direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C3×2- 1+4, C6.20C24, C12.52C23, Q8○(C3×D4), D4○(C3×Q8), C4○D4⋊6C6, (C2×Q8)⋊7C6, (C6×Q8)⋊12C2, D4.4(C2×C6), Q8.7(C2×C6), (C2×C6).8C23, C2.5(C23×C6), C4.10(C22×C6), (C2×C12).71C22, (C3×D4).14C22, C22.3(C22×C6), (C3×Q8).15C22, (C3×D4)○(C3×Q8), (C3×C4○D4)⋊9C2, (C2×C4).12(C2×C6), SmallGroup(96,225)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×2- 1+4
G = < a,b,c,d,e | a3=b4=c2=1, d2=e2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede-1=b2d >
Subgroups: 156 in 146 conjugacy classes, 136 normal (6 characteristic)
C1, C2, C2, C3, C4, C22, C6, C6, C2×C4, D4, Q8, C12, C2×C6, C2×Q8, C4○D4, C2×C12, C3×D4, C3×Q8, 2- 1+4, C6×Q8, C3×C4○D4, C3×2- 1+4
Quotients: C1, C2, C3, C22, C6, C23, C2×C6, C24, C22×C6, 2- 1+4, C23×C6, C3×2- 1+4
►On 48 points
(1 16 11)(2 13 12)(3 14 9)(4 15 10)(5 42 45)(6 43 46)(7 44 47)(8 41 48)(17 23 28)(18 24 25)(19 21 26)(20 22 27)(29 35 37)(30 36 38)(31 33 39)(32 34 40)
(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 4)(2 3)(5 8)(6 7)(9 12)(10 11)(13 14)(15 16)(17 18)(19 20)(21 22)(23 24)(25 28)(26 27)(29 30)(31 32)(33 34)(35 36)(37 38)(39 40)(41 42)(43 44)(45 48)(46 47)
(1 24 3 22)(2 21 4 23)(5 40 7 38)(6 37 8 39)(9 20 11 18)(10 17 12 19)(13 26 15 28)(14 27 16 25)(29 41 31 43)(30 42 32 44)(33 46 35 48)(34 47 36 45)
(1 45 3 47)(2 46 4 48)(5 14 7 16)(6 15 8 13)(9 44 11 42)(10 41 12 43)(17 29 19 31)(18 30 20 32)(21 33 23 35)(22 34 24 36)(25 38 27 40)(26 39 28 37)
G:=sub<Sym(48)| (1,16,11)(2,13,12)(3,14,9)(4,15,10)(5,42,45)(6,43,46)(7,44,47)(8,41,48)(17,23,28)(18,24,25)(19,21,26)(20,22,27)(29,35,37)(30,36,38)(31,33,39)(32,34,40), (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), (1,4)(2,3)(5,8)(6,7)(9,12)(10,11)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,28)(26,27)(29,30)(31,32)(33,34)(35,36)(37,38)(39,40)(41,42)(43,44)(45,48)(46,47), (1,24,3,22)(2,21,4,23)(5,40,7,38)(6,37,8,39)(9,20,11,18)(10,17,12,19)(13,26,15,28)(14,27,16,25)(29,41,31,43)(30,42,32,44)(33,46,35,48)(34,47,36,45), (1,45,3,47)(2,46,4,48)(5,14,7,16)(6,15,8,13)(9,44,11,42)(10,41,12,43)(17,29,19,31)(18,30,20,32)(21,33,23,35)(22,34,24,36)(25,38,27,40)(26,39,28,37)>;
G:=Group( (1,16,11)(2,13,12)(3,14,9)(4,15,10)(5,42,45)(6,43,46)(7,44,47)(8,41,48)(17,23,28)(18,24,25)(19,21,26)(20,22,27)(29,35,37)(30,36,38)(31,33,39)(32,34,40), (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), (1,4)(2,3)(5,8)(6,7)(9,12)(10,11)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,28)(26,27)(29,30)(31,32)(33,34)(35,36)(37,38)(39,40)(41,42)(43,44)(45,48)(46,47), (1,24,3,22)(2,21,4,23)(5,40,7,38)(6,37,8,39)(9,20,11,18)(10,17,12,19)(13,26,15,28)(14,27,16,25)(29,41,31,43)(30,42,32,44)(33,46,35,48)(34,47,36,45), (1,45,3,47)(2,46,4,48)(5,14,7,16)(6,15,8,13)(9,44,11,42)(10,41,12,43)(17,29,19,31)(18,30,20,32)(21,33,23,35)(22,34,24,36)(25,38,27,40)(26,39,28,37) );
