metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C23.53D12, (C2×C8)⋊21D6, (C2×D12)⋊14C4, C4.13(D6⋊C4), (C2×C24)⋊36C22, D12.24(C2×C4), C2.D24⋊39C2, (C2×C12).173D4, (C2×C4).153D12, C12.417(C2×D4), C2.4(C8⋊D6), C6.20(C8⋊C22), (C6×M4(2))⋊19C2, (C2×M4(2))⋊11S3, C4⋊Dic3⋊48C22, (C22×C4).155D6, C22.57(C2×D12), (C22×C6).101D4, C12.27(C22⋊C4), (C2×C12).773C23, C12.115(C22×C4), C22.28(D6⋊C4), (C22×D12).15C2, C3⋊3(C23.37D4), (C2×D12).200C22, C23.26D6⋊16C2, (C22×C12).188C22, C4.73(S3×C2×C4), (C2×C4).53(C4×S3), C2.30(C2×D6⋊C4), (C2×C6).163(C2×D4), C4.110(C2×C3⋊D4), C6.58(C2×C22⋊C4), (C2×C12).108(C2×C4), (C2×C4).76(C3⋊D4), (C2×C6).21(C22⋊C4), (C2×C4).722(C22×S3), SmallGroup(192,690)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C23.53D12
G = < a,b,c,d,e | a2=b2=c2=1, d12=c, e2=cb=bc, ab=ba, dad-1=eae-1=ac=ca, bd=db, be=eb, cd=dc, ce=ec, ede-1=bd11 >
Subgroups: 696 in 190 conjugacy classes, 63 normal (23 characteristic)
C1, C2, C2, C2, C3, C4, C4, C4, C22, C22, C22, S3, C6, C6, C6, C8, C2×C4, C2×C4, C2×C4, D4, C23, C23, Dic3, C12, C12, D6, C2×C6, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C2×C8, M4(2), C22×C4, C2×D4, C24, C24, D12, D12, C2×Dic3, C2×C12, C2×C12, C22×S3, C22×C6, D4⋊C4, C42⋊C2, C2×M4(2), C22×D4, C4×Dic3, C4⋊Dic3, C6.D4, C2×C24, C3×M4(2), C2×D12, C2×D12, C22×C12, S3×C23, C23.37D4, C2.D24, C23.26D6, C6×M4(2), C22×D12, C23.53D12
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, D6, C22⋊C4, C22×C4, C2×D4, C4×S3, D12, C3⋊D4, C22×S3, C2×C22⋊C4, C8⋊C22, D6⋊C4, S3×C2×C4, C2×D12, C2×C3⋊D4, C23.37D4, C8⋊D6, C2×D6⋊C4, C23.53D12
►On 48 points
(2 14)(4 16)(6 18)(8 20)(10 22)(12 24)(25 37)(27 39)(29 41)(31 43)(33 45)(35 47)
(1 44)(2 45)(3 46)(4 47)(5 48)(6 25)(7 26)(8 27)(9 28)(10 29)(11 30)(12 31)(13 32)(14 33)(15 34)(16 35)(17 36)(18 37)(19 38)(20 39)(21 40)(22 41)(23 42)(24 43)
(1 13)(2 14)(3 15)(4 16)(5 17)(6 18)(7 19)(8 20)(9 21)(10 22)(11 23)(12 24)(25 37)(26 38)(27 39)(28 40)(29 41)(30 42)(31 43)(32 44)(33 45)(34 46)(35 47)(36 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 31 32 24)(2 23 33 30)(3 29 34 22)(4 21 35 28)(5 27 36 20)(6 19 37 26)(7 25 38 18)(8 17 39 48)(9 47 40 16)(10 15 41 46)(11 45 42 14)(12 13 43 44)
G:=sub<Sym(48)| (2,14)(4,16)(6,18)(8,20)(10,22)(12,24)(25,37)(27,39)(29,41)(31,43)(33,45)(35,47), (1,44)(2,45)(3,46)(4,47)(5,48)(6,25)(7,26)(8,27)(9,28)(10,29)(11,30)(12,31)(13,32)(14,33)(15,34)(16,35)(17,36)(18,37)(19,38)(20,39)(21,40)(22,41)(23,42)(24,43), (1,13)(2,14)(3,15)(4,16)(5,17)(6,18)(7,19)(8,20)(9,21)(10,22)(11,23)(12,24)(25,37)(26,38)(27,39)(28,40)(29,41)(30,42)(31,43)(32,44)(33,45)(34,46)(35,47)(36,48), (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), (1,31,32,24)(2,23,33,30)(3,29,34,22)(4,21,35,28)(5,27,36,20)(6,19,37,26)(7,25,38,18)(8,17,39,48)(9,47,40,16)(10,15,41,46)(11,45,42,14)(12,13,43,44)>;
