p-group, metabelian, nilpotent (class 2), monomial
Aliases: C23.32C23, C42.34C22, C22.12C24, C2.12- 1+4, (C2×Q8)⋊9C4, (C4×Q8)⋊5C2, Q8○(C22⋊C4), Q8.8(C2×C4), C2.8(C23×C4), C4⋊C4.80C22, C4.20(C22×C4), (C22×Q8).7C2, (C2×C4).128C23, (C2×Q8).72C22, C42⋊C2.10C2, C22⋊C4.28C22, (C22×C4).57C22, C22.12(C22×C4), (C2×C4).31(C2×C4), SmallGroup(64,200)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C23.32C23
G = < a,b,c,d,e,f | a2=b2=c2=1, d2=c, e2=f2=b, dad-1=ab=ba, ac=ca, ae=ea, af=fa, bc=cb, ede-1=bd=db, fef-1=be=eb, bf=fb, cd=dc, ce=ec, cf=fc, df=fd >
Subgroups: 145 in 133 conjugacy classes, 121 normal (6 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C2×C4, Q8, C23, C42, C22⋊C4, C4⋊C4, C22×C4, C2×Q8, C42⋊C2, C4×Q8, C22×Q8, C23.32C23
Quotients: C1, C2, C4, C22, C2×C4, C23, C22×C4, C24, C23×C4, 2- 1+4, C23.32C23
►On 32 points
(2 28)(4 26)(5 30)(7 32)(10 16)(12 14)(18 24)(20 22)
(1 27)(2 28)(3 25)(4 26)(5 30)(6 31)(7 32)(8 29)(9 15)(10 16)(11 13)(12 14)(17 23)(18 24)(19 21)(20 22)
(1 3)(2 4)(5 7)(6 8)(9 11)(10 12)(13 15)(14 16)(17 19)(18 20)(21 23)(22 24)(25 27)(26 28)(29 31)(30 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 31 27 6)(2 7 28 32)(3 29 25 8)(4 5 26 30)(9 17 15 23)(10 24 16 18)(11 19 13 21)(12 22 14 20)
(1 15 27 9)(2 16 28 10)(3 13 25 11)(4 14 26 12)(5 22 30 20)(6 23 31 17)(7 24 32 18)(8 21 29 19)
G:=sub<Sym(32)| (2,28)(4,26)(5,30)(7,32)(10,16)(12,14)(18,24)(20,22), (1,27)(2,28)(3,25)(4,26)(5,30)(6,31)(7,32)(8,29)(9,15)(10,16)(11,13)(12,14)(17,23)(18,24)(19,21)(20,22), (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,15)(14,16)(17,19)(18,20)(21,23)(22,24)(25,27)(26,28)(29,31)(30,32), (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), (1,31,27,6)(2,7,28,32)(3,29,25,8)(4,5,26,30)(9,17,15,23)(10,24,16,18)(11,19,13,21)(12,22,14,20), (1,15,27,9)(2,16,28,10)(3,13,25,11)(4,14,26,12)(5,22,30,20)(6,23,31,17)(7,24,32,18)(8,21,29,19)>;
G:=Group( (2,28)(4,26)(5,30)(7,32)(10,16)(12,14)(18,24)(20,22), (1,27)(2,28)(3,25)(4,26)(5,30)(6,31)(7,32)(8,29)(9,15)(10,16)(11,13)(12,14)(17,23)(18,24)(19,21)(20,22), (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,15)(14,16)(17,19)(18,20)(21,23)(22,24)(25,27)(26,28)(29,31)(30,32), (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), (1,31,27,6)(2,7,28,32)(3,29,25,8)(4,5,26,30)(9,17,15,23)(10,24,16,18)(11,19,13,21)(12,22,14,20), (1,15,27,9)(2,16,28,10)(3,13,25,11)(4,14,26,12)(5,22,30,20)(6,23,31,17)(7,24,32,18)(8,21,29,19) );
G=PermutationGroup([[(2,28),(4,26),(5,30),(7,32),(10,16),(12,14),(18,24),(20,22)], [(1,27),(2,28),(3,25),(4,26),(5,30),(6,31),(7,32),(8,29),(9,15),(10,16),(11,13),(12,14),(17,23),(18,24),(19,21),(20,22)], [(1,3),(2,4),(5,7),(6,8),(9,11),(10,12),(13,15),(14,16),(17,19),(18,20),(21,23),(22,24),(25,27),(26,28),(29,31),(30,32)], [(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)], [(1,31,27,6),(2,7,28,32),(3,29,25,8),(4,5,26,30),(9,17,15,23),(10,24,16,18),(11,19,13,21),(12,22,14,20)], [(1,15,27,9),(2,16,28,10),(3,13,25,11),(4,14,26,12),(5,22,30,20),(6,23,31,17),(7,24,32,18),(8,21,29,19)]])
C23.32C23 is a maximal subgroup of
(C2×Q8).211D4 C4.10D4⋊2C4 C4⋊Q8⋊15C4 C4.4D4⋊13C4 C23.4C24 C42.276C23 C42.278C23 C42.279C23 C42.16C23 C42.17C23 C42.21C23 C42.355C23 C42.360C23 C42.361C23 C22.14C25 C4×2- 1+4 C22.75C25 C22.76C25 C22.84C25 C22.99C25 C22.105C25 C22.113C25 (C2×Q8)⋊C12 D5.2- 1+4
C42.D2p: C42.6D4 C42.7D4 C42.87D6 C42.125D6 C42.87D10 C42.125D10 C42.87D14 C42.125D14 ...
