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## G = C2×D6⋊C4order 96 = 25·3

### Direct product of C2 and D6⋊C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C2×D6⋊C4
 Chief series C1 — C3 — C6 — C2×C6 — C22×S3 — S3×C23 — C2×D6⋊C4
 Lower central C3 — C6 — C2×D6⋊C4
 Upper central C1 — C23 — C22×C4

Generators and relations for C2×D6⋊C4
G = < a,b,c,d | a2=b6=c2=d4=1, ab=ba, ac=ca, ad=da, cbc=b-1, bd=db, dcd-1=b3c >

Subgroups: 322 in 132 conjugacy classes, 57 normal (17 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C3, C4 [×4], C22, C22 [×6], C22 [×16], S3 [×4], C6 [×3], C6 [×4], C2×C4 [×2], C2×C4 [×6], C23, C23 [×10], Dic3 [×2], C12 [×2], D6 [×4], D6 [×12], C2×C6, C2×C6 [×6], C22⋊C4 [×4], C22×C4, C22×C4, C24, C2×Dic3 [×2], C2×Dic3 [×2], C2×C12 [×2], C2×C12 [×2], C22×S3 [×6], C22×S3 [×4], C22×C6, C2×C22⋊C4, D6⋊C4 [×4], C22×Dic3, C22×C12, S3×C23, C2×D6⋊C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], D4 [×4], C23, D6 [×3], C22⋊C4 [×4], C22×C4, C2×D4 [×2], C4×S3 [×2], D12 [×2], C3⋊D4 [×2], C22×S3, C2×C22⋊C4, D6⋊C4 [×4], S3×C2×C4, C2×D12, C2×C3⋊D4, C2×D6⋊C4

Smallest permutation representation of C2×D6⋊C4
On 48 points
Generators in S48
(1 32)(2 33)(3 34)(4 35)(5 36)(6 31)(7 28)(8 29)(9 30)(10 25)(11 26)(12 27)(13 46)(14 47)(15 48)(16 43)(17 44)(18 45)(19 40)(20 41)(21 42)(22 37)(23 38)(24 39)
(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)(2 27)(3 26)(4 25)(5 30)(6 29)(7 32)(8 31)(9 36)(10 35)(11 34)(12 33)(13 41)(14 40)(15 39)(16 38)(17 37)(18 42)(19 47)(20 46)(21 45)(22 44)(23 43)(24 48)
(1 23 11 17)(2 24 12 18)(3 19 7 13)(4 20 8 14)(5 21 9 15)(6 22 10 16)(25 43 31 37)(26 44 32 38)(27 45 33 39)(28 46 34 40)(29 47 35 41)(30 48 36 42)

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

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

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

36 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 3 4A 4B 4C 4D 4E 4F 4G 4H 6A ··· 6G 12A ··· 12H order 1 2 ··· 2 2 2 2 2 3 4 4 4 4 4 4 4 4 6 ··· 6 12 ··· 12 size 1 1 ··· 1 6 6 6 6 2 2 2 2 2 6 6 6 6 2 ··· 2 2 ··· 2

36 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 type + + + + + + + + + + image C1 C2 C2 C2 C2 C4 S3 D4 D6 D6 C4×S3 D12 C3⋊D4 kernel C2×D6⋊C4 D6⋊C4 C22×Dic3 C22×C12 S3×C23 C22×S3 C22×C4 C2×C6 C2×C4 C23 C22 C22 C22 # reps 1 4 1 1 1 8 1 4 2 1 4 4 4

Matrix representation of C2×D6⋊C4 in GL6(𝔽13)

 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12
,
 0 1 0 0 0 0 12 12 0 0 0 0 0 0 0 1 0 0 0 0 12 12 0 0 0 0 0 0 0 12 0 0 0 0 1 1
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 12 0 0 0 0 12 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0
,
 5 0 0 0 0 0 0 5 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 10 7 0 0 0 0 6 3

G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[0,12,0,0,0,0,1,12,0,0,0,0,0,0,0,12,0,0,0,0,1,12,0,0,0,0,0,0,0,1,0,0,0,0,12,1],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[5,0,0,0,0,0,0,5,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,10,6,0,0,0,0,7,3] >;

C2×D6⋊C4 in GAP, Magma, Sage, TeX

C_2\times D_6\rtimes C_4
% in TeX

G:=Group("C2xD6:C4");
// GroupNames label

G:=SmallGroup(96,134);
// by ID

G=gap.SmallGroup(96,134);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-3,362,50,2309]);
// Polycyclic

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

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