G:=Group( (2,14)(4,16)(6,18)(8,20)(10,22)(12,24)(25,37)(27,39)(29,41)(31,43)(33,45)(35,47), (1,44)(2,45)(3,46)(4,47)(5,48)(6,25)(7,26)(8,27)(9,28)(10,29)(11,30)(12,31)(13,32)(14,33)(15,34)(16,35)(17,36)(18,37)(19,38)(20,39)(21,40)(22,41)(23,42)(24,43), (1,13)(2,14)(3,15)(4,16)(5,17)(6,18)(7,19)(8,20)(9,21)(10,22)(11,23)(12,24)(25,37)(26,38)(27,39)(28,40)(29,41)(30,42)(31,43)(32,44)(33,45)(34,46)(35,47)(36,48), (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), (1,31,32,24)(2,23,33,30)(3,29,34,22)(4,21,35,28)(5,27,36,20)(6,19,37,26)(7,25,38,18)(8,17,39,48)(9,47,40,16)(10,15,41,46)(11,45,42,14)(12,13,43,44) );
G=PermutationGroup([[(2,14),(4,16),(6,18),(8,20),(10,22),(12,24),(25,37),(27,39),(29,41),(31,43),(33,45),(35,47)], [(1,44),(2,45),(3,46),(4,47),(5,48),(6,25),(7,26),(8,27),(9,28),(10,29),(11,30),(12,31),(13,32),(14,33),(15,34),(16,35),(17,36),(18,37),(19,38),(20,39),(21,40),(22,41),(23,42),(24,43)], [(1,13),(2,14),(3,15),(4,16),(5,17),(6,18),(7,19),(8,20),(9,21),(10,22),(11,23),(12,24),(25,37),(26,38),(27,39),(28,40),(29,41),(30,42),(31,43),(32,44),(33,45),(34,46),(35,47),(36,48)], [(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)], [(1,31,32,24),(2,23,33,30),(3,29,34,22),(4,21,35,28),(5,27,36,20),(6,19,37,26),(7,25,38,18),(8,17,39,48),(9,47,40,16),(10,15,41,46),(11,45,42,14),(12,13,43,44)]])
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 6A | 6B | 6C | 6D | 6E | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | 12F | 24A | ··· | 24H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 2 | 2 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | ··· | 4 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C4 | S3 | D4 | D4 | D6 | D6 | C4×S3 | D12 | C3⋊D4 | D12 | C8⋊C22 | C8⋊D6 |
| kernel | C23.53D12 | C2.D24 | C23.26D6 | C6×M4(2) | C22×D12 | C2×D12 | C2×M4(2) | C2×C12 | C22×C6 | C2×C8 | C22×C4 | C2×C4 | C2×C4 | C2×C4 | C23 | C6 | C2 |
| # reps | 1 | 4 | 1 | 1 | 1 | 8 | 1 | 3 | 1 | 2 | 1 | 4 | 2 | 4 | 2 | 2 | 4 |
Matrix representation of C23.53D12 ►in GL6(𝔽73)
| 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 | 47 | 26 | 72 | 0 |
| 0 | 0 | 21 | 47 | 0 | 72 |
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 |
| 52 | 70 | 0 | 0 | 0 | 0 |
| 25 | 21 | 0 | 0 | 0 | 0 |
| 0 | 0 | 47 | 47 | 25 | 23 |
| 0 | 0 | 26 | 0 | 25 | 25 |
| 0 | 0 | 19 | 72 | 0 | 26 |
| 0 | 0 | 20 | 19 | 47 | 26 |
| 21 | 3 | 0 | 0 | 0 | 0 |
| 23 | 52 | 0 | 0 | 0 | 0 |
| 0 | 0 | 26 | 26 | 48 | 50 |
| 0 | 0 | 0 | 47 | 50 | 48 |
| 0 | 0 | 64 | 55 | 26 | 47 |
| 0 | 0 | 55 | 64 | 0 | 47 |
G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,47,21,0,0,0,1,26,47,0,0,0,0,72,0,0,0,0,0,0,72],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[52,25,0,0,0,0,70,21,0,0,0,0,0,0,47,26,19,20,0,0,47,0,72,19,0,0,25,25,0,47,0,0,23,25,26,26],[21,23,0,0,0,0,3,52,0,0,0,0,0,0,26,0,64,55,0,0,26,47,55,64,0,0,48,50,26,0,0,0,50,48,47,47] >;
C23.53D12 in GAP, Magma, Sage, TeX
C_2^3._{53}D_{12} % in TeX
G:=Group("C2^3.53D12"); // GroupNames label
G:=SmallGroup(192,690);
// by ID
G=gap.SmallGroup(192,690);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,254,387,142,1123,136,6278]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=1,d^12=c,e^2=c*b=b*c,a*b=b*a,d*a*d^-1=e*a*e^-1=a*c=c*a,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=b*d^11>;
// generators/relations