metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C12⋊7D4, C22⋊2D12, C23.29D6, (C2×C6)⋊5D4, D6⋊C4⋊3C2, (C2×D12)⋊6C2, C3⋊3(C4⋊D4), C4⋊3(C3⋊D4), C4⋊Dic3⋊9C2, (C22×C4)⋊6S3, C6.43(C2×D4), (C2×C4).85D6, (C22×C12)⋊6C2, C2.17(C2×D12), C6.19(C4○D4), (C2×C6).48C23, C2.19(C4○D12), (C2×C12).94C22, C22.56(C22×S3), (C22×C6).40C22, (C22×S3).10C22, (C2×Dic3).16C22, (C2×C3⋊D4)⋊3C2, C2.7(C2×C3⋊D4), SmallGroup(96,137)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C12⋊7D4
G = < a,b,c | a12=b4=c2=1, bab-1=cac=a-1, cbc=b-1 >
Subgroups: 242 in 94 conjugacy classes, 37 normal (21 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C22, S3, C6, C6, C2×C4, C2×C4, D4, C23, C23, Dic3, C12, C12, D6, C2×C6, C2×C6, C2×C6, C22⋊C4, C4⋊C4, C22×C4, C2×D4, D12, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C22×S3, C22×C6, C4⋊D4, C4⋊Dic3, D6⋊C4, C2×D12, C2×C3⋊D4, C22×C12, C12⋊7D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, D12, C3⋊D4, C22×S3, C4⋊D4, C2×D12, C4○D12, C2×C3⋊D4, C12⋊7D4
Character table of C12⋊7D4
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 12 | 12 | 2 | 2 | 2 | 2 | 2 | 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 | 0 | -1 | 2 | 2 | 2 | 2 | 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 | 0 | -1 | 2 | 2 | -2 | -2 | 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 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | -2 | 2 | 0 | 0 | -2 | -2 | 0 | 0 | 2 | 0 | 0 | 0 | 2 | 0 | 0 | 2 | -2 | 0 | -2 | orthogonal lifted from D4 |
| ρ12 | 2 | -2 | 2 | -2 | 2 | -2 | 0 | 0 | 2 | 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 | 0 | -1 | -2 | -2 | 2 | 2 | 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 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 2 | -2 | 0 | 0 | -2 | -2 | 0 | 0 | 2 | 0 | 0 | 0 | -2 | 0 | 0 | -2 | 2 | 0 | 2 | orthogonal lifted from D4 |
| ρ15 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | -1 | -2 | -2 | -2 | -2 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | orthogonal lifted from D6 |
| ρ16 | 2 | -2 | 2 | -2 | -2 | 2 | 0 | 0 | 2 | 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 |
| ρ17 | 2 | -2 | 2 | -2 | -2 | 2 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | -√3 | √3 | -√3 | √3 | -√3 | -√3 | √3 | √3 | orthogonal lifted from D12 |
| ρ18 | 2 | -2 | 2 | -2 | 2 | -2 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | √3 | √3 | √3 | -√3 | -√3 | -√3 | -√3 | √3 | orthogonal lifted from D12 |
| ρ19 | 2 | -2 | 2 | -2 | -2 | 2 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | √3 | -√3 | √3 | -√3 | √3 | √3 | -√3 | -√3 | orthogonal lifted from D12 |
| ρ20 | 2 | -2 | 2 | -2 | 2 | -2 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | -√3 | -√3 | -√3 | √3 | √3 | √3 | √3 | -√3 | orthogonal lifted from D12 |