C2p.2- 1+4: C22.91C25 C22.98C25 C22.107C25 C23.146C24 C6.422- 1+4 C10.422- 1+4 C14.422- 1+4 ...
C23.32C23 is a maximal quotient of
C24.524C23 Q8⋊4C42 C23.192C24 C23.195C24 C24.545C23 C23.202C24 C42⋊4Q8 C23.214C24 C23.218C24 Q8×C22⋊C4 C23.226C24 C23.233C24 C23.238C24 C23.244C24 C23.247C24 C23.253C24 C24.227C23 C23.263C24 C23.264C24 D5.2- 1+4
C42.D2p: C42.159D4 C42.161D4 C42.87D6 C42.125D6 C42.87D10 C42.125D10 C42.87D14 C42.125D14 ...
C2p.2- 1+4: C42.695C23 C42.302C23 Q8.4M4(2) C42.696C23 C42.304C23 C42.305C23 C6.422- 1+4 C10.422- 1+4 ...
34 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | ··· | 4AB |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | ··· | 2 |
34 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 4 |
| type | + | + | + | + | - | |
| image | C1 | C2 | C2 | C2 | C4 | 2- 1+4 |
| kernel | C23.32C23 | C42⋊C2 | C4×Q8 | C22×Q8 | C2×Q8 | C2 |
| # reps | 1 | 6 | 8 | 1 | 16 | 2 |
Matrix representation of C23.32C23 ►in GL5(𝔽5)
| 4 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 |
| 0 | 2 | 2 | 0 | 4 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 4 |
| 4 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 4 |
| 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 2 | 2 | 4 | 3 |
| 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 2 | 0 | 0 | 0 |
| 0 | 0 | 3 | 0 | 0 |
| 0 | 0 | 0 | 3 | 0 |
| 0 | 0 | 1 | 2 | 2 |
| 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 |
| 0 | 2 | 2 | 4 | 3 |
| 0 | 3 | 0 | 1 | 1 |
G:=sub<GL(5,GF(5))| [4,0,0,0,0,0,1,0,0,2,0,0,1,0,2,0,0,0,4,0,0,0,0,0,4],[1,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,4],[4,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,4],[3,0,0,0,0,0,0,2,4,0,0,0,2,0,0,0,1,4,0,0,0,0,3,0,3],[1,0,0,0,0,0,2,0,0,0,0,0,3,0,1,0,0,0,3,2,0,0,0,0,2],[4,0,0,0,0,0,0,4,2,3,0,1,0,2,0,0,0,0,4,1,0,0,0,3,1] >;
C23.32C23 in GAP, Magma, Sage, TeX
C_2^3._{32}C_2^3 % in TeX
G:=Group("C2^3.32C2^3"); // GroupNames label
G:=SmallGroup(64,200);
// by ID
G=gap.SmallGroup(64,200);
# by ID
G:=PCGroup([6,-2,2,2,2,-2,2,192,217,103,188,86,579]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=1,d^2=c,e^2=f^2=b,d*a*d^-1=a*b=b*a,a*c=c*a,a*e=e*a,a*f=f*a,b*c=c*b,e*d*e^-1=b*d=d*b,f*e*f^-1=b*e=e*b,b*f=f*b,c*d=d*c,c*e=e*c,c*f=f*c,d*f=f*d>;
// generators/relations