direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C3xC22.11C24, C6.1492+ 1+4, (C4xD4):5C6, D4:7(C2xC12), (C6xD4):23C4, C42:7(C2xC6), (C2xD4):11C12, (D4xC12):34C2, C23:3(C2xC12), C42:C2:6C6, (C4xC12):38C22, C6.59(C23xC4), C2.7(C23xC12), C24.12(C2xC6), (C2xC6).338C24, C4.19(C22xC12), (C22xC12):5C22, (C22xD4).10C6, (C2xC12).709C23, C12.164(C22xC4), (C6xD4).332C22, (C23xC6).11C22, C22.2(C22xC12), C23.34(C22xC6), C22.11(C23xC6), C2.1(C3x2+ 1+4), (C22xC6).254C23, C4:C4:20(C2xC6), (C2xC4):4(C2xC12), (D4xC2xC6).22C2, (C2xC12):25(C2xC4), (C3xD4):27(C2xC4), (C2xC22:C4):5C6, (C22xC6):5(C2xC4), (C22xC4):4(C2xC6), (C3xC4:C4):77C22, C22:C4:18(C2xC6), (C6xC22:C4):10C2, (C2xD4).78(C2xC6), (C2xC6).33(C22xC4), (C2xC4).56(C22xC6), (C3xC42:C2):27C2, (C3xC22:C4):72C22, SmallGroup(192,1407)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3xC22.11C24
G = < a,b,c,d,e,f,g | a3=b2=c2=e2=f2=g2=1, d2=c, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, bc=cb, ede=gdg=bd=db, fef=be=eb, bf=fb, bg=gb, cd=dc, ce=ec, cf=fc, cg=gc, df=fd, eg=ge, fg=gf >
Subgroups: 514 in 338 conjugacy classes, 242 normal (14 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C22, C6, C6, C6, C2xC4, C2xC4, D4, C23, C23, C23, C12, C12, C2xC6, C2xC6, C2xC6, C42, C22:C4, C4:C4, C22xC4, C22xC4, C2xD4, C24, C2xC12, C2xC12, C3xD4, C22xC6, C22xC6, C22xC6, C2xC22:C4, C42:C2, C4xD4, C22xD4, C4xC12, C3xC22:C4, C3xC4:C4, C22xC12, C22xC12, C6xD4, C23xC6, C22.11C24, C6xC22:C4, C3xC42:C2, D4xC12, D4xC2xC6, C3xC22.11C24
Quotients: C1, C2, C3, C4, C22, C6, C2xC4, C23, C12, C2xC6, C22xC4, C24, C2xC12, C22xC6, C23xC4, 2+ 1+4, C22xC12, C23xC6, C22.11C24, C23xC12, C3x2+ 1+4, C3xC22.11C24
(1 35 11)(2 36 12)(3 33 9)(4 34 10)(5 30 14)(6 31 15)(7 32 16)(8 29 13)(17 37 41)(18 38 42)(19 39 43)(20 40 44)(21 27 45)(22 28 46)(23 25 47)(24 26 48)
(1 39)(2 40)(3 37)(4 38)(5 28)(6 25)(7 26)(8 27)(9 17)(10 18)(11 19)(12 20)(13 21)(14 22)(15 23)(16 24)(29 45)(30 46)(31 47)(32 48)(33 41)(34 42)(35 43)(36 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 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)
(2 40)(4 38)(6 25)(8 27)(10 18)(12 20)(13 21)(15 23)(29 45)(31 47)(34 42)(36 44)
(1 15)(2 16)(3 13)(4 14)(5 34)(6 35)(7 36)(8 33)(9 29)(10 30)(11 31)(12 32)(17 45)(18 46)(19 47)(20 48)(21 37)(22 38)(23 39)(24 40)(25 43)(26 44)(27 41)(28 42)
(2 40)(4 38)(5 28)(7 26)(10 18)(12 20)(14 22)(16 24)(30 46)(32 48)(34 42)(36 44)
