metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C23.16D6, (C2×C4).26D6, Dic3⋊C4⋊7C2, (C4×Dic3)⋊9C2, (C2×Dic3)⋊3C4, C22⋊C4.3S3, C6.5(C22×C4), C22.6(C4×S3), C3⋊2(C42⋊C2), C6.20(C4○D4), (C2×C6).18C23, Dic3○(C22⋊C4), Dic3.8(C2×C4), C2.1(D4⋊2S3), (C2×C12).50C22, C6.D4.1C2, (C22×C6).7C22, C22.12(C22×S3), (C22×Dic3).2C2, (C2×Dic3).46C22, C2.7(S3×C2×C4), (C2×C6).4(C2×C4), (C3×C22⋊C4).3C2, SmallGroup(96,84)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C23.16D6
G = < a,b,c,d,e | a2=b2=c2=1, d6=b, e2=cb=bc, ab=ba, dad-1=eae-1=ac=ca, bd=db, be=eb, cd=dc, ce=ec, ede-1=d5 >
Subgroups: 138 in 76 conjugacy classes, 41 normal (15 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, C6, C6, C6, C2×C4, C2×C4, C23, Dic3, Dic3, C12, C2×C6, C2×C6, C2×C6, C42, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C2×Dic3, C2×Dic3, C2×C12, C22×C6, C42⋊C2, C4×Dic3, Dic3⋊C4, C6.D4, C3×C22⋊C4, C22×Dic3, C23.16D6
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, D6, C22×C4, C4○D4, C4×S3, C22×S3, C42⋊C2, S3×C2×C4, D4⋊2S3, C23.16D6
Character table of C23.16D6
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 6A | 6B | 6C | 6D | 6E | 12A | 12B | 12C | 12D | |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 6 | 6 | 6 | 6 | 6 | 6 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | linear of order 2 |
| ρ3 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | -1 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | linear of order 2 |
| ρ4 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 2 |
| ρ5 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | linear of order 2 |
| ρ6 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ7 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ8 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | linear of order 2 |
| ρ9 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | i | -i | i | -i | -1 | -1 | 1 | 1 | i | -i | 1 | -1 | -i | i | -1 | -1 | 1 | 1 | -1 | i | -i | -i | i | linear of order 4 |
| ρ10 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | i | -i | -i | i | -1 | -1 | 1 | 1 | -i | -i | -1 | 1 | i | i | -1 | -1 | 1 | -1 | 1 | i | -i | i | -i | linear of order 4 |
| ρ11 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | i | -i | -i | i | 1 | 1 | -1 | -1 | i | i | 1 | -1 | -i | -i | -1 | -1 | 1 | -1 | 1 | i | -i | i | -i | linear of order 4 |
| ρ12 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | i | -i | i | -i | 1 | 1 | -1 | -1 | -i | i | -1 | 1 | i | -i | -1 | -1 | 1 | 1 | -1 | i | -i | -i | i | linear of order 4 |
| ρ13 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | -i | i | i | -i | -1 | -1 | 1 | 1 | i | i | -1 | 1 | -i | -i | -1 | -1 | 1 | -1 | 1 | -i | i | -i | i | linear of order 4 |
| ρ14 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | -i | i | -i | i | -1 | -1 | 1 | 1 | -i | i | 1 | -1 | i | -i | -1 | -1 | 1 | 1 | -1 | -i | i | i | -i | linear of order 4 |
| ρ15 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | -i | i | -i | i | 1 | 1 | -1 | -1 | i | -i | -1 | 1 | -i | i | -1 | -1 | 1 | 1 | -1 | -i | i | i | -i | linear of order 4 |
| ρ16 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | -i | i | i | -i | 1 | 1 | -1 | -1 | -i | -i | 1 | -1 | i | i | -1 | -1 | 1 | -1 | 1 | -i | i | -i | i | linear of order 4 |
