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## G = C3×D4.3D4order 192 = 26·3

### Direct product of C3 and D4.3D4

direct product, metabelian, nilpotent (class 3), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C4 — C3×D4.3D4
 Chief series C1 — C2 — C4 — C2×C4 — C2×C12 — C6×D4 — C6×SD16 — C3×D4.3D4
 Lower central C1 — C2 — C2×C4 — C3×D4.3D4
 Upper central C1 — C6 — C2×C12 — C3×D4.3D4

Generators and relations for C3×D4.3D4
G = < a,b,c,d,e | a3=b4=c2=1, d4=e2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc=ebe-1=b-1, bd=db, cd=dc, ece-1=bc, ede-1=d3 >

Subgroups: 194 in 104 conjugacy classes, 50 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C6, C6, C8, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C12, C12, C2×C6, C2×C6, C2×C8, C2×C8, M4(2), M4(2), D8, SD16, Q16, C2×D4, C2×Q8, C4○D4, C24, C24, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C3×Q8, C22×C6, C4.D4, C4.10D4, C8.C4, C8○D4, C2×SD16, C8⋊C22, C8.C22, C2×C24, C2×C24, C3×M4(2), C3×M4(2), C3×D8, C3×SD16, C3×Q16, C6×D4, C6×Q8, C3×C4○D4, D4.3D4, C3×C4.D4, C3×C4.10D4, C3×C8.C4, C3×C8○D4, C6×SD16, C3×C8⋊C22, C3×C8.C22, C3×D4.3D4
Quotients: C1, C2, C3, C22, C6, D4, C23, C2×C6, C2×D4, C4○D4, C3×D4, C22×C6, C4⋊D4, C6×D4, C3×C4○D4, D4.3D4, C3×C4⋊D4, C3×D4.3D4

Smallest permutation representation of C3×D4.3D4
On 48 points
Generators in S48
(1 14 19)(2 15 20)(3 16 21)(4 9 22)(5 10 23)(6 11 24)(7 12 17)(8 13 18)(25 34 46)(26 35 47)(27 36 48)(28 37 41)(29 38 42)(30 39 43)(31 40 44)(32 33 45)
(1 7 5 3)(2 8 6 4)(9 15 13 11)(10 16 14 12)(17 23 21 19)(18 24 22 20)(25 27 29 31)(26 28 30 32)(33 35 37 39)(34 36 38 40)(41 43 45 47)(42 44 46 48)
(1 41)(2 42)(3 43)(4 44)(5 45)(6 46)(7 47)(8 48)(9 31)(10 32)(11 25)(12 26)(13 27)(14 28)(15 29)(16 30)(17 35)(18 36)(19 37)(20 38)(21 39)(22 40)(23 33)(24 34)
(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 4 5 8)(2 7 6 3)(9 10 13 14)(11 16 15 12)(17 24 21 20)(18 19 22 23)(25 28 29 32)(26 31 30 27)(33 34 37 38)(35 40 39 36)(41 42 45 46)(43 48 47 44)

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

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

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

48 conjugacy classes

 class 1 2A 2B 2C 2D 3A 3B 4A 4B 4C 4D 6A 6B 6C 6D 6E 6F 6G 6H 8A 8B 8C 8D 8E 8F 8G 12A 12B 12C 12D 12E 12F 12G 12H 24A 24B 24C 24D 24E ··· 24J 24K 24L 24M 24N order 1 2 2 2 2 3 3 4 4 4 4 6 6 6 6 6 6 6 6 8 8 8 8 8 8 8 12 12 12 12 12 12 12 12 24 24 24 24 24 ··· 24 24 24 24 24 size 1 1 2 4 8 1 1 2 2 4 8 1 1 2 2 4 4 8 8 2 2 4 4 4 8 8 2 2 2 2 4 4 8 8 2 2 2 2 4 ··· 4 8 8 8 8

48 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C3 C6 C6 C6 C6 C6 C6 C6 D4 D4 D4 C4○D4 C3×D4 C3×D4 C3×D4 C3×C4○D4 D4.3D4 C3×D4.3D4 kernel C3×D4.3D4 C3×C4.D4 C3×C4.10D4 C3×C8.C4 C3×C8○D4 C6×SD16 C3×C8⋊C22 C3×C8.C22 D4.3D4 C4.D4 C4.10D4 C8.C4 C8○D4 C2×SD16 C8⋊C22 C8.C22 C24 C3×D4 C3×Q8 C2×C6 C8 D4 Q8 C22 C3 C1 # reps 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 1 1 2 4 2 2 4 2 4

Matrix representation of C3×D4.3D4 in GL4(𝔽73) generated by

 8 0 0 0 0 8 0 0 0 0 8 0 0 0 0 8
,
 0 1 0 0 72 0 0 0 0 0 0 72 0 0 1 0
,
 0 0 0 72 0 0 1 0 0 1 0 0 72 0 0 0
,
 6 6 0 0 67 6 0 0 0 0 6 6 0 0 67 6
,
 6 67 0 0 67 67 0 0 0 0 6 6 0 0 6 67
G:=sub<GL(4,GF(73))| [8,0,0,0,0,8,0,0,0,0,8,0,0,0,0,8],[0,72,0,0,1,0,0,0,0,0,0,1,0,0,72,0],[0,0,0,72,0,0,1,0,0,1,0,0,72,0,0,0],[6,67,0,0,6,6,0,0,0,0,6,67,0,0,6,6],[6,67,0,0,67,67,0,0,0,0,6,6,0,0,6,67] >;

C3×D4.3D4 in GAP, Magma, Sage, TeX

C_3\times D_4._3D_4
% in TeX

G:=Group("C3xD4.3D4");
// GroupNames label

G:=SmallGroup(192,904);
// by ID

G=gap.SmallGroup(192,904);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-2,365,176,1094,4204,172,6053,1531,124]);
// Polycyclic

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

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