G=PermutationGroup([[(1,16,11),(2,13,12),(3,14,9),(4,15,10),(5,42,45),(6,43,46),(7,44,47),(8,41,48),(17,23,28),(18,24,25),(19,21,26),(20,22,27),(29,35,37),(30,36,38),(31,33,39),(32,34,40)], [(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)], [(1,4),(2,3),(5,8),(6,7),(9,12),(10,11),(13,14),(15,16),(17,18),(19,20),(21,22),(23,24),(25,28),(26,27),(29,30),(31,32),(33,34),(35,36),(37,38),(39,40),(41,42),(43,44),(45,48),(46,47)], [(1,24,3,22),(2,21,4,23),(5,40,7,38),(6,37,8,39),(9,20,11,18),(10,17,12,19),(13,26,15,28),(14,27,16,25),(29,41,31,43),(30,42,32,44),(33,46,35,48),(34,47,36,45)], [(1,45,3,47),(2,46,4,48),(5,14,7,16),(6,15,8,13),(9,44,11,42),(10,41,12,43),(17,29,19,31),(18,30,20,32),(21,33,23,35),(22,34,24,36),(25,38,27,40),(26,39,28,37)]])
C3×2- 1+4 is a maximal subgroup of
2- 1+4⋊4S3 2- 1+4.2S3 D12.34C23 D12.35C23 D12.39C23 2- 1+4⋊C9
C3×2- 1+4 is a maximal quotient of C3×D4×Q8
51 conjugacy classes
| class | 1 | 2A | 2B | ··· | 2F | 3A | 3B | 4A | ··· | 4J | 6A | 6B | 6C | ··· | 6L | 12A | ··· | 12T |
| order | 1 | 2 | 2 | ··· | 2 | 3 | 3 | 4 | ··· | 4 | 6 | 6 | 6 | ··· | 6 | 12 | ··· | 12 |
| size | 1 | 1 | 2 | ··· | 2 | 1 | 1 | 2 | ··· | 2 | 1 | 1 | 2 | ··· | 2 | 2 | ··· | 2 |
51 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 4 |
| type | + | + | + | - | ||||
| image | C1 | C2 | C2 | C3 | C6 | C6 | 2- 1+4 | C3×2- 1+4 |
| kernel | C3×2- 1+4 | C6×Q8 | C3×C4○D4 | 2- 1+4 | C2×Q8 | C4○D4 | C3 | C1 |
| # reps | 1 | 5 | 10 | 2 | 10 | 20 | 1 | 2 |
Matrix representation of C3×2- 1+4 ►in GL4(𝔽7) generated by
| 2 | 0 | 0 | 0 |
| 0 | 2 | 0 | 0 |
| 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 2 |
| 5 | 6 | 2 | 6 |
| 5 | 2 | 1 | 4 |
| 0 | 0 | 4 | 6 |
| 0 | 0 | 3 | 3 |
| 5 | 0 | 6 | 0 |
| 1 | 5 | 4 | 1 |
| 3 | 0 | 2 | 0 |
| 6 | 4 | 1 | 2 |
| 5 | 6 | 5 | 0 |
| 5 | 2 | 5 | 5 |
| 0 | 0 | 3 | 1 |
| 0 | 0 | 4 | 4 |
| 1 | 6 | 1 | 4 |
| 1 | 3 | 2 | 6 |
| 2 | 5 | 4 | 6 |
| 1 | 5 | 1 | 6 |
G:=sub<GL(4,GF(7))| [2,0,0,0,0,2,0,0,0,0,2,0,0,0,0,2],[5,5,0,0,6,2,0,0,2,1,4,3,6,4,6,3],[5,1,3,6,0,5,0,4,6,4,2,1,0,1,0,2],[5,5,0,0,6,2,0,0,5,5,3,4,0,5,1,4],[1,1,2,1,6,3,5,5,1,2,4,1,4,6,6,6] >;
C3×2- 1+4 in GAP, Magma, Sage, TeX
C_3\times 2_-^{1+4} % in TeX
G:=Group("C3xES-(2,2)"); // GroupNames label
G:=SmallGroup(96,225);
// by ID
G=gap.SmallGroup(96,225);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-3,-2,601,295,476,230,1347]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^4=c^2=1,d^2=e^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=b^2*d>;
// generators/relations