metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C24.44D4, D12.26D4, Dic6.26D4, C8○D12⋊4C2, C4.66(S3×D4), (C2×C8).92D6, (C6×SD16)⋊1C2, (C2×SD16)⋊3S3, (C2×D4).78D6, (C2×Q8).83D6, C12.D4⋊8C2, C12.180(C2×D4), C3⋊5(D4.3D4), C8.32(C3⋊D4), C24.C4⋊10C2, Q8.11D6⋊4C2, C12.10D4⋊7C2, (C2×C24).48C22, D12⋊6C22.2C2, (C6×Q8).83C22, C2.21(D6⋊3D4), C6.118(C4⋊D4), (C2×C12).454C23, C4○D12.47C22, (C6×D4).103C22, C4.Dic3.20C22, C22.21(D4⋊2S3), C4.84(C2×C3⋊D4), (C2×C6).159(C4○D4), (C2×C4).127(C22×S3), SmallGroup(192,736)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C24.44D4
G = < a,b,c | a24=c2=1, b4=a12, bab-1=a11, cac=a17, cbc=a12b3 >
Subgroups: 280 in 104 conjugacy classes, 37 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C8, C2×C4, C2×C4, D4, Q8, C23, Dic3, C12, C12, D6, C2×C6, C2×C6, C2×C8, C2×C8, M4(2), D8, SD16, Q16, C2×D4, C2×Q8, C4○D4, C3⋊C8, C24, Dic6, C4×S3, D12, C3⋊D4, C2×C12, C2×C12, C3×D4, C3×Q8, C22×C6, C4.D4, C4.10D4, C8.C4, C8○D4, C2×SD16, C8⋊C22, C8.C22, S3×C8, C8⋊S3, C4.Dic3, D4⋊S3, D4.S3, Q8⋊2S3, C3⋊Q16, C2×C24, C3×SD16, C4○D12, C6×D4, C6×Q8, D4.3D4, C24.C4, C12.D4, C12.10D4, C8○D12, D12⋊6C22, Q8.11D6, C6×SD16, C24.44D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, C3⋊D4, C22×S3, C4⋊D4, S3×D4, D4⋊2S3, C2×C3⋊D4, D4.3D4, D6⋊3D4, C24.44D4
Character table of C24.44D4
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | 4B | 4C | 4D | 6A | 6B | 6C | 6D | 6E | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 12A | 12B | 12C | 12D | 24A | 24B | 24C | 24D | |
| size | 1 | 1 | 2 | 8 | 12 | 2 | 2 | 2 | 8 | 12 | 2 | 2 | 2 | 8 | 8 | 2 | 2 | 4 | 12 | 12 | 24 | 24 | 4 | 4 | 8 | 8 | 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 | 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 | 0 | -1 | 2 | 2 | 2 | 0 | -1 | -1 | -1 | -1 | -1 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ10 | 2 | 2 | 2 | -2 | 0 | -1 | 2 | 2 | -2 | 0 | -1 | -1 | -1 | 1 | 1 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -1 | -1 | 1 | 1 | -1 | -1 | -1 | -1 | orthogonal lifted from D6 |
| ρ11 | 2 | 2 | -2 | 0 | 0 | 2 | -2 | 2 | 0 | 0 | -2 | -2 | 2 | 0 | 0 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | -2 | 2 | 2 | -2 | orthogonal lifted from D4 |
| ρ12 | 2 | 2 | -2 | 0 | 2 | 2 | 2 | -2 | 0 | -2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ13 | 2 | 2 | -2 | 0 | -2 | 2 | 2 | -2 | 0 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ14 | 2 | 2 | 2 | 2 | 0 | -1 | 2 | 2 | -2 | 0 | -1 | -1 | -1 | -1 | -1 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | orthogonal lifted from D6 |
| ρ15 | 2 | 2 | 2 | -2 | 0 | -1 | 2 | 2 | 2 | 0 | -1 | -1 | -1 | 1 | 1 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | orthogonal lifted from D6 |
| ρ16 | 2 | 2 | -2 | 0 | 0 | 2 | -2 | 2 | 0 | 0 | -2 | -2 | 2 | 0 | 0 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 2 | -2 | -2 | 2 | orthogonal lifted from D4 |
| ρ17 | 2 | 2 | -2 | 0 | 0 | -1 | -2 | 2 | 0 | 0 | 1 | 1 | -1 | -√-3 | √-3 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | -1 | 1 | -√-3 | √-3 | -1 | 1 | 1 | -1 | complex lifted from C3⋊D4 |
| ρ18 | 2 | 2 | -2 | 0 | 0 | -1 | -2 | 2 | 0 | 0 | 1 | 1 | -1 | √-3 | -√-3 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | -1 | 1 | -√-3 | √-3 | 1 | -1 | -1 | 1 | complex lifted from C3⋊D4 |
| ρ19 | 2 | 2 | -2 | 0 | 0 | -1 | -2 | 2 | 0 | 0 | 1 | 1 | -1 | √-3 | -√-3 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | -1 | 1 | √-3 | -√-3 | -1 | 1 | 1 | -1 | complex lifted from C3⋊D4 |
| ρ20 | 2 | 2 | -2 | 0 | 0 | -1 | -2 | 2 | 0 | 0 | 1 | 1 | -1 | -√-3 | √-3 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | -1 | 1 | √-3 | -√-3 | 1 | -1 | -1 | 1 | complex lifted from C3⋊D4 |
