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## G = C3×C2.C25order 192 = 26·3

### Direct product of C3 and C2.C25

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C3×C2.C25
 Chief series C1 — C2 — C6 — C2×C6 — C3×D4 — C6×D4 — C3×2+ 1+4 — C3×C2.C25
 Lower central C1 — C2 — C3×C2.C25
 Upper central C1 — C12 — C3×C2.C25

Generators and relations for C3×C2.C25
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, dcd=fcf=bc=cb, ede=bd=db, be=eb, bf=fb, bg=gb, ce=ec, cg=gc, df=fd, dg=gd, ef=fe, eg=ge, fg=gf >

Subgroups: 930 in 810 conjugacy classes, 750 normal (8 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C6, C6, C2×C4, D4, Q8, C23, C12, C12, C2×C6, C2×C6, C22×C4, C2×D4, C2×Q8, C4○D4, C2×C12, C3×D4, C3×Q8, C22×C6, C2×C4○D4, 2+ 1+4, 2- 1+4, C22×C12, C6×D4, C6×Q8, C3×C4○D4, C2.C25, C6×C4○D4, C3×2+ 1+4, C3×2- 1+4, C3×C2.C25
Quotients: C1, C2, C3, C22, C6, C23, C2×C6, C24, C22×C6, C25, C23×C6, C2.C25, C24×C6, C3×C2.C25

Smallest permutation representation of C3×C2.C25
On 48 points
Generators in S48
(1 13 9)(2 14 10)(3 15 11)(4 16 12)(5 42 45)(6 43 46)(7 44 47)(8 41 48)(17 24 26)(18 21 27)(19 22 28)(20 23 25)(29 36 38)(30 33 39)(31 34 40)(32 35 37)
(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 35)(2 36)(3 33)(4 34)(5 27)(6 28)(7 25)(8 26)(9 32)(10 29)(11 30)(12 31)(13 37)(14 38)(15 39)(16 40)(17 41)(18 42)(19 43)(20 44)(21 45)(22 46)(23 47)(24 48)
(17 19)(18 20)(21 23)(22 24)(25 27)(26 28)(29 31)(30 32)(33 35)(34 36)(37 39)(38 40)
(1 23)(2 24)(3 21)(4 22)(5 39)(6 40)(7 37)(8 38)(9 20)(10 17)(11 18)(12 19)(13 25)(14 26)(15 27)(16 28)(29 41)(30 42)(31 43)(32 44)(33 45)(34 46)(35 47)(36 48)
(5 7)(6 8)(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)| (1,13,9)(2,14,10)(3,15,11)(4,16,12)(5,42,45)(6,43,46)(7,44,47)(8,41,48)(17,24,26)(18,21,27)(19,22,28)(20,23,25)(29,36,38)(30,33,39)(31,34,40)(32,35,37), (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), (1,35)(2,36)(3,33)(4,34)(5,27)(6,28)(7,25)(8,26)(9,32)(10,29)(11,30)(12,31)(13,37)(14,38)(15,39)(16,40)(17,41)(18,42)(19,43)(20,44)(21,45)(22,46)(23,47)(24,48), (17,19)(18,20)(21,23)(22,24)(25,27)(26,28)(29,31)(30,32)(33,35)(34,36)(37,39)(38,40), (1,23)(2,24)(3,21)(4,22)(5,39)(6,40)(7,37)(8,38)(9,20)(10,17)(11,18)(12,19)(13,25)(14,26)(15,27)(16,28)(29,41)(30,42)(31,43)(32,44)(33,45)(34,46)(35,47)(36,48), (5,7)(6,8)(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( (1,13,9)(2,14,10)(3,15,11)(4,16,12)(5,42,45)(6,43,46)(7,44,47)(8,41,48)(17,24,26)(18,21,27)(19,22,28)(20,23,25)(29,36,38)(30,33,39)(31,34,40)(32,35,37), (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), (1,35)(2,36)(3,33)(4,34)(5,27)(6,28)(7,25)(8,26)(9,32)(10,29)(11,30)(12,31)(13,37)(14,38)(15,39)(16,40)(17,41)(18,42)(19,43)(20,44)(21,45)(22,46)(23,47)(24,48), (17,19)(18,20)(21,23)(22,24)(25,27)(26,28)(29,31)(30,32)(33,35)(34,36)(37,39)(38,40), (1,23)(2,24)(3,21)(4,22)(5,39)(6,40)(7,37)(8,38)(9,20)(10,17)(11,18)(12,19)(13,25)(14,26)(15,27)(16,28)(29,41)(30,42)(31,43)(32,44)(33,45)(34,46)(35,47)(36,48), (5,7)(6,8)(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([[(1,13,9),(2,14,10),(3,15,11),(4,16,12),(5,42,45),(6,43,46),(7,44,47),(8,41,48),(17,24,26),(18,21,27),(19,22,28),(20,23,25),(29,36,38),(30,33,39),(31,34,40),(32,35,37)], [(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)], [(1,35),(2,36),(3,33),(4,34),(5,27),(6,28),(7,25),(8,26),(9,32),(10,29),(11,30),(12,31),(13,37),(14,38),(15,39),(16,40),(17,41),(18,42),(19,43),(20,44),(21,45),(22,46),(23,47),(24,48)], [(17,19),(18,20),(21,23),(22,24),(25,27),(26,28),(29,31),(30,32),(33,35),(34,36),(37,39),(38,40)], [(1,23),(2,24),(3,21),(4,22),(5,39),(6,40),(7,37),(8,38),(9,20),(10,17),(11,18),(12,19),(13,25),(14,26),(15,27),(16,28),(29,41),(30,42),(31,43),(32,44),(33,45),(34,46),(35,47),(36,48)], [(5,7),(6,8),(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)]])

