metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C23.8D6, C4⋊Dic3⋊3C2, (C2×C4).27D6, Dic3⋊C4⋊8C2, C6.7(C4○D4), C22⋊C4.2S3, C3⋊2(C42⋊2C2), (C4×Dic3)⋊10C2, C2.9(C4○D12), (C2×C6).20C23, (C2×C12).2C22, C2.7(D4⋊2S3), C6.D4.3C2, (C22×C6).9C22, C22.40(C22×S3), (C2×Dic3).26C22, (C3×C22⋊C4).2C2, SmallGroup(96,86)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C23.8D6
G = < a,b,c,d,e | a2=b2=c2=1, d6=b, e2=cb=bc, eae-1=ab=ba, dad-1=ac=ca, bd=db, be=eb, cd=dc, ce=ec, ede-1=d5 >
Subgroups: 122 in 60 conjugacy classes, 29 normal (all characteristic)
C1, C2, C2, C3, C4, C22, C22, C6, C6, C2×C4, C2×C4, C23, Dic3, C12, C2×C6, C2×C6, C42, C22⋊C4, C22⋊C4, C4⋊C4, C2×Dic3, C2×C12, C22×C6, C42⋊2C2, C4×Dic3, Dic3⋊C4, C4⋊Dic3, C6.D4, C3×C22⋊C4, C23.8D6
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, C22×S3, C42⋊2C2, C4○D12, D4⋊2S3, C23.8D6
Character table of C23.8D6
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 6A | 6B | 6C | 6D | 6E | 12A | 12B | 12C | 12D | |
| size | 1 | 1 | 1 | 1 | 4 | 2 | 2 | 2 | 4 | 6 | 6 | 6 | 6 | 12 | 12 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | linear of order 2 |
| ρ9 | 2 | 2 | 2 | 2 | -2 | -1 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 1 | 1 | 1 | -1 | 1 | -1 | orthogonal lifted from D6 |
| ρ10 | 2 | 2 | 2 | 2 | 2 | -1 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | orthogonal lifted from D6 |
| ρ11 | 2 | 2 | 2 | 2 | -2 | -1 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 1 | 1 | -1 | 1 | -1 | 1 | orthogonal lifted from D6 |
| ρ12 | 2 | 2 | 2 | 2 | 2 | -1 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ13 | 2 | -2 | 2 | -2 | 0 | 2 | -2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 2 | 0 | 0 | 0 | -2i | 0 | 2i | complex lifted from C4○D4 |
| ρ14 | 2 | -2 | -2 | 2 | 0 | 2 | 0 | 0 | 0 | -2i | 2i | 0 | 0 | 0 | 0 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ15 | 2 | -2 | 2 | -2 | 0 | 2 | 2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 2 | 0 | 0 | 0 | 2i | 0 | -2i | complex lifted from C4○D4 |
| ρ16 | 2 | 2 | -2 | -2 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | -2i | 2i | 0 | 0 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ17 | 2 | -2 | -2 | 2 | 0 | 2 | 0 | 0 | 0 | 2i | -2i | 0 | 0 | 0 | 0 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ18 | 2 | 2 | -2 | -2 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 2i | -2i | 0 | 0 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ19 | 2 | -2 | 2 | -2 | 0 | -1 | 2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | √-3 | -√-3 | √3 | -i | -√3 | i | complex lifted from C4○D12 |
| ρ20 | 2 | -2 | 2 | -2 | 0 | -1 | 2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | -√-3 | √-3 | -√3 | -i | √3 | i | complex lifted from C4○D12 |
| ρ21 | 2 | -2 | 2 | -2 | 0 | -1 | -2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | -√-3 | √-3 | √3 | i | -√3 | -i | complex lifted from C4○D12 |
| ρ22 | 2 | -2 | 2 | -2 | 0 | -1 | -2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | √-3 | -√-3 | -√3 | i | √3 | -i | complex lifted from C4○D12 |
| ρ23 | 4 | 4 | -4 | -4 | 0 | -2 | 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 |
| ρ24 | 4 | -4 | -4 | 4 | 0 | -2 | 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 26)(4 28)(6 30)(8 32)(10 34)(12 36)(13 46)(14 20)(15 48)(16 22)(17 38)(18 24)(19 40)(21 42)(23 44)(37 43)(39 45)(41 47)
(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 25)(2 26)(3 27)(4 28)(5 29)(6 30)(7 31)(8 32)(9 33)(10 34)(11 35)(12 36)(13 40)(14 41)(15 42)(16 43)(17 44)(18 45)(19 46)(20 47)(21 48)(22 37)(23 38)(24 39)
(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 18 31 39)(2 23 32 44)(3 16 33 37)(4 21 34 42)(5 14 35 47)(6 19 36 40)(7 24 25 45)(8 17 26 38)(9 22 27 43)(10 15 28 48)(11 20 29 41)(12 13 30 46)