| ρ21 | 2 | 2 | 2 | 0 | 0 | 2 | -2 | -2 | 0 | 0 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | -2i | 2i | 0 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ22 | 2 | 2 | 2 | 0 | 0 | 2 | -2 | -2 | 0 | 0 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 2i | -2i | 0 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ23 | 4 | 4 | -4 | 0 | 0 | -2 | 4 | -4 | 0 | 0 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S3×D4 |
| ρ24 | 4 | 4 | 4 | 0 | 0 | -2 | -4 | -4 | 0 | 0 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from D4⋊2S3, Schur index 2 |
| ρ25 | 4 | -4 | 0 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | -4 | 0 | 0 | -2√-2 | 2√-2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2√-2 | 0 | 0 | -2√-2 | complex lifted from D4.3D4 |
| ρ26 | 4 | -4 | 0 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | -4 | 0 | 0 | 2√-2 | -2√-2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2√-2 | 0 | 0 | 2√-2 | complex lifted from D4.3D4 |
| ρ27 | 4 | -4 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 2√-3 | -2√-3 | 2 | 0 | 0 | -2√-2 | 2√-2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -√-2 | -√6 | √6 | √-2 | complex faithful |
| ρ28 | 4 | -4 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | -2√-3 | 2√-3 | 2 | 0 | 0 | -2√-2 | 2√-2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -√-2 | √6 | -√6 | √-2 | complex faithful |
| ρ29 | 4 | -4 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 2√-3 | -2√-3 | 2 | 0 | 0 | 2√-2 | -2√-2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | √-2 | √6 | -√6 | -√-2 | complex faithful |
| ρ30 | 4 | -4 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | -2√-3 | 2√-3 | 2 | 0 | 0 | 2√-2 | -2√-2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | √-2 | -√6 | √6 | -√-2 | complex faithful |
►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 19 37 13 43 7 25)(2 42 20 48 14 30 8 36)(3 29 21 35 15 41 9 47)(4 40 22 46 16 28 10 34)(5 27 23 33 17 39 11 45)(6 38 24 44 18 26 12 32)
(1 25)(2 42)(3 35)(4 28)(5 45)(6 38)(7 31)(8 48)(9 41)(10 34)(11 27)(12 44)(13 37)(14 30)(15 47)(16 40)(17 33)(18 26)(19 43)(20 36)(21 29)(22 46)(23 39)(24 32)
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), (1,31,19,37,13,43,7,25)(2,42,20,48,14,30,8,36)(3,29,21,35,15,41,9,47)(4,40,22,46,16,28,10,34)(5,27,23,33,17,39,11,45)(6,38,24,44,18,26,12,32), (1,25)(2,42)(3,35)(4,28)(5,45)(6,38)(7,31)(8,48)(9,41)(10,34)(11,27)(12,44)(13,37)(14,30)(15,47)(16,40)(17,33)(18,26)(19,43)(20,36)(21,29)(22,46)(23,39)(24,32)>;
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), (1,31,19,37,13,43,7,25)(2,42,20,48,14,30,8,36)(3,29,21,35,15,41,9,47)(4,40,22,46,16,28,10,34)(5,27,23,33,17,39,11,45)(6,38,24,44,18,26,12,32), (1,25)(2,42)(3,35)(4,28)(5,45)(6,38)(7,31)(8,48)(9,41)(10,34)(11,27)(12,44)(13,37)(14,30)(15,47)(16,40)(17,33)(18,26)(19,43)(20,36)(21,29)(22,46)(23,39)(24,32) );
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)], [(1,31,19,37,13,43,7,25),(2,42,20,48,14,30,8,36),(3,29,21,35,15,41,9,47),(4,40,22,46,16,28,10,34),(5,27,23,33,17,39,11,45),(6,38,24,44,18,26,12,32)], [(1,25),(2,42),(3,35),(4,28),(5,45),(6,38),(7,31),(8,48),(9,41),(10,34),(11,27),(12,44),(13,37),(14,30),(15,47),(16,40),(17,33),(18,26),(19,43),(20,36),(21,29),(22,46),(23,39),(24,32)]])
Matrix representation of C24.44D4 ►in GL4(𝔽73) generated by
| 54 | 19 | 0 | 0 |
| 54 | 54 | 0 | 0 |
| 0 | 0 | 25 | 48 |
| 0 | 0 | 25 | 25 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 72 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
G:=sub<GL(4,GF(73))| [54,54,0,0,19,54,0,0,0,0,25,25,0,0,48,25],[0,0,0,1,0,0,1,0,1,0,0,0,0,72,0,0],[0,0,1,0,0,0,0,1,1,0,0,0,0,1,0,0] >;
C24.44D4 in GAP, Magma, Sage, TeX
C_{24}._{44}D_4 % in TeX
G:=Group("C24.44D4"); // GroupNames label
G:=SmallGroup(192,736);
// by ID
G=gap.SmallGroup(192,736);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,120,254,555,1123,297,136,438,102,6278]);
// Polycyclic
G:=Group<a,b,c|a^24=c^2=1,b^4=a^12,b*a*b^-1=a^11,c*a*c=a^17,c*b*c=a^12*b^3>;
// generators/relations
Export