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## G = C3×D4○SD16order 192 = 26·3

### Direct product of C3 and D4○SD16

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C4 — C3×D4○SD16
 Chief series C1 — C2 — C4 — C12 — C3×Q8 — C3×SD16 — C6×SD16 — C3×D4○SD16
 Lower central C1 — C2 — C4 — C3×D4○SD16
 Upper central C1 — C6 — C3×C4○D4 — C3×D4○SD16

Generators and relations for C3×D4○SD16
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=d3 >

Subgroups: 410 in 258 conjugacy classes, 158 normal (26 characteristic)
C1, C2, C2, C3, C4, C4, C4, C22, C22, C6, C6, C8, C8, C2×C4, C2×C4, D4, D4, D4, Q8, Q8, Q8, C23, C12, C12, C12, C2×C6, C2×C6, C2×C8, M4(2), D8, SD16, SD16, Q16, C2×D4, C2×D4, C2×Q8, C2×Q8, C4○D4, C4○D4, C4○D4, C24, C24, C2×C12, C2×C12, C3×D4, C3×D4, C3×D4, C3×Q8, C3×Q8, C3×Q8, C22×C6, C8○D4, C2×SD16, C4○D8, C8⋊C22, C8.C22, 2+ 1+4, 2- 1+4, C2×C24, C3×M4(2), C3×D8, C3×SD16, C3×SD16, C3×Q16, C6×D4, C6×D4, C6×Q8, C6×Q8, C3×C4○D4, C3×C4○D4, C3×C4○D4, D4○SD16, C3×C8○D4, C6×SD16, C3×C4○D8, C3×C8⋊C22, C3×C8.C22, C3×2+ 1+4, C3×2- 1+4, C3×D4○SD16
Quotients: C1, C2, C3, C22, C6, D4, C23, C2×C6, C2×D4, C24, C3×D4, C22×C6, C22×D4, C6×D4, C23×C6, D4○SD16, D4×C2×C6, C3×D4○SD16

Smallest permutation representation of C3×D4○SD16
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 47 5 43)(2 48 6 44)(3 41 7 45)(4 42 8 46)(9 17 13 21)(10 18 14 22)(11 19 15 23)(12 20 16 24)(25 36 29 40)(26 37 30 33)(27 38 31 34)(28 39 32 35)
(1 43)(2 44)(3 45)(4 46)(5 47)(6 48)(7 41)(8 42)(9 17)(10 18)(11 19)(12 20)(13 21)(14 22)(15 23)(16 24)(25 36)(26 37)(27 38)(28 39)(29 40)(30 33)(31 34)(32 35)
(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 4)(3 7)(6 8)(9 15)(11 13)(12 16)(17 23)(19 21)(20 24)(25 31)(27 29)(28 32)(34 36)(35 39)(38 40)(41 45)(42 48)(44 46)

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

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

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

66 conjugacy classes

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

66 irreducible representations

 dim 1 1 1 1 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 C2 C2 C3 C6 C6 C6 C6 C6 C6 C6 D4 D4 C3×D4 C3×D4 D4○SD16 C3×D4○SD16 kernel C3×D4○SD16 C3×C8○D4 C6×SD16 C3×C4○D8 C3×C8⋊C22 C3×C8.C22 C3×2+ 1+4 C3×2- 1+4 D4○SD16 C8○D4 C2×SD16 C4○D8 C8⋊C22 C8.C22 2+ 1+4 2- 1+4 C3×D4 C3×Q8 D4 Q8 C3 C1 # reps 1 1 3 3 3 3 1 1 2 2 6 6 6 6 2 2 3 1 6 2 2 4

Matrix representation of C3×D4○SD16 in GL4(𝔽73) generated by

 64 0 0 0 0 64 0 0 0 0 64 0 0 0 0 64
,
 0 0 1 0 0 0 0 1 72 0 0 0 0 72 0 0
,
 0 0 1 0 0 0 0 1 1 0 0 0 0 1 0 0
,
 6 67 0 0 6 6 0 0 0 0 6 67 0 0 6 6
,
 1 0 0 0 0 72 0 0 0 0 1 0 0 0 0 72
G:=sub<GL(4,GF(73))| [64,0,0,0,0,64,0,0,0,0,64,0,0,0,0,64],[0,0,72,0,0,0,0,72,1,0,0,0,0,1,0,0],[0,0,1,0,0,0,0,1,1,0,0,0,0,1,0,0],[6,6,0,0,67,6,0,0,0,0,6,6,0,0,67,6],[1,0,0,0,0,72,0,0,0,0,1,0,0,0,0,72] >;

C3×D4○SD16 in GAP, Magma, Sage, TeX

C_3\times D_4\circ {\rm SD}_{16}
% in TeX

G:=Group("C3xD4oSD16");
// GroupNames label

G:=SmallGroup(192,1466);
// by ID

G=gap.SmallGroup(192,1466);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-2,-2,672,701,745,6053,3036,124]);
// 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=d^3>;
// generators/relations

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