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## G = C2×D4○D12order 192 = 26·3

### Direct product of C2 and D4○D12

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C2×D4○D12
 Chief series C1 — C3 — C6 — D6 — C22×S3 — S3×C23 — C2×S3×D4 — C2×D4○D12
 Lower central C3 — C6 — C2×D4○D12
 Upper central C1 — C22 — C2×C4○D4

Generators and relations for C2×D4○D12
G = < a,b,c,d,e | a2=b4=c2=e2=1, d6=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=b2d5 >

Subgroups: 2184 in 898 conjugacy classes, 447 normal (14 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C22, S3, C6, C6, C6, C2×C4, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C23, Dic3, C12, D6, D6, C2×C6, C2×C6, C2×C6, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C4○D4, C4○D4, C24, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C3×D4, C3×Q8, C22×S3, C22×S3, C22×C6, C22×D4, C2×C4○D4, C2×C4○D4, 2+ 1+4, C2×Dic6, S3×C2×C4, C2×D12, C4○D12, S3×D4, Q83S3, C2×C3⋊D4, C22×C12, C6×D4, C6×Q8, C3×C4○D4, S3×C23, C2×2+ 1+4, C22×D12, C2×C4○D12, C2×S3×D4, C2×Q83S3, D4○D12, C6×C4○D4, C2×D4○D12
Quotients: C1, C2, C22, S3, C23, D6, C24, C22×S3, 2+ 1+4, C25, S3×C23, C2×2+ 1+4, D4○D12, S3×C24, C2×D4○D12

Smallest permutation representation of C2×D4○D12
On 48 points
Generators in S48
(1 26)(2 27)(3 28)(4 29)(5 30)(6 31)(7 32)(8 33)(9 34)(10 35)(11 36)(12 25)(13 38)(14 39)(15 40)(16 41)(17 42)(18 43)(19 44)(20 45)(21 46)(22 47)(23 48)(24 37)
(1 29 7 35)(2 30 8 36)(3 31 9 25)(4 32 10 26)(5 33 11 27)(6 34 12 28)(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 13)(2 14)(3 15)(4 16)(5 17)(6 18)(7 19)(8 20)(9 21)(10 22)(11 23)(12 24)(25 37)(26 38)(27 39)(28 40)(29 41)(30 42)(31 43)(32 44)(33 45)(34 46)(35 47)(36 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 24)(2 23)(3 22)(4 21)(5 20)(6 19)(7 18)(8 17)(9 16)(10 15)(11 14)(12 13)(25 38)(26 37)(27 48)(28 47)(29 46)(30 45)(31 44)(32 43)(33 42)(34 41)(35 40)(36 39)

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

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

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

54 conjugacy classes

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

54 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 4 4 type + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 S3 D6 D6 D6 D6 2+ 1+4 D4○D12 kernel C2×D4○D12 C22×D12 C2×C4○D12 C2×S3×D4 C2×Q8⋊3S3 D4○D12 C6×C4○D4 C2×C4○D4 C22×C4 C2×D4 C2×Q8 C4○D4 C6 C2 # reps 1 3 3 6 2 16 1 1 3 3 1 8 2 4

Matrix representation of C2×D4○D12 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
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 10 7 0 0 0 0 6 3 0 0 0 0 3 6 3 6 0 0 7 10 7 10
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 10 7 7 1 0 0 6 3 12 6 0 0 3 6 3 6 0 0 7 10 7 10
,
 0 1 0 0 0 0 12 12 0 0 0 0 0 0 7 10 0 0 0 0 3 10 0 0 0 0 0 0 7 10 0 0 0 0 3 10
,
 0 12 0 0 0 0 12 0 0 0 0 0 0 0 0 1 0 2 0 0 1 0 2 0 0 0 0 0 0 12 0 0 0 0 12 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],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,10,6,3,7,0,0,7,3,6,10,0,0,0,0,3,7,0,0,0,0,6,10],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,10,6,3,7,0,0,7,3,6,10,0,0,7,12,3,7,0,0,1,6,6,10],[0,12,0,0,0,0,1,12,0,0,0,0,0,0,7,3,0,0,0,0,10,10,0,0,0,0,0,0,7,3,0,0,0,0,10,10],[0,12,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,2,0,12,0,0,2,0,12,0] >;

C2×D4○D12 in GAP, Magma, Sage, TeX

C_2\times D_4\circ D_{12}
% in TeX

G:=Group("C2xD4oD12");
// GroupNames label

G:=SmallGroup(192,1521);
// by ID

G=gap.SmallGroup(192,1521);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,297,1684,102,6278]);
// Polycyclic

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

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