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## G = C3×C23⋊3D4order 192 = 26·3

### Direct product of C3 and C23⋊3D4

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C3×C23⋊3D4
 Chief series C1 — C2 — C22 — C2×C6 — C22×C6 — C6×D4 — C3×C4⋊D4 — C3×C23⋊3D4
 Lower central C1 — C22 — C3×C23⋊3D4
 Upper central C1 — C2×C6 — C3×C23⋊3D4

Generators and relations for C3×C233D4
G = < a,b,c,d,e,f | a3=b2=c2=d2=e4=f2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, fbf=bd=db, be=eb, ece-1=fcf=cd=dc, de=ed, df=fd, fef=e-1 >

Subgroups: 642 in 346 conjugacy classes, 162 normal (14 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, C6, C6, C6, C2×C4, C2×C4, D4, C23, C23, C23, C12, C2×C6, C2×C6, C2×C6, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×D4, C24, C24, C2×C12, C2×C12, C3×D4, C22×C6, C22×C6, C22×C6, C2×C22⋊C4, C22≀C2, C4⋊D4, C22.D4, C22×D4, C3×C22⋊C4, C3×C4⋊C4, C22×C12, C6×D4, C6×D4, C23×C6, C23×C6, C233D4, C6×C22⋊C4, C3×C22≀C2, C3×C4⋊D4, C3×C22.D4, D4×C2×C6, C3×C233D4
Quotients: C1, C2, C3, C22, C6, D4, C23, C2×C6, C2×D4, C24, C3×D4, C22×C6, C22×D4, 2+ 1+4, C6×D4, C23×C6, C233D4, D4×C2×C6, C3×2+ 1+4, C3×C233D4

Smallest permutation representation of C3×C233D4
On 48 points
Generators in S48
(1 29 7)(2 30 8)(3 31 5)(4 32 6)(9 15 33)(10 16 34)(11 13 35)(12 14 36)(17 39 43)(18 40 44)(19 37 41)(20 38 42)(21 25 47)(22 26 48)(23 27 45)(24 28 46)
(1 21)(2 22)(3 23)(4 24)(5 45)(6 46)(7 47)(8 48)(9 38)(10 39)(11 40)(12 37)(13 44)(14 41)(15 42)(16 43)(17 34)(18 35)(19 36)(20 33)(25 29)(26 30)(27 31)(28 32)
(1 3)(2 38)(4 40)(5 7)(6 18)(8 20)(9 22)(10 12)(11 24)(13 28)(14 16)(15 26)(17 19)(21 23)(25 27)(29 31)(30 42)(32 44)(33 48)(34 36)(35 46)(37 39)(41 43)(45 47)
(1 39)(2 40)(3 37)(4 38)(5 19)(6 20)(7 17)(8 18)(9 24)(10 21)(11 22)(12 23)(13 26)(14 27)(15 28)(16 25)(29 43)(30 44)(31 41)(32 42)(33 46)(34 47)(35 48)(36 45)
(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)(2 3)(5 8)(6 7)(9 21)(10 24)(11 23)(12 22)(13 27)(14 26)(15 25)(16 28)(17 20)(18 19)(29 32)(30 31)(33 47)(34 46)(35 45)(36 48)(37 40)(38 39)(41 44)(42 43)

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

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

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

66 conjugacy classes

 class 1 2A 2B 2C 2D ··· 2I 2J 2K 2L 2M 3A 3B 4A ··· 4H 6A ··· 6F 6G ··· 6R 6S ··· 6Z 12A ··· 12P order 1 2 2 2 2 ··· 2 2 2 2 2 3 3 4 ··· 4 6 ··· 6 6 ··· 6 6 ··· 6 12 ··· 12 size 1 1 1 1 2 ··· 2 4 4 4 4 1 1 4 ··· 4 1 ··· 1 2 ··· 2 4 ··· 4 4 ··· 4

66 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 4 4 type + + + + + + + + image C1 C2 C2 C2 C2 C2 C3 C6 C6 C6 C6 C6 D4 C3×D4 2+ 1+4 C3×2+ 1+4 kernel C3×C23⋊3D4 C6×C22⋊C4 C3×C22≀C2 C3×C4⋊D4 C3×C22.D4 D4×C2×C6 C23⋊3D4 C2×C22⋊C4 C22≀C2 C4⋊D4 C22.D4 C22×D4 C22×C6 C23 C6 C2 # reps 1 1 4 4 4 2 2 2 8 8 8 4 4 8 2 4

Matrix representation of C3×C233D4 in GL6(𝔽13)

 9 0 0 0 0 0 0 9 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 11 0 0 0 0 0 12 0 0 0 0 12 1 12 12 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 1 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12
,
 0 1 0 0 0 0 12 0 0 0 0 0 0 0 12 0 11 0 0 0 12 0 11 12 0 0 1 0 1 0 0 0 12 1 0 0
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 12 0 11 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0

G:=sub<GL(6,GF(13))| [9,0,0,0,0,0,0,9,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,12,0,0,0,11,12,1,0,0,0,0,0,12,0,0,0,0,0,12,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,1,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[0,12,0,0,0,0,1,0,0,0,0,0,0,0,12,12,1,12,0,0,0,0,0,1,0,0,11,11,1,0,0,0,0,12,0,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,11,0,1,0,0,0,0,1,0,0] >;

C3×C233D4 in GAP, Magma, Sage, TeX

C_3\times C_2^3\rtimes_3D_4
% in TeX

G:=Group("C3xC2^3:3D4");
// GroupNames label

G:=SmallGroup(192,1423);
// by ID

G=gap.SmallGroup(192,1423);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-2,-2,701,2102,555,1571]);
// Polycyclic

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

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