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## G = C3×C23.9D4order 192 = 26·3

### Direct product of C3 and C23.9D4

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C3×C23.9D4
 Chief series C1 — C2 — C22 — C23 — C24 — C23×C6 — C6×C22⋊C4 — C3×C23.9D4
 Lower central C1 — C2 — C22 — C3×C23.9D4
 Upper central C1 — C2×C6 — C23×C6 — C3×C23.9D4

Generators and relations for C3×C23.9D4
G = < a,b,c,d,e,f | a3=b2=c2=d2=e4=1, f2=bcd, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, fbf-1=bd=db, be=eb, ece-1=fcf-1=cd=dc, de=ed, df=fd, fef-1=bde-1 >

Subgroups: 298 in 142 conjugacy classes, 58 normal (22 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, C6, C6, C6, C2×C4, C23, C23, C23, C12, C2×C6, C2×C6, C2×C6, C22⋊C4, C22⋊C4, C22×C4, C22×C4, C24, C2×C12, C22×C6, C22×C6, C22×C6, C2×C22⋊C4, C2×C22⋊C4, C3×C22⋊C4, C3×C22⋊C4, C22×C12, C22×C12, C23×C6, C23.9D4, C6×C22⋊C4, C6×C22⋊C4, C3×C23.9D4
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, D4, Q8, C12, C2×C6, C42, C22⋊C4, C4⋊C4, C2×C12, C3×D4, C3×Q8, C2.C42, C23⋊C4, C4×C12, C3×C22⋊C4, C3×C4⋊C4, C23.9D4, C3×C2.C42, C3×C23⋊C4, C3×C23.9D4

Smallest permutation representation of C3×C23.9D4
On 48 points
Generators in S48
(1 15 11)(2 16 12)(3 13 9)(4 14 10)(5 33 29)(6 34 30)(7 35 31)(8 36 32)(17 28 21)(18 25 22)(19 26 23)(20 27 24)(37 45 41)(38 46 42)(39 47 43)(40 48 44)
(1 31)(2 32)(3 29)(4 30)(5 13)(6 14)(7 15)(8 16)(9 33)(10 34)(11 35)(12 36)(17 37)(18 38)(19 39)(20 40)(21 41)(22 42)(23 43)(24 44)(25 46)(26 47)(27 48)(28 45)
(1 31)(2 20)(3 29)(4 18)(5 13)(6 46)(7 15)(8 48)(9 33)(10 22)(11 35)(12 24)(14 25)(16 27)(17 37)(19 39)(21 41)(23 43)(26 47)(28 45)(30 38)(32 40)(34 42)(36 44)
(1 39)(2 40)(3 37)(4 38)(5 28)(6 25)(7 26)(8 27)(9 41)(10 42)(11 43)(12 44)(13 45)(14 46)(15 47)(16 48)(17 29)(18 30)(19 31)(20 32)(21 33)(22 34)(23 35)(24 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)
(1 29 39 17)(3 31 37 19)(5 47 28 15)(6 25)(7 45 26 13)(8 27)(9 35 41 23)(11 33 43 21)(18 30)(20 32)(22 34)(24 36)

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

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

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

66 conjugacy classes

 class 1 2A 2B 2C 2D ··· 2I 3A 3B 4A ··· 4L 6A ··· 6F 6G ··· 6R 12A ··· 12X order 1 2 2 2 2 ··· 2 3 3 4 ··· 4 6 ··· 6 6 ··· 6 12 ··· 12 size 1 1 1 1 2 ··· 2 1 1 4 ··· 4 1 ··· 1 2 ··· 2 4 ··· 4

66 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 4 4 type + + + - + image C1 C2 C3 C4 C4 C6 C12 C12 D4 Q8 C3×D4 C3×Q8 C23⋊C4 C3×C23⋊C4 kernel C3×C23.9D4 C6×C22⋊C4 C23.9D4 C3×C22⋊C4 C22×C12 C2×C22⋊C4 C22⋊C4 C22×C4 C22×C6 C22×C6 C23 C23 C6 C2 # reps 1 3 2 8 4 6 16 8 3 1 6 2 2 4

Matrix representation of C3×C23.9D4 in GL6(𝔽13)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 9 0 0 0 0 0 0 9 0 0 0 0 0 0 9 0 0 0 0 0 0 9
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 0 0 0 1 0 1 0 0 0 0 0 12 0 0 0 0 0 11 1
,
 12 0 0 0 0 0 0 12 0 0 0 0 0 0 0 1 12 1 0 0 1 0 12 1 0 0 0 0 1 0 0 0 0 0 2 12
,
 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
,
 12 8 0 0 0 0 3 1 0 0 0 0 0 0 0 1 1 0 0 0 0 1 0 0 0 0 1 12 0 0 0 0 0 11 0 12
,
 5 12 0 0 0 0 0 8 0 0 0 0 0 0 0 12 0 0 0 0 1 0 0 1 0 0 0 0 1 12 0 0 0 0 0 12

G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,9,0,0,0,0,0,0,9,0,0,0,0,0,0,9,0,0,0,0,0,0,9],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,1,1,12,11,0,0,0,0,0,1],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,12,12,1,2,0,0,1,1,0,12],[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],[12,3,0,0,0,0,8,1,0,0,0,0,0,0,0,0,1,0,0,0,1,1,12,11,0,0,1,0,0,0,0,0,0,0,0,12],[5,0,0,0,0,0,12,8,0,0,0,0,0,0,0,1,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,1,12,12] >;

C3×C23.9D4 in GAP, Magma, Sage, TeX

C_3\times C_2^3._9D_4
% in TeX

G:=Group("C3xC2^3.9D4");
// GroupNames label

G:=SmallGroup(192,148);
// by ID

G=gap.SmallGroup(192,148);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-2,168,197,344,3027,2111]);
// Polycyclic

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

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