| ρ17 | 2 | 2 | 2 | 2 | 2 | 2 | -1 | -2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | orthogonal lifted from D6 |
| ρ18 | 2 | 2 | 2 | 2 | 2 | 2 | -1 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ19 | 2 | 2 | 2 | 2 | -2 | -2 | -1 | -2 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | orthogonal lifted from D6 |
| ρ20 | 2 | 2 | 2 | 2 | -2 | -2 | -1 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | orthogonal lifted from D6 |
| ρ21 | 2 | 2 | -2 | -2 | -2 | 2 | -1 | 2i | -2i | 2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | -1 | 1 | -i | i | i | -i | complex lifted from C4×S3 |
| ρ22 | 2 | 2 | -2 | -2 | -2 | 2 | -1 | -2i | 2i | -2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | -1 | 1 | i | -i | -i | i | complex lifted from C4×S3 |
| ρ23 | 2 | -2 | -2 | 2 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | -2i | 2i | 2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ24 | 2 | -2 | 2 | -2 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 2i | -2i | 2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ25 | 2 | 2 | -2 | -2 | 2 | -2 | -1 | 2i | -2i | -2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | 1 | -1 | -i | i | -i | i | complex lifted from C4×S3 |
| ρ26 | 2 | 2 | -2 | -2 | 2 | -2 | -1 | -2i | 2i | 2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | 1 | -1 | i | -i | i | -i | complex lifted from C4×S3 |
| ρ27 | 2 | -2 | 2 | -2 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | -2i | 2i | -2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ28 | 2 | -2 | -2 | 2 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 2i | -2i | -2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ29 | 4 | -4 | -4 | 4 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from D4⋊2S3, Schur index 2 |
| ρ30 | 4 | -4 | 4 | -4 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from D4⋊2S3, Schur index 2 |
►On 48 points
(2 31)(4 33)(6 35)(8 25)(10 27)(12 29)(14 43)(16 45)(18 47)(20 37)(22 39)(24 41)
(1 7)(2 8)(3 9)(4 10)(5 11)(6 12)(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)(25 31)(26 32)(27 33)(28 34)(29 35)(30 36)(37 43)(38 44)(39 45)(40 46)(41 47)(42 48)
(1 30)(2 31)(3 32)(4 33)(5 34)(6 35)(7 36)(8 25)(9 26)(10 27)(11 28)(12 29)(13 42)(14 43)(15 44)(16 45)(17 46)(18 47)(19 48)(20 37)(21 38)(22 39)(23 40)(24 41)
(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 24 36 47)(2 17 25 40)(3 22 26 45)(4 15 27 38)(5 20 28 43)(6 13 29 48)(7 18 30 41)(8 23 31 46)(9 16 32 39)(10 21 33 44)(11 14 34 37)(12 19 35 42)
G:=sub<Sym(48)| (2,31)(4,33)(6,35)(8,25)(10,27)(12,29)(14,43)(16,45)(18,47)(20,37)(22,39)(24,41), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24)(25,31)(26,32)(27,33)(28,34)(29,35)(30,36)(37,43)(38,44)(39,45)(40,46)(41,47)(42,48), (1,30)(2,31)(3,32)(4,33)(5,34)(6,35)(7,36)(8,25)(9,26)(10,27)(11,28)(12,29)(13,42)(14,43)(15,44)(16,45)(17,46)(18,47)(19,48)(20,37)(21,38)(22,39)(23,40)(24,41), (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), (1,24,36,47)(2,17,25,40)(3,22,26,45)(4,15,27,38)(5,20,28,43)(6,13,29,48)(7,18,30,41)(8,23,31,46)(9,16,32,39)(10,21,33,44)(11,14,34,37)(12,19,35,42)>;
