p-group, metabelian, nilpotent (class 3), monomial
Aliases: C23.21C42, C23.5M4(2), C22:C8:5C4, C22:C4:2C8, (C22xC8):1C4, C22.6(C4xC8), C23.5(C2xC8), (C22xC4).4Q8, C22.9(C4:C8), C24.36(C2xC4), C4.32(C23:C4), C23.37(C4:C4), (C22xC4).637D4, (C23xC4).2C22, C22.5(C8:C4), C22.10(C22:C8), C2.2(C23.9D4), C2.2(M4(2):4C4), C23.135(C22:C4), C2.11(C22.7C42), C22.23(C2.C42), (C2xC4).67(C4:C4), (C2xC22:C8).2C2, (C4xC22:C4).1C2, (C2xC22:C4).14C4, (C22xC4).424(C2xC4), (C2xC4).371(C22:C4), SmallGroup(128,14)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C23.21C42
G = < a,b,c,d,e | a2=b2=c2=1, d4=e4=c, dad-1=ab=ba, ac=ca, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede-1=abd >
Subgroups: 240 in 114 conjugacy classes, 46 normal (26 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C8, C2xC4, C2xC4, C23, C23, C23, C42, C22:C4, C22:C4, C2xC8, C22xC4, C22xC4, C24, C2.C42, C22:C8, C22:C8, C2xC42, C2xC22:C4, C22xC8, C22xC8, C23xC4, C4xC22:C4, C2xC22:C8, C23.21C42
Quotients: C1, C2, C4, C22, C8, C2xC4, D4, Q8, C42, C22:C4, C4:C4, C2xC8, M4(2), C2.C42, C4xC8, C8:C4, C22:C8, C23:C4, C4:C8, C22.7C42, C23.9D4, M4(2):4C4, C23.21C42
(1 9)(2 32)(3 11)(4 26)(5 13)(6 28)(7 15)(8 30)(10 19)(12 21)(14 23)(16 17)(18 31)(20 25)(22 27)(24 29)
(1 18)(2 19)(3 20)(4 21)(5 22)(6 23)(7 24)(8 17)(9 31)(10 32)(11 25)(12 26)(13 27)(14 28)(15 29)(16 30)
(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 6 11 16 5 2 15 12)(3 17 13 32 7 21 9 28)(4 31 14 20 8 27 10 24)(18 23 25 30 22 19 29 26)
G:=sub<Sym(32)| (1,9)(2,32)(3,11)(4,26)(5,13)(6,28)(7,15)(8,30)(10,19)(12,21)(14,23)(16,17)(18,31)(20,25)(22,27)(24,29), (1,18)(2,19)(3,20)(4,21)(5,22)(6,23)(7,24)(8,17)(9,31)(10,32)(11,25)(12,26)(13,27)(14,28)(15,29)(16,30), (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), (1,6,11,16,5,2,15,12)(3,17,13,32,7,21,9,28)(4,31,14,20,8,27,10,24)(18,23,25,30,22,19,29,26)>;
G:=Group( (1,9)(2,32)(3,11)(4,26)(5,13)(6,28)(7,15)(8,30)(10,19)(12,21)(14,23)(16,17)(18,31)(20,25)(22,27)(24,29), (1,18)(2,19)(3,20)(4,21)(5,22)(6,23)(7,24)(8,17)(9,31)(10,32)(11,25)(12,26)(13,27)(14,28)(15,29)(16,30), (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), (1,6,11,16,5,2,15,12)(3,17,13,32,7,21,9,28)(4,31,14,20,8,27,10,24)(18,23,25,30,22,19,29,26) );
G=PermutationGroup([[(1,9),(2,32),(3,11),(4,26),(5,13),(6,28),(7,15),(8,30),(10,19),(12,21),(14,23),(16,17),(18,31),(20,25),(22,27),(24,29)], [(1,18),(2,19),(3,20),(4,21),(5,22),(6,23),(7,24),(8,17),(9,31),(10,32),(11,25),(12,26),(13,27),(14,28),(15,29),(16,30)], [(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)], [(1,6,11,16,5,2,15,12),(3,17,13,32,7,21,9,28),(4,31,14,20,8,27,10,24),(18,23,25,30,22,19,29,26)]])
44 conjugacy classes
class | 1 | 2A | 2B | 2C | 2D | ··· | 2I | 4A | 4B | 4C | 4D | 4E | ··· | 4J | 4K | ··· | 4R | 8A | ··· | 8P |
order | 1 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
44 irreducible representations
dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 |
type | + | + | + | + | - | + | ||||||
image | C1 | C2 | C2 | C4 | C4 | C4 | C8 | D4 | Q8 | M4(2) | C23:C4 | M4(2):4C4 |
kernel | C23.21C42 | C4xC22:C4 | C2xC22:C8 | C22:C8 | C2xC22:C4 | C22xC8 | C22:C4 | C22xC4 | C22xC4 | C23 | C4 | C2 |
# reps | 1 | 1 | 2 | 4 | 4 | 4 | 16 | 3 | 1 | 4 | 2 | 2 |
Matrix representation of C23.21C42 ►in GL6(F17)
16 | 0 | 0 | 0 | 0 | 0 |
0 | 16 | 0 | 0 | 0 | 0 |
0 | 0 | 16 | 2 | 0 | 0 |
0 | 0 | 0 | 1 | 0 | 0 |
0 | 0 | 0 | 0 | 16 | 2 |
0 | 0 | 0 | 0 | 0 | 1 |
1 | 0 | 0 | 0 | 0 | 0 |
0 | 1 | 0 | 0 | 0 | 0 |
0 | 0 | 16 | 0 | 0 | 0 |
0 | 0 | 0 | 16 | 0 | 0 |
0 | 0 | 0 | 0 | 16 | 0 |
0 | 0 | 0 | 0 | 0 | 16 |
16 | 0 | 0 | 0 | 0 | 0 |
0 | 16 | 0 | 0 | 0 | 0 |
0 | 0 | 1 | 0 | 0 | 0 |
0 | 0 | 0 | 1 | 0 | 0 |
0 | 0 | 0 | 0 | 1 | 0 |
0 | 0 | 0 | 0 | 0 | 1 |
15 | 9 | 0 | 0 | 0 | 0 |
1 | 2 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 1 | 15 |
0 | 0 | 0 | 0 | 1 | 16 |
0 | 0 | 16 | 0 | 0 | 0 |
0 | 0 | 16 | 1 | 0 | 0 |
15 | 9 | 0 | 0 | 0 | 0 |
0 | 2 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 1 | 0 |
0 | 0 | 0 | 0 | 0 | 1 |
0 | 0 | 16 | 0 | 0 | 0 |
0 | 0 | 0 | 16 | 0 | 0 |
G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,2,1,0,0,0,0,0,0,16,0,0,0,0,0,2,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[15,1,0,0,0,0,9,2,0,0,0,0,0,0,0,0,16,16,0,0,0,0,0,1,0,0,1,1,0,0,0,0,15,16,0,0],[15,0,0,0,0,0,9,2,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,1,0,0,0,0,0,0,1,0,0] >;
C23.21C42 in GAP, Magma, Sage, TeX
C_2^3._{21}C_4^2
% in TeX
G:=Group("C2^3.21C4^2");
// GroupNames label
G:=SmallGroup(128,14);
// by ID
G=gap.SmallGroup(128,14);
# by ID
G:=PCGroup([7,-2,2,-2,2,2,-2,2,56,85,120,758,570,248,172]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=1,d^4=e^4=c,d*a*d^-1=a*b=b*a,a*c=c*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=a*b*d>;
// generators/relations