metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C23.28D6, D6⋊C4⋊2C2, (C22×C4)⋊5S3, (C2×C4).65D6, C6.42(C2×D4), (C2×C6).37D4, Dic3⋊C4⋊3C2, (C22×C12)⋊2C2, C6.18(C4○D4), C6.D4⋊6C2, (C2×C6).47C23, C2.18(C4○D12), (C2×C12).78C22, C3⋊4(C22.D4), C22.9(C3⋊D4), (C22×S3).9C22, C22.55(C22×S3), (C22×C6).39C22, (C2×Dic3).15C22, C2.6(C2×C3⋊D4), (C2×C3⋊D4).6C2, SmallGroup(96,136)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C23.28D6
G = < a,b,c,d,e | a2=b2=c2=1, d6=c, e2=cb=bc, ab=ba, eae-1=ac=ca, ad=da, bd=db, be=eb, cd=dc, ce=ec, ede-1=bd5 >
Subgroups: 178 in 78 conjugacy classes, 33 normal (15 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, S3, C6, C6, C6, C2×C4, C2×C4, D4, C23, C23, Dic3, C12, D6, C2×C6, C2×C6, C2×C6, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C22×S3, C22×C6, C22.D4, Dic3⋊C4, D6⋊C4, C6.D4, C2×C3⋊D4, C22×C12, C23.28D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, C3⋊D4, C22×S3, C22.D4, C4○D12, C2×C3⋊D4, C23.28D6
Character table of C23.28D6
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 12 | 2 | 2 | 2 | 2 | 2 | 12 | 12 | 12 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | |
| 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 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | -1 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ10 | 2 | 2 | 2 | 2 | -2 | -2 | 0 | -1 | -2 | 2 | 2 | -2 | 0 | 0 | 0 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | -1 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | orthogonal lifted from D6 |
| ρ11 | 2 | 2 | 2 | 2 | -2 | -2 | 0 | -1 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | 1 | -1 | -1 | -1 | 1 | orthogonal lifted from D6 |
| ρ12 | 2 | -2 | -2 | 2 | -2 | 2 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | -2 | 2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ13 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | -1 | -2 | -2 | -2 | -2 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | orthogonal lifted from D6 |
| ρ14 | 2 | -2 | -2 | 2 | 2 | -2 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | -2 | -2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ15 | 2 | -2 | -2 | 2 | 2 | -2 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | 1 | 1 | 1 | -1 | 1 | -√-3 | -√-3 | √-3 | -√-3 | √-3 | √-3 | -√-3 | √-3 | complex lifted from C3⋊D4 |
| ρ16 | 2 | -2 | -2 | 2 | -2 | 2 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | -√-3 | √-3 | √-3 | -√-3 | -√-3 | -√-3 | √-3 | √-3 | complex lifted from C3⋊D4 |
| ρ17 | 2 | -2 | -2 | 2 | -2 | 2 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | √-3 | -√-3 | -√-3 | √-3 | √-3 | √-3 | -√-3 | -√-3 | complex lifted from C3⋊D4 |
| ρ18 | 2 | -2 | -2 | 2 | 2 | -2 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | 1 | 1 | 1 | -1 | 1 | √-3 | √-3 | -√-3 | √-3 | -√-3 | -√-3 | √-3 | -√-3 | complex lifted from C3⋊D4 |
| ρ19 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 2 | 2i | 0 | 0 | -2i | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | -2 | -2 | 0 | 2i | 0 | 0 | 2i | -2i | -2i | 0 | complex lifted from C4○D4 |
| ρ20 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 2 | -2i | 0 | 0 | 2i | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | -2 | -2 | 0 | -2i | 0 | 0 | -2i | 2i | 2i | 0 | complex lifted from C4○D4 |