G:=Group( (2,31)(4,33)(6,35)(8,25)(10,27)(12,29)(14,43)(16,45)(18,47)(20,37)(22,39)(24,41), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24)(25,31)(26,32)(27,33)(28,34)(29,35)(30,36)(37,43)(38,44)(39,45)(40,46)(41,47)(42,48), (1,30)(2,31)(3,32)(4,33)(5,34)(6,35)(7,36)(8,25)(9,26)(10,27)(11,28)(12,29)(13,42)(14,43)(15,44)(16,45)(17,46)(18,47)(19,48)(20,37)(21,38)(22,39)(23,40)(24,41), (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), (1,24,36,47)(2,17,25,40)(3,22,26,45)(4,15,27,38)(5,20,28,43)(6,13,29,48)(7,18,30,41)(8,23,31,46)(9,16,32,39)(10,21,33,44)(11,14,34,37)(12,19,35,42) );
G=PermutationGroup([[(2,31),(4,33),(6,35),(8,25),(10,27),(12,29),(14,43),(16,45),(18,47),(20,37),(22,39),(24,41)], [(1,7),(2,8),(3,9),(4,10),(5,11),(6,12),(13,19),(14,20),(15,21),(16,22),(17,23),(18,24),(25,31),(26,32),(27,33),(28,34),(29,35),(30,36),(37,43),(38,44),(39,45),(40,46),(41,47),(42,48)], [(1,30),(2,31),(3,32),(4,33),(5,34),(6,35),(7,36),(8,25),(9,26),(10,27),(11,28),(12,29),(13,42),(14,43),(15,44),(16,45),(17,46),(18,47),(19,48),(20,37),(21,38),(22,39),(23,40),(24,41)], [(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)], [(1,24,36,47),(2,17,25,40),(3,22,26,45),(4,15,27,38),(5,20,28,43),(6,13,29,48),(7,18,30,41),(8,23,31,46),(9,16,32,39),(10,21,33,44),(11,14,34,37),(12,19,35,42)]])
C23.16D6 is a maximal subgroup of
C23⋊C4⋊5S3 C24.35D6 C24.42D6 C42.87D6 S3×C42⋊C2 C42.96D6 C4×D4⋊2S3 C42.105D6 C42.108D6 C42⋊18D6 C24.67D6 C24.43D6 C24.45D6 C4⋊C4.178D6 C6.342+ 1+4 C6.442+ 1+4 C6.452+ 1+4 (Q8×Dic3)⋊C2 C6.752- 1+4 C6.532+ 1+4 C6.772- 1+4 C6.792- 1+4 C4⋊C4.197D6 C6.802- 1+4 C4⋊C4⋊28D6 C6.642+ 1+4 C6.652+ 1+4 C42.137D6 C42.138D6 C42.139D6 C42.234D6 C42.159D6 C42.160D6 C42.189D6 C42.162D6 C23.16D18 C62.47C23 C62.48C23 C62.97C23 C62.99C23 C62.221C23 (D5×Dic3)⋊C4 D10.19(C4×S3) C23.26(S3×D5) C23.48(S3×D5) C23.15D30 C22⋊F5.S3
C23.16D6 is a maximal quotient of
Dic3.5C42 Dic3⋊C42 C3⋊(C42⋊8C4) C3⋊(C42⋊5C4) C2.(C4×D12) C2.(C4×Dic6) Dic3.5M4(2) Dic3.M4(2) C24⋊C4⋊C2 Dic3×C22⋊C4 C24.55D6 C24.56D6 C24.14D6 C24.15D6 C23.16D18 C62.47C23 C62.48C23 C62.97C23 C62.99C23 C62.221C23 (D5×Dic3)⋊C4 D10.19(C4×S3) C23.26(S3×D5) C23.48(S3×D5) C23.15D30 C22⋊F5.S3
Matrix representation of C23.16D6 ►in GL6(𝔽13)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 12 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 5 | 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 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 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 |
| 12 | 11 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 11 | 0 | 0 |
| 0 | 0 | 0 | 8 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 12 | 1 |
| 8 | 3 | 0 | 0 | 0 | 0 |
| 0 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 3 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 1 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(13))| [1,12,0,0,0,0,0,12,0,0,0,0,0,0,1,5,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[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],[12,0,0,0,0,0,11,1,0,0,0,0,0,0,5,0,0,0,0,0,11,8,0,0,0,0,0,0,0,12,0,0,0,0,1,1],[8,0,0,0,0,0,3,5,0,0,0,0,0,0,12,0,0,0,0,0,3,1,0,0,0,0,0,0,12,0,0,0,0,0,1,1] >;
C23.16D6 in GAP, Magma, Sage, TeX
C_2^3._{16}D_6 % in TeX
G:=Group("C2^3.16D6"); // GroupNames label
G:=SmallGroup(96,84);
// by ID
G=gap.SmallGroup(96,84);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-3,96,188,50,2309]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=1,d^6=b,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=d^5>;
// generators/relations
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