| ρ21 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 2 | -2 | 0 | 0 | 1 | 1 | √-3 | -√-3 | -1 | √-3 | -√-3 | -√-3 | 1 | √-3 | -√-3 | 1 | -1 | √-3 | -1 | complex lifted from C3⋊D4 |
| ρ22 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | -2 | 2 | 0 | 0 | 1 | 1 | √-3 | -√-3 | -1 | √-3 | -√-3 | √-3 | -1 | -√-3 | √-3 | -1 | 1 | -√-3 | 1 | complex lifted from C3⋊D4 |
| ρ23 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | -1 | 2i | -2i | 0 | 0 | 0 | 0 | -1 | 1 | √-3 | -√-3 | 1 | -√-3 | √-3 | -i | √3 | i | i | -√3 | √3 | -i | -√3 | complex lifted from C4○D12 |
| ρ24 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | -2 | 2 | 0 | 0 | 1 | 1 | -√-3 | √-3 | -1 | -√-3 | √-3 | -√-3 | -1 | √-3 | -√-3 | -1 | 1 | √-3 | 1 | complex lifted from C3⋊D4 |
| ρ25 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 2 | -2 | 0 | 0 | 1 | 1 | -√-3 | √-3 | -1 | -√-3 | √-3 | √-3 | 1 | -√-3 | √-3 | 1 | -1 | -√-3 | -1 | complex lifted from C3⋊D4 |
| ρ26 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 2 | 2i | -2i | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | -2 | 0 | 0 | 2i | 0 | -2i | -2i | 0 | 0 | 2i | 0 | complex lifted from C4○D4 |
| ρ27 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | -1 | -2i | 2i | 0 | 0 | 0 | 0 | -1 | 1 | √-3 | -√-3 | 1 | -√-3 | √-3 | i | -√3 | -i | -i | √3 | -√3 | i | √3 | complex lifted from C4○D12 |
| ρ28 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | -1 | 2i | -2i | 0 | 0 | 0 | 0 | -1 | 1 | -√-3 | √-3 | 1 | √-3 | -√-3 | -i | -√3 | i | i | √3 | -√3 | -i | √3 | complex lifted from C4○D12 |
| ρ29 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 2 | -2i | 2i | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | -2 | 0 | 0 | -2i | 0 | 2i | 2i | 0 | 0 | -2i | 0 | complex lifted from C4○D4 |
| ρ30 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | -1 | -2i | 2i | 0 | 0 | 0 | 0 | -1 | 1 | -√-3 | √-3 | 1 | √-3 | -√-3 | i | √3 | -i | -i | -√3 | √3 | i | -√3 | complex lifted from C4○D12 |
►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 33 42 14)(2 32 43 13)(3 31 44 24)(4 30 45 23)(5 29 46 22)(6 28 47 21)(7 27 48 20)(8 26 37 19)(9 25 38 18)(10 36 39 17)(11 35 40 16)(12 34 41 15)
(2 12)(3 11)(4 10)(5 9)(6 8)(13 34)(14 33)(15 32)(16 31)(17 30)(18 29)(19 28)(20 27)(21 26)(22 25)(23 36)(24 35)(37 47)(38 46)(39 45)(40 44)(41 43)
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), (1,33,42,14)(2,32,43,13)(3,31,44,24)(4,30,45,23)(5,29,46,22)(6,28,47,21)(7,27,48,20)(8,26,37,19)(9,25,38,18)(10,36,39,17)(11,35,40,16)(12,34,41,15), (2,12)(3,11)(4,10)(5,9)(6,8)(13,34)(14,33)(15,32)(16,31)(17,30)(18,29)(19,28)(20,27)(21,26)(22,25)(23,36)(24,35)(37,47)(38,46)(39,45)(40,44)(41,43)>;