102 conjugacy classes

 class 1 2A 2B ··· 2P 3A 3B 4A 4B 4C ··· 4Q 6A 6B 6C ··· 6AF 12A 12B 12C 12D 12E ··· 12AH order 1 2 2 ··· 2 3 3 4 4 4 ··· 4 6 6 6 ··· 6 12 12 12 12 12 ··· 12 size 1 1 2 ··· 2 1 1 1 1 2 ··· 2 1 1 2 ··· 2 1 1 1 1 2 ··· 2

102 irreducible representations

 dim 1 1 1 1 1 1 1 1 4 4 type + + + + image C1 C2 C2 C2 C3 C6 C6 C6 C2.C25 C3×C2.C25 kernel C3×C2.C25 C6×C4○D4 C3×2+ 1+4 C3×2- 1+4 C2.C25 C2×C4○D4 2+ 1+4 2- 1+4 C3 C1 # reps 1 15 10 6 2 30 20 12 2 4

Matrix representation of C3×C2.C25 in GL4(𝔽13) generated by

 3 0 0 0 0 3 0 0 0 0 3 0 0 0 0 3
,
 12 0 0 0 0 12 0 0 0 0 12 0 0 0 0 12
,
 0 11 1 6 0 1 0 0 1 1 0 6 0 11 0 12
,
 1 0 0 0 0 12 0 12 0 0 12 12 0 0 0 1
,
 0 1 11 6 1 0 1 6 0 0 1 0 0 0 11 12
,
 1 0 0 11 0 1 0 1 0 0 12 0 0 0 0 12
,
 5 0 0 0 0 5 0 0 0 0 5 0 0 0 0 5
G:=sub<GL(4,GF(13))| [3,0,0,0,0,3,0,0,0,0,3,0,0,0,0,3],[12,0,0,0,0,12,0,0,0,0,12,0,0,0,0,12],[0,0,1,0,11,1,1,11,1,0,0,0,6,0,6,12],[1,0,0,0,0,12,0,0,0,0,12,0,0,12,12,1],[0,1,0,0,1,0,0,0,11,1,1,11,6,6,0,12],[1,0,0,0,0,1,0,0,0,0,12,0,11,1,0,12],[5,0,0,0,0,5,0,0,0,0,5,0,0,0,0,5] >;

C3×C2.C25 in GAP, Magma, Sage, TeX

C_3\times C_2.C_2^5
% in TeX

G:=Group("C3xC2.C2^5");
// GroupNames label

G:=SmallGroup(192,1536);
// by ID

G=gap.SmallGroup(192,1536);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-2,1373,1059,2915,242]);
// 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,d*c*d=f*c*f=b*c=c*b,e*d*e=b*d=d*b,b*e=e*b,b*f=f*b,b*g=g*b,c*e=e*c,c*g=g*c,d*f=f*d,d*g=g*d,e*f=f*e,e*g=g*e,f*g=g*f>;
// generators/relations

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