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## G = C3×D4.8D4order 192 = 26·3

### Direct product of C3 and D4.8D4

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C4 — C3×D4.8D4
 Chief series C1 — C2 — C22 — C2×C4 — C2×C12 — C6×D4 — C3×C8⋊C22 — C3×D4.8D4
 Lower central C1 — C2 — C2×C4 — C3×D4.8D4
 Upper central C1 — C6 — C2×C12 — C3×D4.8D4

Generators and relations for C3×D4.8D4
G = < a,b,c,d,e | a3=b4=c2=e2=1, d4=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc=dbd-1=ebe=b-1, dcd-1=bc, ece=b-1c, ede=b2d3 >

Subgroups: 274 in 146 conjugacy classes, 54 normal (22 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C6, C6, C8, C2×C4, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C12, C12, C2×C6, C2×C6, C42, C22⋊C4, M4(2), D8, SD16, C2×D4, C2×Q8, C2×Q8, C4○D4, C4○D4, C24, C2×C12, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C3×Q8, C22×C6, C4.10D4, C4≀C2, C4.4D4, C8⋊C22, 2- 1+4, C4×C12, C3×C22⋊C4, C3×M4(2), C3×D8, C3×SD16, C6×D4, C6×Q8, C6×Q8, C3×C4○D4, C3×C4○D4, D4.8D4, C3×C4.10D4, C3×C4≀C2, C3×C4.4D4, C3×C8⋊C22, C3×2- 1+4, C3×D4.8D4
Quotients: C1, C2, C3, C22, C6, D4, C23, C2×C6, C2×D4, C3×D4, C22×C6, C22≀C2, C6×D4, D4.8D4, C3×C22≀C2, C3×D4.8D4

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

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

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

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

48 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3A 3B 4A 4B 4C ··· 4H 6A 6B 6C 6D 6E 6F 6G 6H 6I 6J 8A 8B 12A 12B 12C 12D 12E ··· 12P 24A 24B 24C 24D order 1 2 2 2 2 2 3 3 4 4 4 ··· 4 6 6 6 6 6 6 6 6 6 6 8 8 12 12 12 12 12 ··· 12 24 24 24 24 size 1 1 2 4 4 8 1 1 2 2 4 ··· 4 1 1 2 2 4 4 4 4 8 8 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 2 2 2 2 2 2 4 4 type + + + + + + + + + image C1 C2 C2 C2 C2 C2 C3 C6 C6 C6 C6 C6 D4 D4 D4 C3×D4 C3×D4 C3×D4 D4.8D4 C3×D4.8D4 kernel C3×D4.8D4 C3×C4.10D4 C3×C4≀C2 C3×C4.4D4 C3×C8⋊C22 C3×2- 1+4 D4.8D4 C4.10D4 C4≀C2 C4.4D4 C8⋊C22 2- 1+4 C2×C12 C3×D4 C3×Q8 C2×C4 D4 Q8 C3 C1 # reps 1 1 2 1 2 1 2 2 4 2 4 2 2 2 2 4 4 4 2 4

Matrix representation of C3×D4.8D4 in GL4(𝔽73) generated by

 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 72 0 0 1 0
,
 0 0 0 27 0 0 27 0 0 46 0 0 46 0 0 0
,
 0 0 1 0 0 0 0 72 0 72 0 0 72 0 0 0
,
 1 0 0 0 0 72 0 0 0 0 0 72 0 0 72 0
G:=sub<GL(4,GF(73))| [8,0,0,0,0,8,0,0,0,0,8,0,0,0,0,8],[0,1,0,0,72,0,0,0,0,0,0,1,0,0,72,0],[0,0,0,46,0,0,46,0,0,27,0,0,27,0,0,0],[0,0,0,72,0,0,72,0,1,0,0,0,0,72,0,0],[1,0,0,0,0,72,0,0,0,0,0,72,0,0,72,0] >;

C3×D4.8D4 in GAP, Magma, Sage, TeX

C_3\times D_4._8D_4
% in TeX

G:=Group("C3xD4.8D4");
// GroupNames label

G:=SmallGroup(192,887);
// by ID

G=gap.SmallGroup(192,887);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-2,365,1094,520,4204,2111,1068,172,3036]);
// Polycyclic

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

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