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## G = C3×C22.45C24order 192 = 26·3

### Direct product of C3 and C22.45C24

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C3×C22.45C24
 Chief series C1 — C2 — C22 — C2×C6 — C2×C12 — C3×C4⋊C4 — C3×C22.D4 — C3×C22.45C24
 Lower central C1 — C22 — C3×C22.45C24
 Upper central C1 — C2×C6 — C3×C22.45C24

Generators and relations for C3×C22.45C24
G = < a,b,c,d,e,f,g | a3=b2=c2=f2=g2=1, d2=b, e2=c, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, bc=cb, ede-1=bd=db, geg=be=eb, bf=fb, bg=gb, fdf=cd=dc, ce=ec, cf=fc, cg=gc, dg=gd, ef=fe, fg=gf >

Subgroups: 394 in 248 conjugacy classes, 150 normal (34 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, C6, C6, C6, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C23, C12, C2×C6, C2×C6, C2×C6, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C24, C2×C12, C2×C12, C2×C12, C3×D4, C3×Q8, C22×C6, C22×C6, C22×C6, C2×C22⋊C4, C42⋊C2, C4×D4, C22≀C2, C22⋊Q8, C22.D4, C22.D4, C4.4D4, C422C2, C4×C12, C4×C12, C3×C22⋊C4, C3×C22⋊C4, C3×C4⋊C4, C22×C12, C22×C12, C6×D4, C6×D4, C6×Q8, C23×C6, C22.45C24, C6×C22⋊C4, C3×C42⋊C2, D4×C12, C3×C22≀C2, C3×C22⋊Q8, C3×C22.D4, C3×C22.D4, C3×C4.4D4, C3×C422C2, C3×C22.45C24
Quotients: C1, C2, C3, C22, C6, C23, C2×C6, C4○D4, C24, C22×C6, C2×C4○D4, 2+ 1+4, C3×C4○D4, C23×C6, C22.45C24, C6×C4○D4, C3×2+ 1+4, C3×C22.45C24

Smallest permutation representation of C3×C22.45C24
On 48 points
Generators in S48
(1 11 31)(2 12 32)(3 9 29)(4 10 30)(5 13 33)(6 14 34)(7 15 35)(8 16 36)(17 37 41)(18 38 42)(19 39 43)(20 40 44)(21 28 45)(22 25 46)(23 26 47)(24 27 48)
(1 3)(2 4)(5 7)(6 8)(9 11)(10 12)(13 15)(14 16)(17 19)(18 20)(21 23)(22 24)(25 27)(26 28)(29 31)(30 32)(33 35)(34 36)(37 39)(38 40)(41 43)(42 44)(45 47)(46 48)
(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 23 39 35)(2 22 40 34)(3 21 37 33)(4 24 38 36)(5 9 28 41)(6 12 25 44)(7 11 26 43)(8 10 27 42)(13 29 45 17)(14 32 46 20)(15 31 47 19)(16 30 48 18)
(2 40)(4 38)(6 25)(8 27)(10 42)(12 44)(14 46)(16 48)(18 30)(20 32)(22 34)(24 36)
(1 39)(2 40)(3 37)(4 38)(5 26)(6 27)(7 28)(8 25)(9 41)(10 42)(11 43)(12 44)(13 47)(14 48)(15 45)(16 46)(17 29)(18 30)(19 31)(20 32)(21 35)(22 36)(23 33)(24 34)

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

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

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

75 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 3A 3B 4A ··· 4H 4I ··· 4O 6A ··· 6F 6G ··· 6N 6O 6P 6Q 6R 12A ··· 12P 12Q ··· 12AD order 1 2 2 2 2 2 2 2 2 2 3 3 4 ··· 4 4 ··· 4 6 ··· 6 6 ··· 6 6 6 6 6 12 ··· 12 12 ··· 12 size 1 1 1 1 2 2 2 2 4 4 1 1 2 ··· 2 4 ··· 4 1 ··· 1 2 ··· 2 4 4 4 4 2 ··· 2 4 ··· 4

75 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 4 4 type + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 C3 C6 C6 C6 C6 C6 C6 C6 C6 C4○D4 C3×C4○D4 2+ 1+4 C3×2+ 1+4 kernel C3×C22.45C24 C6×C22⋊C4 C3×C42⋊C2 D4×C12 C3×C22≀C2 C3×C22⋊Q8 C3×C22.D4 C3×C4.4D4 C3×C42⋊2C2 C22.45C24 C2×C22⋊C4 C42⋊C2 C4×D4 C22≀C2 C22⋊Q8 C22.D4 C4.4D4 C42⋊2C2 C2×C6 C22 C6 C2 # reps 1 2 2 2 1 2 3 1 2 2 4 4 4 2 4 6 2 4 8 16 1 2

Matrix representation of C3×C22.45C24 in GL4(𝔽13) generated by

 9 0 0 0 0 9 0 0 0 0 3 0 0 0 0 3
,
 1 0 0 0 0 1 0 0 0 0 12 0 0 0 0 12
,
 12 0 0 0 0 12 0 0 0 0 1 0 0 0 0 1
,
 1 2 0 0 0 12 0 0 0 0 8 0 0 0 5 5
,
 8 0 0 0 0 8 0 0 0 0 12 11 0 0 0 1
,
 1 0 0 0 12 12 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 1 0 0 0 0 1 0 0 0 12 12
G:=sub<GL(4,GF(13))| [9,0,0,0,0,9,0,0,0,0,3,0,0,0,0,3],[1,0,0,0,0,1,0,0,0,0,12,0,0,0,0,12],[12,0,0,0,0,12,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,2,12,0,0,0,0,8,5,0,0,0,5],[8,0,0,0,0,8,0,0,0,0,12,0,0,0,11,1],[1,12,0,0,0,12,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,1,12,0,0,0,12] >;

C3×C22.45C24 in GAP, Magma, Sage, TeX

C_3\times C_2^2._{45}C_2^4
% in TeX

G:=Group("C3xC2^2.45C2^4");
// GroupNames label

G:=SmallGroup(192,1440);
// by ID

G=gap.SmallGroup(192,1440);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-2,-2,336,701,680,2102,794]);
// Polycyclic

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

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