metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C12.17D4, C23.12D6, (C2×D4).6S3, (C6×D4).5C2, (C2×C4).50D6, C6.48(C2×D4), (C4×Dic3)⋊5C2, C3⋊3(C4.4D4), C4.7(C3⋊D4), (C2×Dic6)⋊10C2, C6.30(C4○D4), C6.D4⋊9C2, (C2×C6).51C23, (C2×C12).33C22, C2.16(D4⋊2S3), C22.58(C22×S3), (C22×C6).19C22, (C2×Dic3).18C22, C2.12(C2×C3⋊D4), SmallGroup(96,143)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C23.12D6
G = < a,b,c,d,e | a2=b2=c2=1, d6=b, e2=cb=bc, dad-1=ab=ba, eae-1=ac=ca, bd=db, be=eb, cd=dc, ce=ec, ede-1=d5 >
Subgroups: 162 in 76 conjugacy classes, 33 normal (13 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C6, C6, C6, C2×C4, C2×C4, D4, Q8, C23, Dic3, C12, C2×C6, C2×C6, C42, C22⋊C4, C2×D4, C2×Q8, Dic6, C2×Dic3, C2×C12, C3×D4, C22×C6, C4.4D4, C4×Dic3, C6.D4, C2×Dic6, C6×D4, C23.12D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, C3⋊D4, C22×S3, C4.4D4, D4⋊2S3, C2×C3⋊D4, C23.12D6
Character table of C23.12D6
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 12A | 12B | |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 2 | 2 | 2 | 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 | 0 | 0 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 2 | -2 | orthogonal lifted from D4 |
| ρ10 | 2 | 2 | 2 | 2 | 2 | 2 | -1 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ11 | 2 | 2 | -2 | -2 | 0 | 0 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | -2 | 2 | orthogonal lifted from D4 |
| ρ12 | 2 | 2 | 2 | 2 | -2 | 2 | -1 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 1 | -1 | -1 | 1 | 1 | 1 | orthogonal lifted from D6 |
| ρ13 | 2 | 2 | 2 | 2 | -2 | -2 | -1 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | orthogonal lifted from D6 |
| ρ14 | 2 | 2 | 2 | 2 | 2 | -2 | -1 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | 1 | 1 | -1 | 1 | 1 | orthogonal lifted from D6 |
| ρ15 | 2 | 2 | -2 | -2 | 0 | 0 | -1 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | √-3 | -√-3 | √-3 | -√-3 | -1 | 1 | complex lifted from C3⋊D4 |
| ρ16 | 2 | 2 | -2 | -2 | 0 | 0 | -1 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | -√-3 | √-3 | -√-3 | √-3 | -1 | 1 | complex lifted from C3⋊D4 |
| ρ17 | 2 | 2 | -2 | -2 | 0 | 0 | -1 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | √-3 | √-3 | -√-3 | -√-3 | 1 | -1 | complex lifted from C3⋊D4 |
| ρ18 | 2 | 2 | -2 | -2 | 0 | 0 | -1 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | -√-3 | -√-3 | √-3 | √-3 | 1 | -1 | complex lifted from C3⋊D4 |
| ρ19 | 2 | -2 | -2 | 2 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 2i | -2i | 0 | 0 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ20 | 2 | -2 | -2 | 2 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | -2i | 2i | 0 | 0 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ21 | 2 | -2 | 2 | -2 | 0 | 0 | 2 | 0 | 0 | -2i | 2i | 0 | 0 | 0 | 0 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ22 | 2 | -2 | 2 | -2 | 0 | 0 | 2 | 0 | 0 | 2i | -2i | 0 | 0 | 0 | 0 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ23 | 4 | -4 | -4 | 4 | 0 | 0 | -2 | 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 | 0 | -2 | 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 8)(4 10)(6 12)(13 26)(14 33)(15 28)(16 35)(17 30)(18 25)(19 32)(20 27)(21 34)(22 29)(23 36)(24 31)(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 46)(2 47)(3 48)(4 37)(5 38)(6 39)(7 40)(8 41)(9 42)(10 43)(11 44)(12 45)(13 26)(14 27)(15 28)(16 29)(17 30)(18 31)(19 32)(20 33)(21 34)(22 35)(23 36)(24 25)
(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 13 40 32)(2 18 41 25)(3 23 42 30)(4 16 43 35)(5 21 44 28)(6 14 45 33)(7 19 46 26)(8 24 47 31)(9 17 48 36)(10 22 37 29)(11 15 38 34)(12 20 39 27)