G:=sub<Sym(48)| (1,35,11)(2,36,12)(3,33,9)(4,34,10)(5,30,14)(6,31,15)(7,32,16)(8,29,13)(17,37,41)(18,38,42)(19,39,43)(20,40,44)(21,27,45)(22,28,46)(23,25,47)(24,26,48), (1,39)(2,40)(3,37)(4,38)(5,28)(6,25)(7,26)(8,27)(9,17)(10,18)(11,19)(12,20)(13,21)(14,22)(15,23)(16,24)(29,45)(30,46)(31,47)(32,48)(33,41)(34,42)(35,43)(36,44), (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), (1,2,3,4)(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), (2,40)(4,38)(6,25)(8,27)(10,18)(12,20)(13,21)(15,23)(29,45)(31,47)(34,42)(36,44), (1,15)(2,16)(3,13)(4,14)(5,34)(6,35)(7,36)(8,33)(9,29)(10,30)(11,31)(12,32)(17,45)(18,46)(19,47)(20,48)(21,37)(22,38)(23,39)(24,40)(25,43)(26,44)(27,41)(28,42), (2,40)(4,38)(5,28)(7,26)(10,18)(12,20)(14,22)(16,24)(30,46)(32,48)(34,42)(36,44)>;
G:=Group( (1,35,11)(2,36,12)(3,33,9)(4,34,10)(5,30,14)(6,31,15)(7,32,16)(8,29,13)(17,37,41)(18,38,42)(19,39,43)(20,40,44)(21,27,45)(22,28,46)(23,25,47)(24,26,48), (1,39)(2,40)(3,37)(4,38)(5,28)(6,25)(7,26)(8,27)(9,17)(10,18)(11,19)(12,20)(13,21)(14,22)(15,23)(16,24)(29,45)(30,46)(31,47)(32,48)(33,41)(34,42)(35,43)(36,44), (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), (1,2,3,4)(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), (2,40)(4,38)(6,25)(8,27)(10,18)(12,20)(13,21)(15,23)(29,45)(31,47)(34,42)(36,44), (1,15)(2,16)(3,13)(4,14)(5,34)(6,35)(7,36)(8,33)(9,29)(10,30)(11,31)(12,32)(17,45)(18,46)(19,47)(20,48)(21,37)(22,38)(23,39)(24,40)(25,43)(26,44)(27,41)(28,42), (2,40)(4,38)(5,28)(7,26)(10,18)(12,20)(14,22)(16,24)(30,46)(32,48)(34,42)(36,44) );
G=PermutationGroup([[(1,35,11),(2,36,12),(3,33,9),(4,34,10),(5,30,14),(6,31,15),(7,32,16),(8,29,13),(17,37,41),(18,38,42),(19,39,43),(20,40,44),(21,27,45),(22,28,46),(23,25,47),(24,26,48)], [(1,39),(2,40),(3,37),(4,38),(5,28),(6,25),(7,26),(8,27),(9,17),(10,18),(11,19),(12,20),(13,21),(14,22),(15,23),(16,24),(29,45),(30,46),(31,47),(32,48),(33,41),(34,42),(35,43),(36,44)], [(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)], [(1,2,3,4),(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)], [(2,40),(4,38),(6,25),(8,27),(10,18),(12,20),(13,21),(15,23),(29,45),(31,47),(34,42),(36,44)], [(1,15),(2,16),(3,13),(4,14),(5,34),(6,35),(7,36),(8,33),(9,29),(10,30),(11,31),(12,32),(17,45),(18,46),(19,47),(20,48),(21,37),(22,38),(23,39),(24,40),(25,43),(26,44),(27,41),(28,42)], [(2,40),(4,38),(5,28),(7,26),(10,18),(12,20),(14,22),(16,24),(30,46),(32,48),(34,42),(36,44)]])
102 conjugacy classes
