p-group, metabelian, nilpotent (class 4), monomial
Aliases: C23.3C42, (C2xD4).2Q8, (C22xC8).9C4, (C2xQ8).39D4, C23.1(C4:C4), C4.D4.3C4, (C22xC4).37D4, C4.26(C23:C4), (C2xM4(2)).9C4, C2.14(C23.9D4), C22.3(C2.C42), M4(2).8C22.3C2, (C2xC4).8(C4:C4), (C2xD4).45(C2xC4), (C2xC4).2(C22:C4), (C22xC4).65(C2xC4), (C2xC4oD4).2C22, (C22xC8):C2.11C2, SmallGroup(128,124)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C23.3C42
G = < a,b,c,d,e | a2=b2=c2=1, d4=e4=c, ab=ba, ac=ca, dad-1=abc, ae=ea, dbd-1=bc=cb, be=eb, cd=dc, ce=ec, ede-1=abcd >
Subgroups: 176 in 79 conjugacy classes, 30 normal (16 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2xC4, C2xC4, C2xC4, D4, Q8, C23, C23, C2xC8, M4(2), C22xC4, C22xC4, C2xD4, C2xD4, C2xQ8, C4oD4, C22:C8, C4.D4, C4.10D4, C22xC8, C2xM4(2), C2xM4(2), C2xC4oD4, (C22xC8):C2, M4(2).8C22, C23.3C42
Quotients: C1, C2, C4, C22, C2xC4, D4, Q8, C42, C22:C4, C4:C4, C2.C42, C23:C4, C23.9D4, C23.3C42
Character table of C23.3C42
class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 8I | 8J | 8K | 8L | 8M | 8N | |
size | 1 | 1 | 2 | 4 | 4 | 4 | 1 | 1 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | |
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 | 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 | 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 | 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 | linear of order 2 |
ρ5 | 1 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | 1 | -i | i | -i | i | i | 1 | 1 | -1 | i | -1 | -i | i | -i | -i | linear of order 4 |
ρ6 | 1 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | 1 | i | -i | i | -i | -i | 1 | 1 | -1 | -i | -1 | i | -i | i | i | linear of order 4 |
ρ7 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | -1 | -1 | -i | i | -i | i | 1 | i | -i | i | -1 | -i | i | -i | 1 | -1 | linear of order 4 |
ρ8 | 1 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | 1 | -i | i | -i | i | -i | -1 | -1 | 1 | -i | 1 | -i | i | i | i | linear of order 4 |
ρ9 | 1 | 1 | 1 | -1 | 1 | -1 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | -1 | -1 | -1 | i | i | -i | -i | -i | i | 1 | 1 | -i | i | linear of order 4 |
ρ10 | 1 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | 1 | i | -i | i | -i | i | -1 | -1 | 1 | i | 1 | i | -i | -i | -i | linear of order 4 |
ρ11 | 1 | 1 | 1 | -1 | 1 | -1 | 1 | 1 | 1 | -1 | 1 | -1 | 1 | 1 | 1 | 1 | -i | i | -i | -i | i | i | -1 | -1 | i | -i | linear of order 4 |
ρ12 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | -1 | -1 | i | -i | i | -i | -1 | i | -i | i | 1 | -i | -i | i | -1 | 1 | linear of order 4 |
ρ13 | 1 | 1 | 1 | -1 | 1 | -1 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | -1 | -1 | -1 | -i | -i | i | i | i | -i | 1 | 1 | i | -i | linear of order 4 |
ρ14 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | -1 | -1 | -i | i | -i | i | -1 | -i | i | -i | 1 | i | i | -i | -1 | 1 | linear of order 4 |
ρ15 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | -1 | -1 | i | -i | i | -i | 1 | -i | i | -i | -1 | i | -i | i | 1 | -1 | linear of order 4 |
ρ16 | 1 | 1 | 1 | -1 | 1 | -1 | 1 | 1 | 1 | -1 | 1 | -1 | 1 | 1 | 1 | 1 | i | -i | i | i | -i | -i | -1 | -1 | -i | i | linear of order 4 |
ρ17 | 2 | 2 | 2 | 2 | -2 | -2 | 2 | 2 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
ρ18 | 2 | 2 | 2 | -2 | -2 | 2 | 2 | 2 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
ρ19 | 2 | 2 | 2 | -2 | -2 | -2 | -2 | -2 | -2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
ρ20 | 2 | 2 | 2 | 2 | -2 | 2 | -2 | -2 | -2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