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), (1,33,42,14)(2,32,43,13)(3,31,44,24)(4,30,45,23)(5,29,46,22)(6,28,47,21)(7,27,48,20)(8,26,37,19)(9,25,38,18)(10,36,39,17)(11,35,40,16)(12,34,41,15), (2,12)(3,11)(4,10)(5,9)(6,8)(13,34)(14,33)(15,32)(16,31)(17,30)(18,29)(19,28)(20,27)(21,26)(22,25)(23,36)(24,35)(37,47)(38,46)(39,45)(40,44)(41,43) );
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)], [(1,33,42,14),(2,32,43,13),(3,31,44,24),(4,30,45,23),(5,29,46,22),(6,28,47,21),(7,27,48,20),(8,26,37,19),(9,25,38,18),(10,36,39,17),(11,35,40,16),(12,34,41,15)], [(2,12),(3,11),(4,10),(5,9),(6,8),(13,34),(14,33),(15,32),(16,31),(17,30),(18,29),(19,28),(20,27),(21,26),(22,25),(23,36),(24,35),(37,47),(38,46),(39,45),(40,44),(41,43)]])
C12⋊7D4 is a maximal subgroup of
C6.C4≀C2 C22.2D24 D12⋊13D4 D12⋊14D4 C23.43D12 C22.D24 C23.18D12 Dic6⋊14D4 (C2×C6).40D8 C4⋊C4.228D6 C4⋊C4.236D6 C3⋊C8⋊22D4 C4⋊D4⋊S3 C3⋊C8⋊24D4 C3⋊C8⋊6D4 C24⋊30D4 C24⋊29D4 C24⋊2D4 C24⋊3D4 (C2×C6)⋊8D8 (C3×Q8)⋊13D4 (C3×D4)⋊14D4 C42.276D6 C42.277D6 C24.38D6 C23⋊4D12 C24.41D6 C6.2- 1+4 C6.2+ 1+4 C6.112+ 1+4 C6.62- 1+4 C42.95D6 C42.97D6 C42.99D6 C42.100D6 C42.104D6 D4×D12 D12⋊23D4 Dic6⋊23D4 Dic6⋊24D4 D4⋊5D12 D4⋊6D12 C42⋊19D6 C42.116D6 C42.117D6 C42.119D6 C12⋊(C4○D4) C6.322+ 1+4 S3×C4⋊D4 C6.372+ 1+4 C6.382+ 1+4 C6.472+ 1+4 C6.482+ 1+4 C4⋊C4.187D6 C4⋊C4⋊26D6 C6.172- 1+4 C6.242- 1+4 C6.562+ 1+4 C6.782- 1+4 C6.592+ 1+4 C6.612+ 1+4 C6.662+ 1+4 C6.682+ 1+4 C6.692+ 1+4 C24.83D6 D4×C3⋊D4 C6.452- 1+4 C6.1452+ 1+4 C6.1462+ 1+4 C6.1082- 1+4 C6.1482+ 1+4 C36⋊7D4 C22⋊D36 D6⋊D12 D6⋊2D12 C12⋊7D12 C62⋊6D4 C62⋊19D4 C12⋊S4 D10⋊D12 C60⋊D4 C12⋊7D20 (C2×C10)⋊4D12 C60⋊29D4
C12⋊7D4 is a maximal quotient of
C12⋊4(C4⋊C4) (C2×C4)⋊6D12 (C2×C42)⋊3S3 C24.21D6 C23⋊3D12 C24.27D6 (C2×C4).44D12 (C2×C4)⋊3D12 (C2×C12).56D4 C12⋊7D8 D4.1D12 D4.2D12 Q8⋊2D12 Q8.6D12 C12⋊7Q16 C24⋊30D4 C24⋊29D4 C24.82D4 C24⋊2D4 C24⋊3D4 C24.4D4 Q8.8D12 Q8.9D12 Q8.10D12 C24.75D6 C24.76D6 C36⋊7D4 D6⋊D12 D6⋊2D12 C12⋊7D12 C62⋊6D4 C62⋊19D4 D10⋊D12 C60⋊D4 C12⋊7D20 (C2×C10)⋊4D12 C60⋊29D4
Matrix representation of C12⋊7D4 ►in GL4(𝔽13) generated by
| 10 | 10 | 0 | 0 |
| 3 | 7 | 0 | 0 |
| 0 | 0 | 6 | 3 |
| 0 | 0 | 10 | 3 |
| 9 | 2 | 0 | 0 |
| 11 | 4 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 12 | 12 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 12 | 12 |
G:=sub<GL(4,GF(13))| [10,3,0,0,10,7,0,0,0,0,6,10,0,0,3,3],[9,11,0,0,2,4,0,0,0,0,1,12,0,0,0,12],[0,1,0,0,1,0,0,0,0,0,1,12,0,0,0,12] >;
C12⋊7D4 in GAP, Magma, Sage, TeX
C_{12}\rtimes_7D_4 % in TeX
G:=Group("C12:7D4"); // GroupNames label
G:=SmallGroup(96,137);
// by ID
G=gap.SmallGroup(96,137);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-3,217,103,218,2309]);
// Polycyclic
G:=Group<a,b,c|a^12=b^4=c^2=1,b*a*b^-1=c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations
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