metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C12.23D4, (C2×Q8)⋊6S3, (C6×Q8)⋊4C2, D6⋊C4⋊16C2, C6.58(C2×D4), (C2×C4).57D6, (C4×Dic3)⋊7C2, (C2×D12).9C2, C3⋊4(C4.4D4), C6.37(C4○D4), C4.11(C3⋊D4), (C2×C6).59C23, (C2×C12).40C22, C2.9(Q8⋊3S3), C22.65(C22×S3), (C22×S3).13C22, (C2×Dic3).41C22, C2.22(C2×C3⋊D4), SmallGroup(96,154)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C12.23D4
G = < a,b,c | a12=b4=c2=1, bab-1=a5, cac=a-1, cbc=a6b-1 >
Subgroups: 194 in 76 conjugacy classes, 33 normal (13 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, C2×C4, D4, Q8, C23, Dic3, C12, C12, D6, C2×C6, C42, C22⋊C4, C2×D4, C2×Q8, D12, C2×Dic3, C2×C12, C2×C12, C3×Q8, C22×S3, C4.4D4, C4×Dic3, D6⋊C4, C2×D12, C6×Q8, C12.23D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, C3⋊D4, C22×S3, C4.4D4, Q8⋊3S3, C2×C3⋊D4, C12.23D4
Character table of C12.23D4
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 6A | 6B | 6C | 12A | 12B | 12C | 12D | 12E | 12F | |
| size | 1 | 1 | 1 | 1 | 12 | 12 | 2 | 2 | 2 | 4 | 4 | 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 | 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 | -1 | -2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 1 | 1 | -1 | 1 | -1 | 1 | orthogonal lifted from D6 |
| ρ10 | 2 | 2 | -2 | -2 | 0 | 0 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | -2 | 2 | 0 | 0 | -2 | 0 | 0 | orthogonal lifted from D4 |
| ρ11 | 2 | 2 | 2 | 2 | 0 | 0 | -1 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | 1 | 1 | -1 | 1 | 1 | orthogonal lifted from D6 |
| ρ12 | 2 | 2 | -2 | -2 | 0 | 0 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | -2 | -2 | 0 | 0 | 2 | 0 | 0 | orthogonal lifted from D4 |
| ρ13 | 2 | 2 | 2 | 2 | 0 | 0 | -1 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ14 | 2 | 2 | 2 | 2 | 0 | 0 | -1 | -2 | -2 | -2 | 2 | 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 | 1 | -√-3 | √-3 | -1 | -√-3 | √-3 | complex lifted from C3⋊D4 |
| ρ16 | 2 | 2 | -2 | -2 | 0 | 0 | -1 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | -1 | -√-3 | -√-3 | 1 | √-3 | √-3 | complex lifted from C3⋊D4 |
| ρ17 | 2 | 2 | -2 | -2 | 0 | 0 | -1 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | 1 | √-3 | -√-3 | -1 | √-3 | -√-3 | complex lifted from C3⋊D4 |
| ρ18 | 2 | 2 | -2 | -2 | 0 | 0 | -1 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | -1 | √-3 | √-3 | 1 | -√-3 | -√-3 | complex lifted from C3⋊D4 |
| ρ19 | 2 | -2 | 2 | -2 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 2i | 0 | -2i | 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 | 0 | -2i | 0 | 2i | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ21 | 2 | -2 | -2 | 2 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | -2i | 0 | 2i | 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 | 0 | 0 | 2i | 0 | -2i | 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 | orthogonal lifted from Q8⋊3S3, 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 | orthogonal lifted from Q8⋊3S3, Schur index 2 |
►On 48 points
