metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C24.44D4, D12.26D4, Dic6.26D4, C8oD12:4C2, C4.66(S3xD4), (C2xC8).92D6, (C6xSD16):1C2, (C2xSD16):3S3, (C2xD4).78D6, (C2xQ8).83D6, C12.D4:8C2, C12.180(C2xD4), C3:5(D4.3D4), C8.32(C3:D4), C24.C4:10C2, Q8.11D6:4C2, C12.10D4:7C2, (C2xC24).48C22, D12:6C22.2C2, (C6xQ8).83C22, C2.21(D6:3D4), C6.118(C4:D4), (C2xC12).454C23, C4oD12.47C22, (C6xD4).103C22, C4.Dic3.20C22, C22.21(D4:2S3), C4.84(C2xC3:D4), (C2xC6).159(C4oD4), (C2xC4).127(C22xS3), 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, C2xC4, C2xC4, D4, Q8, C23, Dic3, C12, C12, D6, C2xC6, C2xC6, C2xC8, C2xC8, M4(2), D8, SD16, Q16, C2xD4, C2xQ8, C4oD4, C3:C8, C24, Dic6, C4xS3, D12, C3:D4, C2xC12, C2xC12, C3xD4, C3xQ8, C22xC6, C4.D4, C4.10D4, C8.C4, C8oD4, C2xSD16, C8:C22, C8.C22, S3xC8, C8:S3, C4.Dic3, D4:S3, D4.S3, Q8:2S3, C3:Q16, C2xC24, C3xSD16, C4oD12, C6xD4, C6xQ8, D4.3D4, C24.C4, C12.D4, C12.10D4, C8oD12, D12:6C22, Q8.11D6, C6xSD16, C24.44D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2xD4, C4oD4, C3:D4, C22xS3, C4:D4, S3xD4, D4:2S3, C2xC3: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 C4oD4 |
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 C4oD4 |
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 S3xD4 |
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 |
(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
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