direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C3×C24⋊3C4, C24⋊7C12, C25.5C6, (C23×C6)⋊5C4, (C24×C6).1C2, C6.87C22≀C2, C24.29(C2×C6), C23.38(C3×D4), C22.29(C6×D4), C23.29(C2×C12), (C22×C12)⋊3C22, (C22×C6).153D4, (C23×C6).83C22, C23.56(C22×C6), (C22×C6).443C23, C22.28(C22×C12), (C2×C22⋊C4)⋊1C6, (C6×C22⋊C4)⋊5C2, (C22×C4)⋊2(C2×C6), C2.4(C6×C22⋊C4), (C2×C6)⋊7(C22⋊C4), C2.1(C3×C22≀C2), (C2×C6).596(C2×D4), C6.91(C2×C22⋊C4), C22⋊3(C3×C22⋊C4), (C22×C6).110(C2×C4), (C2×C6).215(C22×C4), SmallGroup(192,812)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C24⋊3C4
G = < a,b,c,d,e,f | a3=b2=c2=d2=e2=f4=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, fbf-1=bd=db, be=eb, cd=dc, fcf-1=ce=ec, de=ed, df=fd, ef=fe >
Subgroups: 898 in 506 conjugacy classes, 130 normal (10 characteristic)
C1, C2 [×7], C2 [×12], C3, C4 [×4], C22, C22 [×18], C22 [×68], C6 [×7], C6 [×12], C2×C4 [×12], C23, C23 [×18], C23 [×68], C12 [×4], C2×C6, C2×C6 [×18], C2×C6 [×68], C22⋊C4 [×12], C22×C4 [×4], C24 [×7], C24 [×12], C2×C12 [×12], C22×C6, C22×C6 [×18], C22×C6 [×68], C2×C22⋊C4 [×6], C25, C3×C22⋊C4 [×12], C22×C12 [×4], C23×C6 [×7], C23×C6 [×12], C24⋊3C4, C6×C22⋊C4 [×6], C24×C6, C3×C24⋊3C4
Quotients: C1, C2 [×7], C3, C4 [×4], C22 [×7], C6 [×7], C2×C4 [×6], D4 [×12], C23, C12 [×4], C2×C6 [×7], C22⋊C4 [×12], C22×C4, C2×D4 [×6], C2×C12 [×6], C3×D4 [×12], C22×C6, C2×C22⋊C4 [×3], C22≀C2 [×4], C3×C22⋊C4 [×12], C22×C12, C6×D4 [×6], C24⋊3C4, C6×C22⋊C4 [×3], C3×C22≀C2 [×4], C3×C24⋊3C4
(1 31 35)(2 32 36)(3 29 33)(4 30 34)(5 14 10)(6 15 11)(7 16 12)(8 13 9)(17 27 21)(18 28 22)(19 25 23)(20 26 24)(37 45 41)(38 46 42)(39 47 43)(40 48 44)
(2 12)(4 10)(5 30)(7 32)(14 34)(16 36)(18 38)(20 40)(22 42)(24 44)(26 48)(28 46)
(2 40)(4 38)(5 28)(7 26)(10 18)(12 20)(14 22)(16 24)(30 46)(32 48)(34 42)(36 44)
(1 11)(2 12)(3 9)(4 10)(5 30)(6 31)(7 32)(8 29)(13 33)(14 34)(15 35)(16 36)(17 37)(18 38)(19 39)(20 40)(21 41)(22 42)(23 43)(24 44)(25 47)(26 48)(27 45)(28 46)
(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 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)| (1,31,35)(2,32,36)(3,29,33)(4,30,34)(5,14,10)(6,15,11)(7,16,12)(8,13,9)(17,27,21)(18,28,22)(19,25,23)(20,26,24)(37,45,41)(38,46,42)(39,47,43)(40,48,44), (2,12)(4,10)(5,30)(7,32)(14,34)(16,36)(18,38)(20,40)(22,42)(24,44)(26,48)(28,46), (2,40)(4,38)(5,28)(7,26)(10,18)(12,20)(14,22)(16,24)(30,46)(32,48)(34,42)(36,44), (1,11)(2,12)(3,9)(4,10)(5,30)(6,31)(7,32)(8,29)(13,33)(14,34)(15,35)(16,36)(17,37)(18,38)(19,39)(20,40)(21,41)(22,42)(23,43)(24,44)(25,47)(26,48)(27,45)(28,46), (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), (1,2,3,4)(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( (1,31,35)(2,32,36)(3,29,33)(4,30,34)(5,14,10)(6,15,11)(7,16,12)(8,13,9)(17,27,21)(18,28,22)(19,25,23)(20,26,24)(37,45,41)(38,46,42)(39,47,43)(40,48,44), (2,12)(4,10)(5,30)(7,32)(14,34)(16,36)(18,38)(20,40)(22,42)(24,44)(26,48)(28,46), (2,40)(4,38)(5,28)(7,26)(10,18)(12,20)(14,22)(16,24)(30,46)(32,48)(34,42)(36,44), (1,11)(2,12)(3,9)(4,10)(5,30)(6,31)(7,32)(8,29)(13,33)(14,34)(15,35)(16,36)(17,37)(18,38)(19,39)(20,