metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C12.37C24, D12.32C23, 2+ 1+4⋊8S3, Dic6.32C23, C4○D4⋊8D6, C3⋊5(D4○D8), D4○D12⋊9C2, (C2×D4)⋊16D6, (C3×D4).36D4, C3⋊C8.16C23, (C3×Q8).36D4, D4⋊D6⋊11C2, D4⋊S3⋊20C22, Q8.13D6⋊8C2, C12.269(C2×D4), (C6×D4)⋊24C22, C4.37(S3×C23), D12⋊6C22⋊11C2, D4.18(C3⋊D4), C4○D12⋊10C22, (C2×D12)⋊39C22, D4.Dic3⋊10C2, D4.S3⋊19C22, Q8.25(C3⋊D4), D4.25(C22×S3), C3⋊Q16⋊21C22, (C3×D4).25C23, C6.171(C22×D4), (C3×Q8).25C23, Q8.35(C22×S3), (C2×C12).118C23, Q8⋊2S3⋊19C22, C4.Dic3⋊16C22, (C3×2+ 1+4)⋊2C2, (C2×D4⋊S3)⋊32C2, (C2×C3⋊C8)⋊24C22, (C2×C6).85(C2×D4), C4.75(C2×C3⋊D4), (C3×C4○D4)⋊8C22, C22.6(C2×C3⋊D4), C2.44(C22×C3⋊D4), (C2×C4).102(C22×S3), SmallGroup(192,1394)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D12.32C23
G = < a,b,c,d,e | a12=b2=c2=d2=e2=1, bab=dad=a-1, ac=ca, eae=a7, cbc=a6b, dbd=a10b, ebe=a3b, cd=dc, ce=ec, ede=a9d >
Subgroups: 728 in 268 conjugacy classes, 107 normal (20 characteristic)
C1, C2, C2, C3, C4, C4, C4, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, Dic3, C12, C12, C12, D6, C2×C6, C2×C6, C2×C8, M4(2), D8, SD16, Q16, C2×D4, C2×D4, C4○D4, C4○D4, C4○D4, C3⋊C8, C3⋊C8, Dic6, C4×S3, D12, D12, C3⋊D4, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C22×S3, C22×C6, C8○D4, C2×D8, C4○D8, C8⋊C22, 2+ 1+4, 2+ 1+4, C2×C3⋊C8, C4.Dic3, D4⋊S3, D4.S3, Q8⋊2S3, C3⋊Q16, C2×D12, C4○D12, S3×D4, Q8⋊3S3, C6×D4, C6×D4, C3×C4○D4, C3×C4○D4, C3×C4○D4, D4○D8, C2×D4⋊S3, D12⋊6C22, D4.Dic3, D4⋊D6, Q8.13D6, D4○D12, C3×2+ 1+4, D12.32C23
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C24, C3⋊D4, C22×S3, C22×D4, C2×C3⋊D4, S3×C23, D4○D8, C22×C3⋊D4, D12.32C23
►On 48 points
(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 33)(2 32)(3 31)(4 30)(5 29)(6 28)(7 27)(8 26)(9 25)(10 36)(11 35)(12 34)(13 45)(14 44)(15 43)(16 42)(17 41)(18 40)(19 39)(20 38)(21 37)(22 48)(23 47)(24 46)
(25 31)(26 32)(27 33)(28 34)(29 35)(30 36)(37 43)(38 44)(39 45)(40 46)(41 47)(42 48)
(2 12)(3 11)(4 10)(5 9)(6 8)(13 22)(14 21)(15 20)(16 19)(17 18)(23 24)(25 31)(26 30)(27 29)(32 36)(33 35)(37 46)(38 45)(39 44)(40 43)(41 42)(47 48)
(1 19)(2 14)(3 21)(4 16)(5 23)(6 18)(7 13)(8 20)(9 15)(10 22)(11 17)(12 24)(25 40)(26 47)(27 42)(28 37)(29 44)(30 39)(31 46)(32 41)(33 48)(34 43)(35 38)(36 45)
G:=sub<Sym(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), (1,33)(2,32)(3,31)(4,30)(5,29)(6,28)(7,27)(8,26)(9,25)(10,36)(11,35)(12,34)(13,45)(14,44)(15,43)(16,42)(17,41)(18,40)(19,39)(20,38)(21,37)(22,48)(23,47)(24,46), (25,31)(26,32)(27,33)(28,34)(29,35)(30,36)(37,43)(38,44)(39,45)(40,46)(41,47)(42,48), (2,12)(3,11)(4,10)(5,9)(6,8)(13,22)(14,21)(15,20)(16,19)(17,18)(23,24)(25,31)(26,30)(27,29)(32,36)(33,35)(37,46)(38,45)(39,44)(40,43)(41,42)(47,48), (1,19)(2,14)(3,21)(4,16)(5,23)(6,18)(7,13)(8,20)(9,15)(10,22)(11,17)(12,24)(25,40)(26,47)(27,42)(28,37)(29,44)(30,39)(31,46)(32,41)(33,48)(34,43)(35,38)(36,45)>;
G:=Group( (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), (1,33)(2,32)(3,31)(4,30)(5,29)(6,28)(7,27)(8,26)(9,25)(10,36)(11,35)(12,34)(13,45)(14,44)(15,43)(16,42)(17,41)(18,40)(19,39)(20,38)(21,37)(22,48)(23,47)(24,46), (25,31)(26,32)(27,33)(28,34)(29,35)(30,36)(37,43)(38,44)(39,45)(40,46)(41,47)(42,48), (2,12)(3,11)(4,10)(5,9)(6,8)(13,22)(14,21)(15,20)(16,19)(17,18)(23,24)(25,31)(26,30)(27,29)(32,36)(33,35)(37,46)(38,45)(39,44)(40,43)(41,42)(47,48), (1,19)(2,14)(3,21)(4,16)(5,23)(6,18)(7,13)(8,20)(9,15)(10,22)(11,17)(12,24)(25,40)(26,47)(27,42)(28,37)(29,44)(30,39)(31,46)(32,41)(33,48)(34,43)(35,38)(36,45) );
