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## G = C3×C22.54C24order 192 = 26·3

### Direct product of C3 and C22.54C24

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C3×C22.54C24
 Chief series C1 — C2 — C22 — C2×C6 — C22×C6 — C6×D4 — C3×C4⋊1D4 — C3×C22.54C24
 Lower central C1 — C22 — C3×C22.54C24
 Upper central C1 — C2×C6 — C3×C22.54C24

Generators and relations for C3×C22.54C24
G = < a,b,c,d,e,f,g | a3=b2=c2=d2=e2=f2=g2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, bc=cb, ede=bd=db, geg=be=eb, bf=fb, bg=gb, fdf=cd=dc, ce=ec, cf=fc, cg=gc, gdg=bcd, fef=bce, fg=gf >

Subgroups: 474 in 252 conjugacy classes, 142 normal (14 characteristic)
C1, C2, C2, C3, C4, C22, C22, C6, C6, C2×C4, C2×C4, D4, C23, C23, C23, C12, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C24, C2×C12, C2×C12, C3×D4, C22×C6, C22×C6, C22×C6, C22≀C2, C4⋊D4, C22.D4, C422C2, C41D4, C4×C12, C3×C22⋊C4, C3×C4⋊C4, C22×C12, C6×D4, C23×C6, C22.54C24, C3×C22≀C2, C3×C4⋊D4, C3×C22.D4, C3×C422C2, C3×C41D4, C3×C22.54C24
Quotients: C1, C2, C3, C22, C6, C23, C2×C6, C24, C22×C6, 2+ 1+4, C23×C6, C22.54C24, C3×2+ 1+4, C3×C22.54C24

Smallest permutation representation of C3×C22.54C24
On 48 points
Generators in S48
(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 15)(2 13)(3 14)(4 44)(5 45)(6 43)(7 48)(8 46)(9 47)(10 18)(11 16)(12 17)(19 26)(20 27)(21 25)(22 30)(23 28)(24 29)(31 38)(32 39)(33 37)(34 42)(35 40)(36 41)
(1 10)(2 11)(3 12)(4 46)(5 47)(6 48)(7 43)(8 44)(9 45)(13 16)(14 17)(15 18)(19 29)(20 30)(21 28)(22 27)(23 25)(24 26)(31 41)(32 42)(33 40)(34 39)(35 37)(36 38)
(1 39)(2 37)(3 38)(4 27)(5 25)(6 26)(7 29)(8 30)(9 28)(10 34)(11 35)(12 36)(13 33)(14 31)(15 32)(16 40)(17 41)(18 42)(19 43)(20 44)(21 45)(22 46)(23 47)(24 48)
(1 27)(2 25)(3 26)(4 32)(5 33)(6 31)(7 36)(8 34)(9 35)(10 22)(11 23)(12 24)(13 21)(14 19)(15 20)(16 28)(17 29)(18 30)(37 45)(38 43)(39 44)(40 47)(41 48)(42 46)
(1 10)(2 11)(3 12)(4 8)(5 9)(6 7)(13 16)(14 17)(15 18)(19 26)(20 27)(21 25)(22 30)(23 28)(24 29)(43 48)(44 46)(45 47)
(1 10)(2 11)(3 12)(13 16)(14 17)(15 18)(19 24)(20 22)(21 23)(25 28)(26 29)(27 30)(31 38)(32 39)(33 37)(34 42)(35 40)(36 41)

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

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

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

57 conjugacy classes

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

57 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 4 4 type + + + + + + + image C1 C2 C2 C2 C2 C2 C3 C6 C6 C6 C6 C6 2+ 1+4 C3×2+ 1+4 kernel C3×C22.54C24 C3×C22≀C2 C3×C4⋊D4 C3×C22.D4 C3×C42⋊2C2 C3×C4⋊1D4 C22.54C24 C22≀C2 C4⋊D4 C22.D4 C42⋊2C2 C4⋊1D4 C6 C2 # reps 1 3 6 3 2 1 2 6 12 6 4 2 3 6

Matrix representation of C3×C22.54C24 in GL8(𝔽13)

 3 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 12
,
 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 0 10 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 3 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 1 12 11 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 12
,
 0 1 3 0 0 0 0 0 1 0 0 3 0 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 12 12 1 2 0 0 0 0 1 1 0 12
,
 12 0 0 7 0 0 0 0 0 12 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 12 12
,
 12 0 0 7 0 0 0 0 0 1 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 12 1 1

G:=sub<GL(8,GF(13))| [3,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12],[12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,0,1,0,0,0,0,0,10,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,3,0,0,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,12,0,0,0,0,0,0,0,11,0,12],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,3,0,0,12,0,0,0,0,0,3,12,0,0,0,0,0,0,0,0,0,0,1,12,1,0,0,0,0,1,0,12,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,2,12],[12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,7,0,0,1,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,12,0,0,0,0,0,0,0,12],[12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,7,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,12,0,0,0,0,0,0,12,1,0,0,0,0,0,0,0,1] >;

C3×C22.54C24 in GAP, Magma, Sage, TeX

C_3\times C_2^2._{54}C_2^4
% in TeX

G:=Group("C3xC2^2.54C2^4");
// GroupNames label

G:=SmallGroup(192,1449);
// by ID

G=gap.SmallGroup(192,1449);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-2,-2,701,2102,1563,4259,794]);
// Polycyclic

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

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