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## G = C2×C4○D12order 96 = 25·3

### Direct product of C2 and C4○D12

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C2×C4○D12
 Chief series C1 — C3 — C6 — D6 — C22×S3 — S3×C2×C4 — C2×C4○D12
 Lower central C3 — C6 — C2×C4○D12
 Upper central C1 — C2×C4 — C22×C4

Generators and relations for C2×C4○D12
G = < a,b,c,d | a2=b4=d2=1, c6=b2, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=b2c5 >

Subgroups: 322 in 164 conjugacy classes, 89 normal (17 characteristic)
C1, C2, C2 [×2], C2 [×6], C3, C4 [×4], C4 [×4], C22, C22 [×2], C22 [×10], S3 [×4], C6, C6 [×2], C6 [×2], C2×C4 [×2], C2×C4 [×4], C2×C4 [×10], D4 [×12], Q8 [×4], C23, C23 [×2], Dic3 [×4], C12 [×4], D6 [×4], D6 [×4], C2×C6, C2×C6 [×2], C2×C6 [×2], C22×C4, C22×C4 [×2], C2×D4 [×3], C2×Q8, C4○D4 [×8], Dic6 [×4], C4×S3 [×8], D12 [×4], C2×Dic3 [×2], C3⋊D4 [×8], C2×C12 [×2], C2×C12 [×4], C22×S3 [×2], C22×C6, C2×C4○D4, C2×Dic6, S3×C2×C4 [×2], C2×D12, C4○D12 [×8], C2×C3⋊D4 [×2], C22×C12, C2×C4○D12
Quotients: C1, C2 [×15], C22 [×35], S3, C23 [×15], D6 [×7], C4○D4 [×2], C24, C22×S3 [×7], C2×C4○D4, C4○D12 [×2], S3×C23, C2×C4○D12

Smallest permutation representation of C2×C4○D12
On 48 points
Generators in S48
(1 47)(2 48)(3 37)(4 38)(5 39)(6 40)(7 41)(8 42)(9 43)(10 44)(11 45)(12 46)(13 31)(14 32)(15 33)(16 34)(17 35)(18 36)(19 25)(20 26)(21 27)(22 28)(23 29)(24 30)
(1 25 7 31)(2 26 8 32)(3 27 9 33)(4 28 10 34)(5 29 11 35)(6 30 12 36)(13 47 19 41)(14 48 20 42)(15 37 21 43)(16 38 22 44)(17 39 23 45)(18 40 24 46)
(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 46)(2 45)(3 44)(4 43)(5 42)(6 41)(7 40)(8 39)(9 38)(10 37)(11 48)(12 47)(13 30)(14 29)(15 28)(16 27)(17 26)(18 25)(19 36)(20 35)(21 34)(22 33)(23 32)(24 31)

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

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

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

36 conjugacy classes

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

36 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 type + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 S3 D6 D6 C4○D4 C4○D12 kernel C2×C4○D12 C2×Dic6 S3×C2×C4 C2×D12 C4○D12 C2×C3⋊D4 C22×C12 C22×C4 C2×C4 C23 C6 C2 # reps 1 1 2 1 8 2 1 1 6 1 4 8

Matrix representation of C2×C4○D12 in GL3(𝔽13) generated by

 12 0 0 0 1 0 0 0 1
,
 1 0 0 0 8 0 0 0 8
,
 1 0 0 0 7 10 0 3 10
,
 1 0 0 0 7 10 0 3 6
G:=sub<GL(3,GF(13))| [12,0,0,0,1,0,0,0,1],[1,0,0,0,8,0,0,0,8],[1,0,0,0,7,3,0,10,10],[1,0,0,0,7,3,0,10,6] >;

C2×C4○D12 in GAP, Magma, Sage, TeX

C_2\times C_4\circ D_{12}
% in TeX

G:=Group("C2xC4oD12");
// GroupNames label

G:=SmallGroup(96,208);
// by ID

G=gap.SmallGroup(96,208);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-3,86,579,2309]);
// Polycyclic

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

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