G=PermutationGroup([[(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)], [(1,33),(2,32),(3,31),(4,30),(5,29),(6,28),(7,27),(8,26),(9,25),(10,36),(11,35),(12,34),(13,45),(14,44),(15,43),(16,42),(17,41),(18,40),(19,39),(20,38),(21,37),(22,48),(23,47),(24,46)], [(25,31),(26,32),(27,33),(28,34),(29,35),(30,36),(37,43),(38,44),(39,45),(40,46),(41,47),(42,48)], [(2,12),(3,11),(4,10),(5,9),(6,8),(13,22),(14,21),(15,20),(16,19),(17,18),(23,24),(25,31),(26,30),(27,29),(32,36),(33,35),(37,46),(38,45),(39,44),(40,43),(41,42),(47,48)], [(1,19),(2,14),(3,21),(4,16),(5,23),(6,18),(7,13),(8,20),(9,15),(10,22),(11,17),(12,24),(25,40),(26,47),(27,42),(28,37),(29,44),(30,39),(31,46),(32,41),(33,48),(34,43),(35,38),(36,45)]])
39 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 6A | 6B | ··· | 6J | 8A | 8B | 8C | 8D | 8E | 12A | ··· | 12F |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | ··· | 6 | 8 | 8 | 8 | 8 | 8 | 12 | ··· | 12 |
| size | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 12 | 12 | 12 | 2 | 2 | 2 | 2 | 2 | 4 | 12 | 2 | 4 | ··· | 4 | 6 | 6 | 12 | 12 | 12 | 4 | ··· | 4 |
39 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 8 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D6 | D6 | C3⋊D4 | C3⋊D4 | D4○D8 | D12.32C23 |
| kernel | D12.32C23 | C2×D4⋊S3 | D12⋊6C22 | D4.Dic3 | D4⋊D6 | Q8.13D6 | D4○D12 | C3×2+ 1+4 | 2+ 1+4 | C3×D4 | C3×Q8 | C2×D4 | C4○D4 | D4 | Q8 | C3 | C1 |
| # reps | 1 | 3 | 3 | 1 | 3 | 3 | 1 | 1 | 1 | 3 | 1 | 3 | 4 | 6 | 2 | 2 | 1 |
Matrix representation of D12.32C23 ►in GL6(𝔽73)
| 72 | 72 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 72 | 72 | 1 | 2 |
| 0 | 0 | 1 | 0 | 72 | 72 |
| 72 | 72 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 72 | 72 | 1 | 2 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 1 | 1 |
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 0 | 0 | 1 | 1 | 0 | 72 |
| 72 | 0 | 0 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 72 | 71 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 57 | 57 | 0 | 0 |
| 0 | 0 | 57 | 16 | 0 | 0 |
| 0 | 0 | 57 | 57 | 0 | 32 |
| 0 | 0 | 57 | 0 | 16 | 0 |
G:=sub<GL(6,GF(73))| [72,1,0,0,0,0,72,0,0,0,0,0,0,0,0,1,72,1,0,0,72,0,72,0,0,0,0,0,1,72,0,0,0,0,2,72],[72,0,0,0,0,0,72,1,0,0,0,0,0,0,0,72,1,72,0,0,0,72,0,0,0,0,1,1,0,1,0,0,0,2,0,1],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,1,0,0,0,1,0,1,0,0,0,0,72,0,0,0,0,0,0,72],[72,1,0,0,0,0,0,1,0,0,0,0,0,0,0,1,1,0,0,0,1,0,1,0,0,0,0,0,72,0,0,0,0,0,71,1],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,57,57,57,57,0,0,57,16,57,0,0,0,0,0,0,16,0,0,0,0,32,0] >;
D12.32C23 in GAP, Magma, Sage, TeX
D_{12}._{32}C_2^3 % in TeX
G:=Group("D12.32C2^3"); // GroupNames label
G:=SmallGroup(192,1394);
// by ID
G=gap.SmallGroup(192,1394);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,387,675,1684,235,102,6278]);
// Polycyclic
G:=Group<a,b,c,d,e|a^12=b^2=c^2=d^2=e^2=1,b*a*b=d*a*d=a^-1,a*c=c*a,e*a*e=a^7,c*b*c=a^6*b,d*b*d=a^10*b,e*b*e=a^3*b,c*d=d*c,c*e=e*c,e*d*e=a^9*d>;
// generators/relations