G:=sub<Sym(48)| (2,26)(4,28)(6,30)(8,32)(10,34)(12,36)(13,46)(14,20)(15,48)(16,22)(17,38)(18,24)(19,40)(21,42)(23,44)(37,43)(39,45)(41,47), (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), (1,25)(2,26)(3,27)(4,28)(5,29)(6,30)(7,31)(8,32)(9,33)(10,34)(11,35)(12,36)(13,40)(14,41)(15,42)(16,43)(17,44)(18,45)(19,46)(20,47)(21,48)(22,37)(23,38)(24,39), (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), (1,18,31,39)(2,23,32,44)(3,16,33,37)(4,21,34,42)(5,14,35,47)(6,19,36,40)(7,24,25,45)(8,17,26,38)(9,22,27,43)(10,15,28,48)(11,20,29,41)(12,13,30,46)>;
G:=Group( (2,26)(4,28)(6,30)(8,32)(10,34)(12,36)(13,46)(14,20)(15,48)(16,22)(17,38)(18,24)(19,40)(21,42)(23,44)(37,43)(39,45)(41,47), (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), (1,25)(2,26)(3,27)(4,28)(5,29)(6,30)(7,31)(8,32)(9,33)(10,34)(11,35)(12,36)(13,40)(14,41)(15,42)(16,43)(17,44)(18,45)(19,46)(20,47)(21,48)(22,37)(23,38)(24,39), (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), (1,18,31,39)(2,23,32,44)(3,16,33,37)(4,21,34,42)(5,14,35,47)(6,19,36,40)(7,24,25,45)(8,17,26,38)(9,22,27,43)(10,15,28,48)(11,20,29,41)(12,13,30,46) );
G=PermutationGroup([[(2,26),(4,28),(6,30),(8,32),(10,34),(12,36),(13,46),(14,20),(15,48),(16,22),(17,38),(18,24),(19,40),(21,42),(23,44),(37,43),(39,45),(41,47)], [(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)], [(1,25),(2,26),(3,27),(4,28),(5,29),(6,30),(7,31),(8,32),(9,33),(10,34),(11,35),(12,36),(13,40),(14,41),(15,42),(16,43),(17,44),(18,45),(19,46),(20,47),(21,48),(22,37),(23,38),(24,39)], [(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)], [(1,18,31,39),(2,23,32,44),(3,16,33,37),(4,21,34,42),(5,14,35,47),(6,19,36,40),(7,24,25,45),(8,17,26,38),(9,22,27,43),(10,15,28,48),(11,20,29,41),(12,13,30,46)]])
C23.8D6 is a maximal subgroup of
C24.41D6 C24.42D6 C42.89D6 C42.93D6 C42.94D6 C42.98D6 C42.102D6 C42.104D6 C42.105D6 C42.106D6 C42.229D6 C42.113D6 C42.115D6 C42.118D6 C24.43D6 C24.46D6 C24.47D6 C4⋊C4.178D6 C6.342+ 1+4 C6.702- 1+4 C6.712- 1+4 C6.422+ 1+4 C6.432+ 1+4 C6.1152+ 1+4 C6.482+ 1+4 C6.152- 1+4 C6.202- 1+4 C6.212- 1+4 C6.222- 1+4 C6.232- 1+4 C6.252- 1+4 C4⋊C4.197D6 C6.802- 1+4 C6.812- 1+4 C6.612+ 1+4 C6.622+ 1+4 C6.632+ 1+4 C6.642+ 1+4 C6.652+ 1+4 C6.852- 1+4 C42.137D6 C42.139D6 C42.140D6 C42⋊22D6 C42⋊23D6 C42.234D6 C42.144D6 C42.159D6 C42.160D6 S3×C42⋊2C2 C42⋊26D6 C42.162D6 C42.165D6 C23.8D18 C62.28C23 C62.29C23 C62.31C23 C62.32C23 C62.98C23 C62.223C23 (C2×C12).D10 (C2×C60).C22 (C4×Dic3)⋊D5 (C4×Dic15)⋊C2 C23.13(S3×D5) C23.14(S3×D5) C23.8D30
C23.8D6 is a maximal quotient of
C3⋊(C42⋊5C4) C6.(C4×D4) C2.(C4×D12) Dic3⋊C4⋊C4 (C2×Dic3).9D4 (C2×C4).17D12 (C2×C4).Dic6 (C22×C4).30D6 C24.14D6 C24.15D6 C24.17D6 C24.19D6 C24.20D6 C24.21D6 C23.8D18 C62.28C23 C62.29C23 C62.31C23 C62.32C23 C62.98C23 C62.223C23 (C2×C12).D10 (C2×C60).C22 (C4×Dic3)⋊D5 (C4×Dic15)⋊C2 C23.13(S3×D5) C23.14(S3×D5) C23.8D30
Matrix representation of C23.8D6 ►in GL4(𝔽13) generated by
| 1 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 12 | 12 |
| 12 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 12 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 12 |
| 6 | 0 | 0 | 0 |
| 0 | 2 | 0 | 0 |
| 0 | 0 | 5 | 10 |
| 0 | 0 | 0 | 8 |
| 0 | 2 | 0 | 0 |
| 6 | 0 | 0 | 0 |
| 0 | 0 | 12 | 11 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(13))| [1,0,0,0,0,12,0,0,0,0,1,12,0,0,0,12],[12,0,0,0,0,12,0,0,0,0,12,0,0,0,0,12],[1,0,0,0,0,1,0,0,0,0,12,0,0,0,0,12],[6,0,0,0,0,2,0,0,0,0,5,0,0,0,10,8],[0,6,0,0,2,0,0,0,0,0,12,0,0,0,11,1] >;
C23.8D6 in GAP, Magma, Sage, TeX
C_2^3._8D_6
% in TeX
G:=Group("C2^3.8D6"); // GroupNames label
G:=SmallGroup(96,86);
// by ID
G=gap.SmallGroup(96,86);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-3,96,55,506,188,2309]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=1,d^6=b,e^2=c*b=b*c,e*a*e^-1=a*b=b*a,d*a*d^-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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