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## G = C3×D4○D8order 192 = 26·3

### Direct product of C3 and D4○D8

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C4 — C3×D4○D8
 Chief series C1 — C2 — C4 — C12 — C3×D4 — C3×D8 — C6×D8 — C3×D4○D8
 Lower central C1 — C2 — C4 — C3×D4○D8
 Upper central C1 — C6 — C3×C4○D4 — C3×D4○D8

Generators and relations for C3×D4○D8
G = < a,b,c,d,e | a3=b4=c2=e2=1, d4=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=b2d3 >

Subgroups: 474 in 268 conjugacy classes, 158 normal (18 characteristic)
C1, C2, C2, C3, C4, C4, C4, C22, C22, C6, C6, C8, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C12, C12, C12, C2×C6, C2×C6, C2×C8, M4(2), D8, SD16, Q16, C2×D4, C2×D4, C4○D4, C4○D4, C4○D4, C24, C24, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C3×Q8, C22×C6, C8○D4, C2×D8, C4○D8, C8⋊C22, 2+ 1+4, C2×C24, C3×M4(2), C3×D8, C3×SD16, C3×Q16, C6×D4, C6×D4, C3×C4○D4, C3×C4○D4, C3×C4○D4, D4○D8, C3×C8○D4, C6×D8, C3×C4○D8, C3×C8⋊C22, C3×2+ 1+4, C3×D4○D8
Quotients: C1, C2, C3, C22, C6, D4, C23, C2×C6, C2×D4, C24, C3×D4, C22×C6, C22×D4, C6×D4, C23×C6, D4○D8, D4×C2×C6, C3×D4○D8

Smallest permutation representation of C3×D4○D8
On 48 points
Generators in S48
(1 35 19)(2 36 20)(3 37 21)(4 38 22)(5 39 23)(6 40 24)(7 33 17)(8 34 18)(9 41 26)(10 42 27)(11 43 28)(12 44 29)(13 45 30)(14 46 31)(15 47 32)(16 48 25)
(1 45 5 41)(2 46 6 42)(3 47 7 43)(4 48 8 44)(9 19 13 23)(10 20 14 24)(11 21 15 17)(12 22 16 18)(25 34 29 38)(26 35 30 39)(27 36 31 40)(28 37 32 33)
(9 13)(10 14)(11 15)(12 16)(25 29)(26 30)(27 31)(28 32)(41 45)(42 46)(43 47)(44 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)
(2 8)(3 7)(4 6)(10 16)(11 15)(12 14)(17 21)(18 20)(22 24)(25 27)(28 32)(29 31)(33 37)(34 36)(38 40)(42 48)(43 47)(44 46)

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

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

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

66 conjugacy classes

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

66 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 4 4 type + + + + + + + + + image C1 C2 C2 C2 C2 C2 C3 C6 C6 C6 C6 C6 D4 D4 C3×D4 C3×D4 D4○D8 C3×D4○D8 kernel C3×D4○D8 C3×C8○D4 C6×D8 C3×C4○D8 C3×C8⋊C22 C3×2+ 1+4 D4○D8 C8○D4 C2×D8 C4○D8 C8⋊C22 2+ 1+4 C3×D4 C3×Q8 D4 Q8 C3 C1 # reps 1 1 3 3 6 2 2 2 6 6 12 4 3 1 6 2 2 4

Matrix representation of C3×D4○D8 in GL4(𝔽7) generated by

 4 0 0 0 0 4 0 0 0 0 4 0 0 0 0 4
,
 6 6 3 5 3 6 5 1 1 0 0 1 1 1 3 2
,
 6 0 1 4 0 5 4 5 0 4 5 5 0 6 6 5
,
 5 0 5 1 1 5 2 1 1 6 2 5 5 5 1 1
,
 0 4 5 0 4 6 1 0 4 5 2 0 6 1 4 6
G:=sub<GL(4,GF(7))| [4,0,0,0,0,4,0,0,0,0,4,0,0,0,0,4],[6,3,1,1,6,6,0,1,3,5,0,3,5,1,1,2],[6,0,0,0,0,5,4,6,1,4,5,6,4,5,5,5],[5,1,1,5,0,5,6,5,5,2,2,1,1,1,5,1],[0,4,4,6,4,6,5,1,5,1,2,4,0,0,0,6] >;

C3×D4○D8 in GAP, Magma, Sage, TeX

C_3\times D_4\circ D_8
% in TeX

G:=Group("C3xD4oD8");
// GroupNames label

G:=SmallGroup(192,1465);
// by ID

G=gap.SmallGroup(192,1465);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-2,-2,701,745,6053,3036,124]);
// 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=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=b^2*d^3>;
// generators/relations

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