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## G = C2×C12.D4order 192 = 26·3

### Direct product of C2 and C12.D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C2×C12.D4
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C4.Dic3 — C2×C4.Dic3 — C2×C12.D4
 Lower central C3 — C6 — C2×C6 — C2×C12.D4
 Upper central C1 — C22 — C22×C4 — C22×D4

Generators and relations for C2×C12.D4
G = < a,b,c,d | a2=b12=1, c4=b6, d2=b9, ab=ba, ac=ca, ad=da, cbc-1=b-1, dbd-1=b5, dcd-1=b3c3 >

Subgroups: 424 in 186 conjugacy classes, 71 normal (21 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C6, C6, C6, C8, C2×C4, C2×C4, D4, C23, C23, C23, C12, C2×C6, C2×C6, C2×C8, M4(2), C22×C4, C2×D4, C2×D4, C24, C3⋊C8, C2×C12, C2×C12, C3×D4, C22×C6, C22×C6, C22×C6, C4.D4, C2×M4(2), C22×D4, C2×C3⋊C8, C4.Dic3, C4.Dic3, C22×C12, C6×D4, C6×D4, C23×C6, C2×C4.D4, C12.D4, C2×C4.Dic3, D4×C2×C6, C2×C12.D4
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, Dic3, D6, C22⋊C4, C22×C4, C2×D4, C2×Dic3, C3⋊D4, C22×S3, C4.D4, C2×C22⋊C4, C6.D4, C22×Dic3, C2×C3⋊D4, C2×C4.D4, C12.D4, C2×C6.D4, C2×C12.D4

Smallest permutation representation of C2×C12.D4
On 48 points
Generators in S48
(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 40)(26 41)(27 42)(28 43)(29 44)(30 45)(31 46)(32 47)(33 48)(34 37)(35 38)(36 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 41 10 44 7 47 4 38)(2 40 11 43 8 46 5 37)(3 39 12 42 9 45 6 48)(13 26 22 29 19 32 16 35)(14 25 23 28 20 31 17 34)(15 36 24 27 21 30 18 33)
(1 44 10 41 7 38 4 47)(2 37 11 46 8 43 5 40)(3 42 12 39 9 48 6 45)(13 29 22 26 19 35 16 32)(14 34 23 31 20 28 17 25)(15 27 24 36 21 33 18 30)

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

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

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

42 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 3 4A 4B 4C 4D 6A ··· 6G 6H ··· 6O 8A ··· 8H 12A 12B 12C 12D order 1 2 2 2 2 2 2 2 2 2 3 4 4 4 4 6 ··· 6 6 ··· 6 8 ··· 8 12 12 12 12 size 1 1 1 1 2 2 4 4 4 4 2 2 2 2 2 2 ··· 2 4 ··· 4 12 ··· 12 4 4 4 4

42 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 type + + + + + + + - + - + image C1 C2 C2 C2 C4 C4 S3 D4 D6 Dic3 D6 Dic3 C3⋊D4 C4.D4 C12.D4 kernel C2×C12.D4 C12.D4 C2×C4.Dic3 D4×C2×C6 C6×D4 C23×C6 C22×D4 C2×C12 C22×C4 C2×D4 C2×D4 C24 C2×C4 C6 C2 # reps 1 4 2 1 4 4 1 4 1 2 2 2 8 2 4

Matrix representation of C2×C12.D4 in GL8(𝔽73)

 72 0 0 0 0 0 0 0 0 72 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 72 0 0 0 0 0 0 0 0 72 0 0 0 0 0 0 0 0 72 0 0 0 0 0 0 0 0 72
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 37 28 0 0 0 0 0 0 28 37 0 0 0 0 0 0 0 0 72 70 0 0 0 0 0 0 25 1 0 0 0 0 0 0 0 0 1 3 0 0 0 0 0 0 48 72
,
 46 0 0 0 0 0 0 0 0 27 0 0 0 0 0 0 0 0 66 11 0 0 0 0 0 0 62 7 0 0 0 0 0 0 0 0 0 0 72 0 0 0 0 0 0 0 0 72 0 0 0 0 1 3 0 0 0 0 0 0 48 72 0 0
,
 0 27 0 0 0 0 0 0 46 0 0 0 0 0 0 0 0 0 11 66 0 0 0 0 0 0 7 62 0 0 0 0 0 0 0 0 0 0 72 70 0 0 0 0 0 0 0 1 0 0 0 0 72 0 0 0 0 0 0 0 25 1 0 0

G:=sub<GL(8,GF(73))| [72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,37,28,0,0,0,0,0,0,28,37,0,0,0,0,0,0,0,0,72,25,0,0,0,0,0,0,70,1,0,0,0,0,0,0,0,0,1,48,0,0,0,0,0,0,3,72],[46,0,0,0,0,0,0,0,0,27,0,0,0,0,0,0,0,0,66,62,0,0,0,0,0,0,11,7,0,0,0,0,0,0,0,0,0,0,1,48,0,0,0,0,0,0,3,72,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0],[0,46,0,0,0,0,0,0,27,0,0,0,0,0,0,0,0,0,11,7,0,0,0,0,0,0,66,62,0,0,0,0,0,0,0,0,0,0,72,25,0,0,0,0,0,0,0,1,0,0,0,0,72,0,0,0,0,0,0,0,70,1,0,0] >;

C2×C12.D4 in GAP, Magma, Sage, TeX

C_2\times C_{12}.D_4
% in TeX

G:=Group("C2xC12.D4");
// GroupNames label

G:=SmallGroup(192,775);
// by ID

G=gap.SmallGroup(192,775);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,56,422,297,136,1684,6278]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^12=1,c^4=b^6,d^2=b^9,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,d*b*d^-1=b^5,d*c*d^-1=b^3*c^3>;
// generators/relations

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