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## G = C3×C22.19C24order 192 = 26·3

### Direct product of C3 and C22.19C24

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C3×C22.19C24
 Chief series C1 — C2 — C22 — C2×C6 — C22×C6 — C6×D4 — C3×C22≀C2 — C3×C22.19C24
 Lower central C1 — C22 — C3×C22.19C24
 Upper central C1 — C2×C12 — C3×C22.19C24

Generators and relations for C3×C22.19C24
G = < a,b,c,d,e,f,g | a3=b2=c2=d2=e2=f2=1, g2=b, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, bc=cb, ede=bd=db, be=eb, bf=fb, bg=gb, fdf=cd=dc, ce=ec, cf=fc, cg=gc, dg=gd, ef=fe, eg=ge, fg=gf >

Subgroups: 498 in 330 conjugacy classes, 170 normal (30 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C22, C6, C6, C6, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C23, C12, C12, C2×C6, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C24, C2×C12, C2×C12, C2×C12, C3×D4, C3×Q8, C22×C6, C22×C6, C22×C6, C42⋊C2, C4×D4, C22≀C2, C4⋊D4, C22⋊Q8, C22.D4, C23×C4, C2×C4○D4, C4×C12, C3×C22⋊C4, C3×C4⋊C4, C22×C12, C22×C12, C22×C12, C6×D4, C6×D4, C6×Q8, C3×C4○D4, C23×C6, C22.19C24, C3×C42⋊C2, D4×C12, C3×C22≀C2, C3×C4⋊D4, C3×C22⋊Q8, C3×C22.D4, C23×C12, C6×C4○D4, C3×C22.19C24
Quotients: C1, C2, C3, C22, C6, D4, C23, C2×C6, C2×D4, C4○D4, C24, C3×D4, C22×C6, C22×D4, C2×C4○D4, C6×D4, C3×C4○D4, C23×C6, C22.19C24, D4×C2×C6, C6×C4○D4, C3×C22.19C24

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

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

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

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

84 conjugacy classes

 class 1 2A 2B 2C 2D ··· 2I 2J 2K 3A 3B 4A 4B 4C 4D 4E ··· 4J 4K ··· 4P 6A ··· 6F 6G ··· 6R 6S 6T 6U 6V 12A ··· 12H 12I ··· 12T 12U ··· 12AF order 1 2 2 2 2 ··· 2 2 2 3 3 4 4 4 4 4 ··· 4 4 ··· 4 6 ··· 6 6 ··· 6 6 6 6 6 12 ··· 12 12 ··· 12 12 ··· 12 size 1 1 1 1 2 ··· 2 4 4 1 1 1 1 1 1 2 ··· 2 4 ··· 4 1 ··· 1 2 ··· 2 4 4 4 4 1 ··· 1 2 ··· 2 4 ··· 4

84 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 type + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 C3 C6 C6 C6 C6 C6 C6 C6 C6 D4 C4○D4 C3×D4 C3×C4○D4 kernel C3×C22.19C24 C3×C42⋊C2 D4×C12 C3×C22≀C2 C3×C4⋊D4 C3×C22⋊Q8 C3×C22.D4 C23×C12 C6×C4○D4 C22.19C24 C42⋊C2 C4×D4 C22≀C2 C4⋊D4 C22⋊Q8 C22.D4 C23×C4 C2×C4○D4 C2×C12 C2×C6 C2×C4 C22 # reps 1 1 4 2 2 2 2 1 1 2 2 8 4 4 4 4 2 2 4 8 8 16

Matrix representation of C3×C22.19C24 in GL4(𝔽13) generated by

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

C3×C22.19C24 in GAP, Magma, Sage, TeX

C_3\times C_2^2._{19}C_2^4
% in TeX

G:=Group("C3xC2^2.19C2^4");
// GroupNames label

G:=SmallGroup(192,1414);
// by ID

G=gap.SmallGroup(192,1414);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-2,-2,701,2102,192]);
// Polycyclic

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

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