class | 1 | 2A | 2B | 2C | 2D | ··· | 2M | 3A | 3B | 4A | ··· | 4T | 6A | ··· | 6F | 6G | ··· | 6Z | 12A | ··· | 12AN |
order | 1 | 2 | 2 | 2 | 2 | ··· | 2 | 3 | 3 | 4 | ··· | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 12 | ··· | 12 |
size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 |
102 irreducible representations
dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 4 |
type | + | + | + | + | + | + | ||||||||
image | C1 | C2 | C2 | C2 | C2 | C3 | C4 | C6 | C6 | C6 | C6 | C12 | 2+ 1+4 | C3x2+ 1+4 |
kernel | C3xC22.11C24 | C6xC22:C4 | C3xC42:C2 | D4xC12 | D4xC2xC6 | C22.11C24 | C6xD4 | C2xC22:C4 | C42:C2 | C4xD4 | C22xD4 | C2xD4 | C6 | C2 |
# reps | 1 | 4 | 2 | 8 | 1 | 2 | 16 | 8 | 4 | 16 | 2 | 32 | 2 | 4 |
Matrix representation of C3xC22.11C24 ►in GL5(F13)
3 | 0 | 0 | 0 | 0 |
0 | 9 | 0 | 0 | 0 |
0 | 0 | 9 | 0 | 0 |
0 | 0 | 0 | 9 | 0 |
0 | 0 | 0 | 0 | 9 |
1 | 0 | 0 | 0 | 0 |
0 | 12 | 0 | 0 | 0 |
0 | 0 | 12 | 0 | 0 |
0 | 0 | 0 | 12 | 0 |
0 | 0 | 0 | 0 | 12 |
12 | 0 | 0 | 0 | 0 |
0 | 1 | 0 | 0 | 0 |
0 | 0 | 1 | 0 | 0 |
0 | 0 | 0 | 1 | 0 |
0 | 0 | 0 | 0 | 1 |
5 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 1 | 0 |
0 | 0 | 0 | 0 | 1 |
0 | 1 | 0 | 0 | 0 |
0 | 0 | 1 | 0 | 0 |
12 | 0 | 0 | 0 | 0 |
0 | 1 | 0 | 0 | 0 |
0 | 0 | 12 | 0 | 0 |
0 | 0 | 0 | 12 | 0 |
0 | 0 | 0 | 0 | 1 |
1 | 0 | 0 | 0 | 0 |
0 | 0 | 1 | 0 | 0 |
0 | 1 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 1 |
0 | 0 | 0 | 1 | 0 |
1 | 0 | 0 | 0 | 0 |
0 | 1 | 0 | 0 | 0 |
0 | 0 | 1 | 0 | 0 |
0 | 0 | 0 | 12 | 0 |
0 | 0 | 0 | 0 | 12 |
G:=sub<GL(5,GF(13))| [3,0,0,0,0,0,9,0,0,0,0,0,9,0,0,0,0,0,9,0,0,0,0,0,9],[1,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12],[12,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[5,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,1,0,0,0,0,0,1,0,0],[12,0,0,0,0,0,1,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,1,0],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,12,0,0,0,0,0,12] >;
C3xC22.11C24 in GAP, Magma, Sage, TeX
C_3\times C_2^2._{11}C_2^4
% in TeX
G:=Group("C3xC2^2.11C2^4");
// GroupNames label
G:=SmallGroup(192,1407);
// by ID
G=gap.SmallGroup(192,1407);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-2,-2,672,701,555,1571]);
// Polycyclic
G:=Group<a,b,c,d,e,f,g|a^3=b^2=c^2=e^2=f^2=g^2=1,d^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=g*d*g=b*d=d*b,f*e*f=b*e=e*b,b*f=f*b,b*g=g*b,c*d=d*c,c*e=e*c,c*f=f*c,c*g=g*c,d*f=f*d,e*g=g*e,f*g=g*f>;
// generators/relations