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## G = C3×C23.7D4order 192 = 26·3

### Direct product of C3 and C23.7D4

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C23 — C3×C23.7D4
 Chief series C1 — C2 — C22 — C23 — C22×C6 — C22×C12 — C3×C22.D4 — C3×C23.7D4
 Lower central C1 — C2 — C23 — C3×C23.7D4
 Upper central C1 — C6 — C22×C6 — C3×C23.7D4

Generators and relations for C3×C23.7D4
G = < a,b,c,d,e,f | a3=b2=c2=d2=e4=1, f2=d, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, ebe-1=fbf-1=bcd, ece-1=cd=dc, cf=fc, de=ed, df=fd, fef-1=de-1 >

Subgroups: 322 in 160 conjugacy classes, 54 normal (12 characteristic)
C1, C2, C2, C3, C4, C22, C22, C6, C6, C2×C4, C2×C4, D4, Q8, C23, C23, C23, C12, C2×C6, C2×C6, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×D4, C4○D4, C2×C12, C2×C12, C3×D4, C3×Q8, C22×C6, C22×C6, C22×C6, C23⋊C4, C22.D4, 2+ 1+4, C3×C22⋊C4, C3×C22⋊C4, C3×C4⋊C4, C22×C12, C6×D4, C6×D4, C3×C4○D4, C23.7D4, C3×C23⋊C4, C3×C22.D4, C3×2+ 1+4, C3×C23.7D4
Quotients: C1, C2, C3, C22, C6, D4, C23, C2×C6, C2×D4, C3×D4, C22×C6, C22≀C2, C6×D4, C23.7D4, C3×C22≀C2, C3×C23.7D4

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

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

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

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

48 conjugacy classes

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

48 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 4 4 type + + + + + + image C1 C2 C2 C2 C3 C6 C6 C6 D4 D4 C3×D4 C3×D4 C23.7D4 C3×C23.7D4 kernel C3×C23.7D4 C3×C23⋊C4 C3×C22.D4 C3×2+ 1+4 C23.7D4 C23⋊C4 C22.D4 2+ 1+4 C2×C12 C22×C6 C2×C4 C23 C3 C1 # reps 1 3 3 1 2 6 6 2 3 3 6 6 2 4

Matrix representation of C3×C23.7D4 in GL4(𝔽13) generated by

 3 0 0 0 0 3 0 0 0 0 3 0 0 0 0 3
,
 1 0 0 0 0 12 0 0 0 0 12 0 0 0 0 1
,
 1 0 0 11 0 1 11 0 0 0 12 0 0 0 0 12
,
 12 0 0 0 0 12 0 0 0 0 12 0 0 0 0 12
,
 0 8 5 8 5 0 5 8 5 0 0 8 0 8 5 0
,
 0 8 5 8 8 0 8 5 0 0 8 0 0 0 0 8
G:=sub<GL(4,GF(13))| [3,0,0,0,0,3,0,0,0,0,3,0,0,0,0,3],[1,0,0,0,0,12,0,0,0,0,12,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,11,12,0,11,0,0,12],[12,0,0,0,0,12,0,0,0,0,12,0,0,0,0,12],[0,5,5,0,8,0,0,8,5,5,0,5,8,8,8,0],[0,8,0,0,8,0,0,0,5,8,8,0,8,5,0,8] >;

C3×C23.7D4 in GAP, Magma, Sage, TeX

C_3\times C_2^3._7D_4
% in TeX

G:=Group("C3xC2^3.7D4");
// GroupNames label

G:=SmallGroup(192,891);
// by ID

G=gap.SmallGroup(192,891);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-2,672,365,1094,1068,3036]);
// Polycyclic

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

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