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## G = C3×C23.C23order 192 = 26·3

### Direct product of C3 and C23.C23

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C3×C23.C23
 Chief series C1 — C2 — C22 — C23 — C22×C6 — C3×C22⋊C4 — C3×C23⋊C4 — C3×C23.C23
 Lower central C1 — C2 — C22 — C3×C23.C23
 Upper central C1 — C12 — C22×C12 — C3×C23.C23

Generators and relations for C3×C23.C23
G = < a,b,c,d,e,f,g | a3=b2=c2=d2=f2=1, e2=c, g2=d, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, bc=cb, ebe-1=bd=db, bf=fb, bg=gb, fcf=cd=dc, ce=ec, cg=gc, de=ed, df=fd, dg=gd, fef=bde, eg=ge, fg=gf >

Subgroups: 274 in 158 conjugacy classes, 78 normal (26 characteristic)
C1, C2, C2, C3, C4, C4, C4, C22, C22, C22, C6, C6, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C12, C12, C12, C2×C6, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C2×C12, C2×C12, C2×C12, C3×D4, C3×Q8, C22×C6, C22×C6, C23⋊C4, C42⋊C2, C2×C4○D4, C4×C12, C3×C22⋊C4, C3×C4⋊C4, C22×C12, C22×C12, C6×D4, C6×D4, C6×Q8, C3×C4○D4, C23.C23, C3×C23⋊C4, C3×C42⋊C2, C6×C4○D4, C3×C23.C23
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, D4, C23, C12, C2×C6, C22⋊C4, C22×C4, C2×D4, C2×C12, C3×D4, C22×C6, C2×C22⋊C4, C3×C22⋊C4, C22×C12, C6×D4, C23.C23, C6×C22⋊C4, C3×C23.C23

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

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

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

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

66 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 3A 3B 4A 4B 4C 4D 4E 4F ··· 4O 6A 6B 6C ··· 6H 6I 6J 6K 6L 12A 12B 12C 12D 12E ··· 12J 12K ··· 12AD order 1 2 2 2 2 2 2 3 3 4 4 4 4 4 4 ··· 4 6 6 6 ··· 6 6 6 6 6 12 12 12 12 12 ··· 12 12 ··· 12 size 1 1 2 2 2 4 4 1 1 1 1 2 2 2 4 ··· 4 1 1 2 ··· 2 4 4 4 4 1 1 1 1 2 ··· 2 4 ··· 4

66 irreducible representations

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

Matrix representation of C3×C23.C23 in GL6(𝔽13)

 3 0 0 0 0 0 0 3 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 12 0 0 0 0 0 0 12 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0
,
 12 0 0 0 0 0 0 12 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 12 0 0 0 0 0 0 12
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12
,
 8 0 0 0 0 0 0 5 0 0 0 0 0 0 1 0 0 0 0 0 0 12 0 0 0 0 0 0 0 12 0 0 0 0 1 0
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 1 0 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 5 0 0 0 0 0 0 5 0 0 0 0 0 0 5 0 0 0 0 0 0 5

G:=sub<GL(6,GF(13))| [3,0,0,0,0,0,0,3,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[8,0,0,0,0,0,0,5,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,12,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,5,0,0,0,0,0,0,5,0,0,0,0,0,0,5,0,0,0,0,0,0,5] >;

C3×C23.C23 in GAP, Magma, Sage, TeX

C_3\times C_2^3.C_2^3
% in TeX

G:=Group("C3xC2^3.C2^3");
// GroupNames label

G:=SmallGroup(192,843);
// by ID

G=gap.SmallGroup(192,843);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-2,336,365,520,4204,3036]);
// Polycyclic

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

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