G:=sub<Sym(48)| (2,8)(4,10)(6,12)(13,26)(14,33)(15,28)(16,35)(17,30)(18,25)(19,32)(20,27)(21,34)(22,29)(23,36)(24,31)(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), (1,46)(2,47)(3,48)(4,37)(5,38)(6,39)(7,40)(8,41)(9,42)(10,43)(11,44)(12,45)(13,26)(14,27)(15,28)(16,29)(17,30)(18,31)(19,32)(20,33)(21,34)(22,35)(23,36)(24,25), (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), (1,13,40,32)(2,18,41,25)(3,23,42,30)(4,16,43,35)(5,21,44,28)(6,14,45,33)(7,19,46,26)(8,24,47,31)(9,17,48,36)(10,22,37,29)(11,15,38,34)(12,20,39,27)>;
G:=Group( (2,8)(4,10)(6,12)(13,26)(14,33)(15,28)(16,35)(17,30)(18,25)(19,32)(20,27)(21,34)(22,29)(23,36)(24,31)(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), (1,46)(2,47)(3,48)(4,37)(5,38)(6,39)(7,40)(8,41)(9,42)(10,43)(11,44)(12,45)(13,26)(14,27)(15,28)(16,29)(17,30)(18,31)(19,32)(20,33)(21,34)(22,35)(23,36)(24,25), (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), (1,13,40,32)(2,18,41,25)(3,23,42,30)(4,16,43,35)(5,21,44,28)(6,14,45,33)(7,19,46,26)(8,24,47,31)(9,17,48,36)(10,22,37,29)(11,15,38,34)(12,20,39,27) );
G=PermutationGroup([[(2,8),(4,10),(6,12),(13,26),(14,33),(15,28),(16,35),(17,30),(18,25),(19,32),(20,27),(21,34),(22,29),(23,36),(24,31),(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)], [(1,46),(2,47),(3,48),(4,37),(5,38),(6,39),(7,40),(8,41),(9,42),(10,43),(11,44),(12,45),(13,26),(14,27),(15,28),(16,29),(17,30),(18,31),(19,32),(20,33),(21,34),(22,35),(23,36),(24,25)], [(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)], [(1,13,40,32),(2,18,41,25),(3,23,42,30),(4,16,43,35),(5,21,44,28),(6,14,45,33),(7,19,46,26),(8,24,47,31),(9,17,48,36),(10,22,37,29),(11,15,38,34),(12,20,39,27)]])
C23.12D6 is a maximal subgroup of
C23.D12 M4(2)⋊D6 C12⋊Q8⋊C2 Dic6.D4 (C2×C8).200D6 D12.D4 (C6×D8).C2 C24⋊11D4 C24.22D4 (C3×Q8).D4 C24.31D4 C24.43D4 D12.38D4 2+ 1+4.4S3 C42.106D6 C42.229D6 C42.114D6 C42.115D6 C24.43D6 C24.46D6 C24⋊9D6 Dic6⋊19D4 C4⋊C4.178D6 C6.712- 1+4 D12⋊20D4 C6.422+ 1+4 C6.452+ 1+4 C6.492+ 1+4 C6.812- 1+4 C6.622+ 1+4 C6.652+ 1+4 C42.139D6 S3×C4.4D4 C42.141D6 C42.166D6 C42⋊28D6 C42.238D6 C24.52D6 C24.53D6 C6.1052- 1+4 C6.1072- 1+4 (C2×C12)⋊17D4 C36.17D4 C12.27D12 C62.33C23 C62.101C23 C62.254C23 C60.44D4 C60.88D4 C6.(D4×D5) C60.17D4
C23.12D6 is a maximal quotient of
C24.15D6 C23⋊2Dic6 C24.21D6 C4.(D6⋊C4) (C4×Dic3)⋊9C4 (C2×C12).288D4 C42.62D6 C42.213D6 C12.16D8 C42.72D6 C12.9Q16 C42.77D6 C24.30D6 C24.31D6 C36.17D4 C12.27D12 C62.33C23 C62.101C23 C62.254C23 C60.44D4 C60.88D4 C6.(D4×D5) C60.17D4
Matrix representation of C23.12D6 ►in GL4(𝔽13) generated by
| 1 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 12 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 12 |
| 12 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 9 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 |
| 0 | 0 | 0 | 12 |
| 0 | 0 | 1 | 0 |
| 0 | 3 | 0 | 0 |
| 4 | 0 | 0 | 0 |
| 0 | 0 | 5 | 0 |
| 0 | 0 | 0 | 5 |
G:=sub<GL(4,GF(13))| [1,0,0,0,0,12,0,0,0,0,1,0,0,0,0,12],[1,0,0,0,0,1,0,0,0,0,12,0,0,0,0,12],[12,0,0,0,0,12,0,0,0,0,1,0,0,0,0,1],[9,0,0,0,0,3,0,0,0,0,0,1,0,0,12,0],[0,4,0,0,3,0,0,0,0,0,5,0,0,0,0,5] >;
C23.12D6 in GAP, Magma, Sage, TeX
C_2^3._{12}D_6 % in TeX
G:=Group("C2^3.12D6"); // GroupNames label
G:=SmallGroup(96,143);
// by ID
G=gap.SmallGroup(96,143);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-3,96,55,506,116,2309]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=1,d^6=b,e^2=c*b=b*c,d*a*d^-1=a*b=b*a,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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