| ρ21 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 2 | 0 | 2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | -2 | 2 | 2i | 0 | -2i | -2i | 0 | 0 | 0 | 2i | complex lifted from C4○D4 |
| ρ22 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 2 | 0 | -2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | -2 | 2 | -2i | 0 | 2i | 2i | 0 | 0 | 0 | -2i | complex lifted from C4○D4 |
| ρ23 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | -1 | -2i | 0 | 0 | 2i | 0 | 0 | 0 | √-3 | -√-3 | -1 | √-3 | -√-3 | 1 | 1 | -√3 | i | -√3 | √3 | i | -i | -i | √3 | complex lifted from C4○D12 |
| ρ24 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | -1 | -2i | 0 | 0 | 2i | 0 | 0 | 0 | -√-3 | √-3 | -1 | -√-3 | √-3 | 1 | 1 | √3 | i | √3 | -√3 | i | -i | -i | -√3 | complex lifted from C4○D12 |
| ρ25 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | -1 | 0 | -2i | 2i | 0 | 0 | 0 | 0 | √-3 | -√-3 | 1 | -√-3 | √-3 | 1 | -1 | i | -√3 | -i | -i | √3 | -√3 | √3 | i | complex lifted from C4○D12 |
| ρ26 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | -1 | 0 | -2i | 2i | 0 | 0 | 0 | 0 | -√-3 | √-3 | 1 | √-3 | -√-3 | 1 | -1 | i | √3 | -i | -i | -√3 | √3 | -√3 | i | complex lifted from C4○D12 |
| ρ27 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | -1 | 2i | 0 | 0 | -2i | 0 | 0 | 0 | √-3 | -√-3 | -1 | √-3 | -√-3 | 1 | 1 | √3 | -i | √3 | -√3 | -i | i | i | -√3 | complex lifted from C4○D12 |
| ρ28 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | -1 | 0 | 2i | -2i | 0 | 0 | 0 | 0 | -√-3 | √-3 | 1 | √-3 | -√-3 | 1 | -1 | -i | -√3 | i | i | √3 | -√3 | √3 | -i | complex lifted from C4○D12 |
| ρ29 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | -1 | 0 | 2i | -2i | 0 | 0 | 0 | 0 | √-3 | -√-3 | 1 | -√-3 | √-3 | 1 | -1 | -i | √3 | i | i | -√3 | √3 | -√3 | -i | complex lifted from C4○D12 |
| ρ30 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | -1 | 2i | 0 | 0 | -2i | 0 | 0 | 0 | -√-3 | √-3 | -1 | -√-3 | √-3 | 1 | 1 | -√3 | -i | -√3 | √3 | -i | i | i | √3 | complex lifted from C4○D12 |
►On 48 points
(1 43)(2 44)(3 45)(4 46)(5 47)(6 48)(7 37)(8 38)(9 39)(10 40)(11 41)(12 42)(13 31)(14 32)(15 33)(16 34)(17 35)(18 36)(19 25)(20 26)(21 27)(22 28)(23 29)(24 30)
(1 31)(2 32)(3 33)(4 34)(5 35)(6 36)(7 25)(8 26)(9 27)(10 28)(11 29)(12 30)(13 43)(14 44)(15 45)(16 46)(17 47)(18 48)(19 37)(20 38)(21 39)(22 40)(23 41)(24 42)
(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 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 6 25 30)(2 29 26 5)(3 4 27 28)(7 12 31 36)(8 35 32 11)(9 10 33 34)(13 24 37 48)(14 47 38 23)(15 22 39 46)(16 45 40 21)(17 20 41 44)(18 43 42 19)
G:=sub<Sym(48)| (1,43)(2,44)(3,45)(4,46)(5,47)(6,48)(7,37)(8,38)(9,39)(10,40)(11,41)(12,42)(13,31)(14,32)(15,33)(16,34)(17,35)(18,36)(19,25)(20,26)(21,27)(22,28)(23,29)(24,30), (1,31)(2,32)(3,33)(4,34)(5,35)(6,36)(7,25)(8,26)(9,27)(10,28)(11,29)(12,30)(13,43)(14,44)(15,45)(16,46)(17,47)(18,48)(19,37)(20,38)(21,39)(22,40)(23,41)(24,42), (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), (1,2,3,4,5,6,7,8,9,10,11,12)(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), (1,6,25,30)(2,29,26,5)(3,4,27,28)(7,12,31,36)(8,35,32,11)(9,10,33,34)(13,24,37,48)(14,47,38,23)(15,22,39,46)(16,45,40,21)(17,20,41,44)(18,43,42,19)>;