(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 23 38)(2 36 24 43)(3 29 13 48)(4 34 14 41)(5 27 15 46)(6 32 16 39)(7 25 17 44)(8 30 18 37)(9 35 19 42)(10 28 20 47)(11 33 21 40)(12 26 22 45)
(2 12)(3 11)(4 10)(5 9)(6 8)(13 21)(14 20)(15 19)(16 18)(22 24)(25 38)(26 37)(27 48)(28 47)(29 46)(30 45)(31 44)(32 43)(33 42)(34 41)(35 40)(36 39)
G:=sub<Sym(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), (1,31,23,38)(2,36,24,43)(3,29,13,48)(4,34,14,41)(5,27,15,46)(6,32,16,39)(7,25,17,44)(8,30,18,37)(9,35,19,42)(10,28,20,47)(11,33,21,40)(12,26,22,45), (2,12)(3,11)(4,10)(5,9)(6,8)(13,21)(14,20)(15,19)(16,18)(22,24)(25,38)(26,37)(27,48)(28,47)(29,46)(30,45)(31,44)(32,43)(33,42)(34,41)(35,40)(36,39)>;
G:=Group( (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), (1,31,23,38)(2,36,24,43)(3,29,13,48)(4,34,14,41)(5,27,15,46)(6,32,16,39)(7,25,17,44)(8,30,18,37)(9,35,19,42)(10,28,20,47)(11,33,21,40)(12,26,22,45), (2,12)(3,11)(4,10)(5,9)(6,8)(13,21)(14,20)(15,19)(16,18)(22,24)(25,38)(26,37)(27,48)(28,47)(29,46)(30,45)(31,44)(32,43)(33,42)(34,41)(35,40)(36,39) );
G=PermutationGroup([[(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)], [(1,31,23,38),(2,36,24,43),(3,29,13,48),(4,34,14,41),(5,27,15,46),(6,32,16,39),(7,25,17,44),(8,30,18,37),(9,35,19,42),(10,28,20,47),(11,33,21,40),(12,26,22,45)], [(2,12),(3,11),(4,10),(5,9),(6,8),(13,21),(14,20),(15,19),(16,18),(22,24),(25,38),(26,37),(27,48),(28,47),(29,46),(30,45),(31,44),(32,43),(33,42),(34,41),(35,40),(36,39)]])
C12.23D4 is a maximal subgroup of
(C2×C4).D12 D12.5D4 (C2×C8).D6 Dic6.11D4 Q8⋊C4⋊S3 D12.12D4 (C3×D4).D4 C24.43D4 C24⋊9D4 (C2×Q16)⋊S3 C24.37D4 C24.28D4 D12.39D4 2- 1+4⋊4S3 C42.122D6 C42.131D6 C42.133D6 C42.136D6 C4⋊C4.187D6 D12⋊21D4 Dic6⋊22D4 C6.532+ 1+4 C6.222- 1+4 C6.242- 1+4 C6.562+ 1+4 C6.592+ 1+4 C42.138D6 S3×C4.4D4 C42⋊20D6 C42⋊22D6 C42.143D6 C42⋊24D6 C42.171D6 C42.240D6 C42.177D6 C42.178D6 C42.179D6 C42.180D6 C6.442- 1+4 C6.452- 1+4 C6.1452+ 1+4 C6.1462+ 1+4 (C2×C12)⋊17D4 C36.23D4 C62.33C23 C12.28D12 C62.77C23 C62.262C23 (C2×C20).D6 C60.89D4 C60.47D4 C60.23D4
C12.23D4 is a maximal quotient of
(C4×Dic3)⋊9C4 (C2×C12).55D4 (C2×D12)⋊10C4 D6⋊C4⋊7C4 (C2×C4)⋊3D12 (C2×C12).289D4 C42.70D6 C42.216D6 C42.71D6 C12.D8 C42.82D6 C12.Q16 (C6×Q8)⋊7C4 (C22×Q8)⋊9S3 C36.23D4 C62.33C23 C12.28D12 C62.77C23 C62.262C23 (C2×C20).D6 C60.89D4 C60.47D4 C60.23D4
Matrix representation of C12.23D4 ►in GL4(𝔽13) generated by
| 0 | 1 | 0 | 0 |
| 12 | 12 | 0 | 0 |
| 0 | 0 | 5 | 3 |
| 0 | 0 | 0 | 8 |
| 11 | 9 | 0 | 0 |
| 11 | 2 | 0 | 0 |
| 0 | 0 | 12 | 2 |
| 0 | 0 | 0 | 1 |
| 12 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 1 | 12 |
G:=sub<GL(4,GF(13))| [0,12,0,0,1,12,0,0,0,0,5,0,0,0,3,8],[11,11,0,0,9,2,0,0,0,0,12,0,0,0,2,1],[12,1,0,0,0,1,0,0,0,0,1,1,0,0,0,12] >;
C12.23D4 in GAP, Magma, Sage, TeX
C_{12}._{23}D_4 % in TeX
G:=Group("C12.23D4"); // GroupNames label
G:=SmallGroup(96,154);
// by ID
G=gap.SmallGroup(96,154);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-3,217,103,218,188,86,2309]);
// Polycyclic
G:=Group<a,b,c|a^12=b^4=c^2=1,b*a*b^-1=a^5,c*a*c=a^-1,c*b*c=a^6*b^-1>;
// generators/relations
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