On 48 points
Generators in S48
(1 34)(2 35)(3 36)(4 31)(5 32)(6 33)(7 39)(8 40)(9 41)(10 42)(11 37)(12 38)(13 27)(14 28)(15 29)(16 30)(17 25)(18 26)(19 43)(20 44)(21 45)(22 46)(23 47)(24 48)
(1 31 23 44)(2 45 24 32)(3 33 19 46)(4 47 20 34)(5 35 21 48)(6 43 22 36)(7 18 29 42)(8 37 30 13)(9 14 25 38)(10 39 26 15)(11 16 27 40)(12 41 28 17)
(1 38)(2 15)(3 40)(4 17)(5 42)(6 13)(7 48)(8 36)(9 44)(10 32)(11 46)(12 34)(14 23)(16 19)(18 21)(20 41)(22 37)(24 39)(25 31)(26 45)(27 33)(28 47)(29 35)(30 43)
(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 36)(2 35)(3 34)(4 33)(5 32)(6 31)(7 15)(8 14)(9 13)(10 18)(11 17)(12 16)(19 47)(20 46)(21 45)(22 44)(23 43)(24 48)(25 37)(26 42)(27 41)(28 40)(29 39)(30 38)

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

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

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

54 conjugacy classes

 class 1 2A 2B 2C 2D ··· 2M 2N ··· 2U 3 4A 4B 4C 4D 4E ··· 4L 6A ··· 6G 6H ··· 6O 12A 12B 12C 12D order 1 2 2 2 2 ··· 2 2 ··· 2 3 4 4 4 4 4 ··· 4 6 ··· 6 6 ··· 6 12 12 12 12 size 1 1 1 1 2 ··· 2 6 ··· 6 2 2 2 2 2 6 ··· 6 2 ··· 2 4 ··· 4 4 4 4 4

54 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 4 4 type + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 S3 D6 D6 D6 2+ 1+4 D4⋊6D6 kernel C2×D4⋊6D6 C2×C4○D12 C2×S3×D4 C2×D4⋊2S3 D4⋊6D6 C22×C3⋊D4 D4×C2×C6 C22×D4 C22×C4 C2×D4 C24 C6 C2 # reps 1 2 4 4 16 4 1 1 1 12 2 2 4

Matrix representation of C2×D46D6 in GL6(𝔽13)

 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 1 0 0 0 0 0 0 1
,
 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 2 0 0 0 0 12 0 2 0 0 12 0 1 0 0 0 0 12 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 2 4 0 0 0 0 9 11 0 0 0 0 2 4 11 9 0 0 9 11 4 2
,
 1 12 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 11 0 0 12 12 2 2 0 0 0 0 0 12 0 0 0 0 1 1
,
 12 0 0 0 0 0 12 1 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0

G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,12,0,0,0,0,12,0,12,0,0,2,0,1,0,0,0,0,2,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,2,9,2,9,0,0,4,11,4,11,0,0,0,0,11,4,0,0,0,0,9,2],[1,1,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,1,12,0,0,0,0,0,2,0,1,0,0,11,2,12,1],[12,12,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;

C2×D46D6 in GAP, Magma, Sage, TeX

C_2\times D_4\rtimes_6D_6
% in TeX

G:=Group("C2xD4:6D6");
// GroupNames label

G:=SmallGroup(192,1516);
// by ID

G=gap.SmallGroup(192,1516);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,297,1684,6278]);
// Polycyclic

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

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