G:=Group( (1,43)(2,44)(3,45)(4,46)(5,47)(6,48)(7,37)(8,38)(9,39)(10,40)(11,41)(12,42)(13,31)(14,32)(15,33)(16,34)(17,35)(18,36)(19,25)(20,26)(21,27)(22,28)(23,29)(24,30), (1,31)(2,32)(3,33)(4,34)(5,35)(6,36)(7,25)(8,26)(9,27)(10,28)(11,29)(12,30)(13,43)(14,44)(15,45)(16,46)(17,47)(18,48)(19,37)(20,38)(21,39)(22,40)(23,41)(24,42), (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), (1,2,3,4,5,6,7,8,9,10,11,12)(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), (1,6,25,30)(2,29,26,5)(3,4,27,28)(7,12,31,36)(8,35,32,11)(9,10,33,34)(13,24,37,48)(14,47,38,23)(15,22,39,46)(16,45,40,21)(17,20,41,44)(18,43,42,19) );
G=PermutationGroup([[(1,43),(2,44),(3,45),(4,46),(5,47),(6,48),(7,37),(8,38),(9,39),(10,40),(11,41),(12,42),(13,31),(14,32),(15,33),(16,34),(17,35),(18,36),(19,25),(20,26),(21,27),(22,28),(23,29),(24,30)], [(1,31),(2,32),(3,33),(4,34),(5,35),(6,36),(7,25),(8,26),(9,27),(10,28),(11,29),(12,30),(13,43),(14,44),(15,45),(16,46),(17,47),(18,48),(19,37),(20,38),(21,39),(22,40),(23,41),(24,42)], [(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)], [(1,2,3,4,5,6,7,8,9,10,11,12),(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)], [(1,6,25,30),(2,29,26,5),(3,4,27,28),(7,12,31,36),(8,35,32,11),(9,10,33,34),(13,24,37,48),(14,47,38,23),(15,22,39,46),(16,45,40,21),(17,20,41,44),(18,43,42,19)]])
C23.28D6 is a maximal subgroup of
C22⋊C4⋊D6 C42.277D6 C24.38D6 C24.42D6 C6.2- 1+4 C6.52- 1+4 C6.62- 1+4 C42⋊12D6 C42.96D6 C42.104D6 C42⋊18D6 C42.113D6 C42.114D6 C42⋊19D6 C42.115D6 C42.116D6 C42.118D6 C6.422+ 1+4 C6.442+ 1+4 C6.482+ 1+4 C6.492+ 1+4 C6.202- 1+4 C6.222- 1+4 C6.252- 1+4 C6.592+ 1+4 C6.792- 1+4 C4⋊C4.197D6 S3×C22.D4 C6.1202+ 1+4 C4⋊C4⋊28D6 C6.852- 1+4 C24.83D6 C24⋊12D6 C6.442- 1+4 C6.1042- 1+4 C6.1452+ 1+4 C23.28D18 C62.20C23 C62.75C23 C62.56D4 C62.60D4 C62.129D4 Dic3⋊C4⋊D5 D6⋊C4⋊D5 C30.(C2×D4) C6.D4⋊D5 C23.28D30
C23.28D6 is a maximal quotient of
(C2×C42).6S3 (C2×C42)⋊3S3 C24.20D6 C24.25D6 (C2×C6).40D8 C4⋊C4.228D6 C4⋊C4.230D6 C4⋊C4.231D6 (C2×Dic3).Q8 (C2×C12).288D4 (C2×C12).289D4 (C2×C12).290D4 C4⋊C4.233D6 C4⋊C4.236D6 C24.73D6 C24.74D6 C24.76D6 C23.28D18 C62.20C23 C62.75C23 C62.56D4 C62.60D4 C62.129D4 Dic3⋊C4⋊D5 D6⋊C4⋊D5 C30.(C2×D4) C6.D4⋊D5 C23.28D30
Matrix representation of C23.28D6 ►in GL6(𝔽13)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 5 | 0 | 0 |
| 0 | 0 | 8 | 0 | 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 |
| 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 | 11 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 12 | 1 |
| 1 | 2 | 0 | 0 | 0 | 0 |
| 12 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 1 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,8,0,0,0,0,5,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[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],[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],[12,0,0,0,0,0,11,1,0,0,0,0,0,0,0,12,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,1,1],[1,12,0,0,0,0,2,12,0,0,0,0,0,0,0,12,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,1,1] >;
C23.28D6 in GAP, Magma, Sage, TeX
C_2^3._{28}D_6 % in TeX
G:=Group("C2^3.28D6"); // GroupNames label
G:=SmallGroup(96,136);
// by ID
G=gap.SmallGroup(96,136);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-3,217,218,86,2309]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=1,d^6=c,e^2=c*b=b*c,a*b=b*a,e*a*e^-1=a*c=c*a,a*d=d*a,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=b*d^5>;
// generators/relations
Export