40)(21,41)(22,42)(23,43)(24,44)(25,47)(26,48)(27,45)(28,46), (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), (1,2,3,4)(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([(1,31,35),(2,32,36),(3,29,33),(4,30,34),(5,14,10),(6,15,11),(7,16,12),(8,13,9),(17,27,21),(18,28,22),(19,25,23),(20,26,24),(37,45,41),(38,46,42),(39,47,43),(40,48,44)], [(2,12),(4,10),(5,30),(7,32),(14,34),(16,36),(18,38),(20,40),(22,42),(24,44),(26,48),(28,46)], [(2,40),(4,38),(5,28),(7,26),(10,18),(12,20),(14,22),(16,24),(30,46),(32,48),(34,42),(36,44)], [(1,11),(2,12),(3,9),(4,10),(5,30),(6,31),(7,32),(8,29),(13,33),(14,34),(15,35),(16,36),(17,37),(18,38),(19,39),(20,40),(21,41),(22,42),(23,43),(24,44),(25,47),(26,48),(27,45),(28,46)], [(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)], [(1,2,3,4),(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 | ··· | 2G | 2H | ··· | 2S | 3A | 3B | 4A | ··· | 4H | 6A | ··· | 6N | 6O | ··· | 6AL | 12A | ··· | 12P |
order | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 3 | 3 | 4 | ··· | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 12 | ··· | 12 |
size | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | 1 | 4 | ··· | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 |
84 irreducible representations
dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 |
type | + | + | + | + | ||||||
image | C1 | C2 | C2 | C3 | C4 | C6 | C6 | C12 | D4 | C3×D4 |
kernel | C3×C24⋊3C4 | C6×C22⋊C4 | C24×C6 | C24⋊3C4 | C23×C6 | C2×C22⋊C4 | C25 | C24 | C22×C6 | C23 |
# reps | 1 | 6 | 1 | 2 | 8 | 12 | 2 | 16 | 12 | 24 |
Matrix representation of C3×C24⋊3C4 ►in GL6(𝔽13)
1 | 0 | 0 | 0 | 0 | 0 |
0 | 1 | 0 | 0 | 0 | 0 |
0 | 0 | 9 | 0 | 0 | 0 |
0 | 0 | 0 | 9 | 0 | 0 |
0 | 0 | 0 | 0 | 9 | 0 |
0 | 0 | 0 | 0 | 0 | 9 |
1 | 0 | 0 | 0 | 0 | 0 |
0 | 12 | 0 | 0 | 0 | 0 |
0 | 0 | 1 | 0 | 0 | 0 |
0 | 0 | 0 | 12 | 0 | 0 |
0 | 0 | 0 | 0 | 1 | 0 |
0 | 0 | 0 | 0 | 0 | 1 |
1 | 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 | 0 |
0 | 0 | 0 | 0 | 1 | 12 |
12 | 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 | 1 | 0 |
0 | 0 | 0 | 0 | 0 | 1 |
1 | 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 | 12 | 0 |
0 | 0 | 0 | 0 | 0 | 12 |
0 | 1 | 0 | 0 | 0 | 0 |
1 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 1 | 0 | 0 |
0 | 0 | 12 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 1 | 11 |
0 | 0 | 0 | 0 | 0 | 12 |
G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,9,0,0,0,0,0,0,9,0,0,0,0,0,0,9,0,0,0,0,0,0,9],[1,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,0,12],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,11,12] >;
C3×C24⋊3C4 in GAP, Magma, Sage, TeX
C_3\times C_2^4\rtimes_3C_4
% in TeX
G:=Group("C3xC2^4:3C4");
// GroupNames label
G:=SmallGroup(192,812);
// by ID
G=gap.SmallGroup(192,812);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-2,672,365,1094]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^3=b^2=c^2=d^2=e^2=f^4=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,f*b*f^-1=b*d=d*b,b*e=e*b,c*d=d*c,f*c*f^-1=c*e=e*c,d*e=e*d,d*f=f*d,e*f=f*e>;
// generators/relations