ρ21 | 4 | 4 | -4 | 0 | 0 | 0 | 4 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C23:C4 |
ρ22 | 4 | 4 | -4 | 0 | 0 | 0 | -4 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C23:C4 |
ρ23 | 4 | -4 | 0 | 0 | 0 | 0 | -4i | 4i | 0 | 0 | 0 | 0 | 2ζ85 | 2ζ87 | 2ζ8 | 2ζ83 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
ρ24 | 4 | -4 | 0 | 0 | 0 | 0 | -4i | 4i | 0 | 0 | 0 | 0 | 2ζ8 | 2ζ83 | 2ζ85 | 2ζ87 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
ρ25 | 4 | -4 | 0 | 0 | 0 | 0 | 4i | -4i | 0 | 0 | 0 | 0 | 2ζ87 | 2ζ85 | 2ζ83 | 2ζ8 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
ρ26 | 4 | -4 | 0 | 0 | 0 | 0 | 4i | -4i | 0 | 0 | 0 | 0 | 2ζ83 | 2ζ8 | 2ζ87 | 2ζ85 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
(1 31)(2 28)(3 29)(4 26)(5 27)(6 32)(7 25)(8 30)(9 17)(10 18)(11 23)(12 24)(13 21)(14 22)(15 19)(16 20)
(2 6)(4 8)(9 13)(11 15)(17 21)(19 23)(26 30)(28 32)
(1 5)(2 6)(3 7)(4 8)(9 13)(10 14)(11 15)(12 16)(17 21)(18 22)(19 23)(20 24)(25 29)(26 30)(27 31)(28 32)
(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)
(1 16 7 14 5 12 3 10)(2 17 8 19 6 21 4 23)(9 30 15 32 13 26 11 28)(18 31 20 25 22 27 24 29)
G:=sub<Sym(32)| (1,31)(2,28)(3,29)(4,26)(5,27)(6,32)(7,25)(8,30)(9,17)(10,18)(11,23)(12,24)(13,21)(14,22)(15,19)(16,20), (2,6)(4,8)(9,13)(11,15)(17,21)(19,23)(26,30)(28,32), (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16)(17,21)(18,22)(19,23)(20,24)(25,29)(26,30)(27,31)(28,32), (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), (1,16,7,14,5,12,3,10)(2,17,8,19,6,21,4,23)(9,30,15,32,13,26,11,28)(18,31,20,25,22,27,24,29)>;
G:=Group( (1,31)(2,28)(3,29)(4,26)(5,27)(6,32)(7,25)(8,30)(9,17)(10,18)(11,23)(12,24)(13,21)(14,22)(15,19)(16,20), (2,6)(4,8)(9,13)(11,15)(17,21)(19,23)(26,30)(28,32), (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16)(17,21)(18,22)(19,23)(20,24)(25,29)(26,30)(27,31)(28,32), (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), (1,16,7,14,5,12,3,10)(2,17,8,19,6,21,4,23)(9,30,15,32,13,26,11,28)(18,31,20,25,22,27,24,29) );
G=PermutationGroup([[(1,31),(2,28),(3,29),(4,26),(5,27),(6,32),(7,25),(8,30),(9,17),(10,18),(11,23),(12,24),(13,21),(14,22),(15,19),(16,20)], [(2,6),(4,8),(9,13),(11,15),(17,21),(19,23),(26,30),(28,32)], [(1,5),(2,6),(3,7),(4,8),(9,13),(10,14),(11,15),(12,16),(17,21),(18,22),(19,23),(20,24),(25,29),(26,30),(27,31),(28,32)], [(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)], [(1,16,7,14,5,12,3,10),(2,17,8,19,6,21,4,23),(9,30,15,32,13,26,11,28),(18,31,20,25,22,27,24,29)]])
Matrix representation of C23.3C42 ►in GL4(F17) generated by
1 | 15 | 0 | 0 |
0 | 16 | 0 | 0 |
0 | 0 | 1 | 15 |
0 | 0 | 0 | 16 |
1 | 0 | 0 | 0 |
0 | 1 | 0 | 0 |
0 | 0 | 16 | 0 |
0 | 0 | 0 | 16 |
16 | 0 | 0 | 0 |
0 | 16 | 0 | 0 |
0 | 0 | 16 | 0 |
0 | 0 | 0 | 16 |
0 | 0 | 1 | 0 |
0 | 0 | 0 | 1 |
13 | 0 | 0 | 0 |
13 | 4 | 0 | 0 |
2 | 0 | 0 | 0 |
0 | 2 | 0 | 0 |
0 | 0 | 2 | 13 |
0 | 0 | 0 | 15 |
G:=sub<GL(4,GF(17))| [1,0,0,0,15,16,0,0,0,0,1,0,0,0,15,16],[1,0,0,0,0,1,0,0,0,0,16,0,0,0,0,16],[16,0,0,0,0,16,0,0,0,0,16,0,0,0,0,16],[0,0,13,13,0,0,0,4,1,0,0,0,0,1,0,0],[2,0,0,0,0,2,0,0,0,0,2,0,0,0,13,15] >;
C23.3C42 in GAP, Magma, Sage, TeX
C_2^3._3C_4^2
% in TeX
G:=Group("C2^3.3C4^2");
// GroupNames label
G:=SmallGroup(128,124);
// by ID
G=gap.SmallGroup(128,124);
# by ID
G:=PCGroup([7,-2,2,-2,2,2,-2,-2,56,85,120,758,570,521,248,172,4037]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=1,d^4=e^4=c,a*b=b*a,a*c=c*a,d*a*d^-1=a*b*c,a*e=e*a,d*b*d^-1=b*c=c*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=a*b*c*d>;
// generators/relations
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