p-group, metabelian, nilpotent (class 2), monomial
Aliases: C23.33C23, C22.13C24, C42.35C22, C2.22+ 1+4, C2.22- 1+4, D4○(C4⋊C4), Q8○(C4⋊C4), C4○D4⋊5C4, (C4×D4)⋊6C2, D4⋊8(C2×C4), (C4×Q8)⋊6C2, Q8⋊7(C2×C4), C2.9(C23×C4), C42⋊C2⋊7C2, C4⋊C4.81C22, (C2×C4).51C23, C4.21(C22×C4), (C2×D4).78C22, C22.3(C22×C4), (C2×Q8).73C22, C22⋊C4.29C22, (C22×C4).58C22, C4⋊C4○(C2×D4), (C2×C4)⋊5(C2×C4), (C2×C4⋊C4)⋊13C2, C22⋊C4○(C4⋊C4), (C2×C4○D4).9C2, SmallGroup(64,201)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C23.33C23
G = < a,b,c,d,e,f | a2=b2=c2=e2=1, d2=c, f2=b, eae=ab=ba, ac=ca, ad=da, af=fa, bc=cb, fdf-1=bd=db, fef-1=be=eb, bf=fb, cd=dc, ce=ec, cf=fc, de=ed >
Subgroups: 185 in 147 conjugacy classes, 121 normal (10 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C42, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C2×D4, C2×Q8, C4○D4, C2×C4⋊C4, C42⋊C2, C4×D4, C4×Q8, C2×C4○D4, C23.33C23
Quotients: C1, C2, C4, C22, C2×C4, C23, C22×C4, C24, C23×C4, 2+ 1+4, 2- 1+4, C23.33C23
►On 32 points
(5 30)(6 31)(7 32)(8 29)(17 23)(18 24)(19 21)(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)(2 32)(3 29)(4 30)(5 26)(6 27)(7 28)(8 25)(9 17)(10 18)(11 19)(12 20)(13 21)(14 22)(15 23)(16 24)
(1 15 27 9)(2 10 28 16)(3 13 25 11)(4 12 26 14)(5 20 30 22)(6 23 31 17)(7 18 32 24)(8 21 29 19)
G:=sub<Sym(32)| (5,30)(6,31)(7,32)(8,29)(17,23)(18,24)(19,21)(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)(2,32)(3,29)(4,30)(5,26)(6,27)(7,28)(8,25)(9,17)(10,18)(11,19)(12,20)(13,21)(14,22)(15,23)(16,24), (1,15,27,9)(2,10,28,16)(3,13,25,11)(4,12,26,14)(5,20,30,22)(6,23,31,17)(7,18,32,24)(8,21,29,19)>;
G:=Group( (5,30)(6,31)(7,32)(8,29)(17,23)(18,24)(19,21)(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)(2,32)(3,29)(4,30)(5,26)(6,27)(7,28)(8,25)(9,17)(10,18)(11,19)(12,20)(13,21)(14,22)(15,23)(16,24), (1,15,27,9)(2,10,28,16)(3,13,25,11)(4,12,26,14)(5,20,30,22)(6,23,31,17)(7,18,32,24)(8,21,29,19) );
G=PermutationGroup([[(5,30),(6,31),(7,32),(8,29),(17,23),(18,24),(19,21),(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),(2,32),(3,29),(4,30),(5,26),(6,27),(7,28),(8,25),(9,17),(10,18),(11,19),(12,20),(13,21),(14,22),(15,23),(16,24)], [(1,15,27,9),(2,10,28,16),(3,13,25,11),(4,12,26,14),(5,20,30,22),(6,23,31,17),(7,18,32,24),(8,21,29,19)]])
C23.33C23 is a maximal subgroup of
C4≀C2⋊C4 C42⋊9(C2×C4) C8.C22⋊C4 C8⋊C22⋊C4 2+ 1+4⋊5C4 2- 1+4⋊4C4 C4○D4.7Q8 C4○D4.8Q8 C42.275C23 C42.276C23 C42.280C23 C42.281C23 C42.18C23 C42.19C23 C42.22C23 C42.23C23 (C2×D4).301D4 (C2×D4).302D4 (C2×D4).303D4 (C2×D4).304D4 C42.353C23 C42.354C23 C42.358C23 C42.359C23 C22.14C25 C4×2+ 1+4 C4×2- 1+4 C22.77C25 C22.78C25 C22.81C25 C22.82C25 C22.83C25 C22.84C25 C22.93C25 C22.96C25 C22.101C25 C22.102C25 C22.104C25 C22.105C25 C22.110C25 C22.111C25 C22.113C25 C4○D4⋊C12 D5.2+ 1+4
C2p.2+ 1+4: C22.92C25 C22.95C25 C22.100C25 C22.106C25 C23.144C24 C6.82+ 1+4 C42.91D6 C42.108D6 ...
C23.33C23 is a maximal quotient of
C24.524C23 D4⋊4C42 Q8⋊4C42 C24.542C23 C24.192C23 C23.199C24 C24.547C23 C23.201C24 C23.202C24 C24.195C23 C42⋊13D4 C24.198C23 C23.211C24 C42.33Q8 C24.203C23 C24.204C23 C23.218C24 C24.205C23 C24.549C23 C23.223C24 C23.225C24 C24.208C23 C23.229C24 D4×C4⋊C4 Q8×C4⋊C4 C23.234C24 C23.236C24 C23.237C24 C23.241C24 C24.558C23 C24.215C23 C24.220C23 C23.252C24 C23.255C24 C24.225C23 C23.259C24 C24.227C23 C23.261C24 C23.264C24 C24.230C23 D5.2+ 1+4
C42.D2p: C42.159D4 C42.160D4 C42.91D6 C42.108D6 C42.126D6 C42.91D10 C42.108D10 C42.126D10 ...
C2p.2+ 1+4: C42.697C23 C42.698C23 D4⋊8M4(2) Q8⋊7M4(2) C42.307C23 C42.308C23 C42.309C23 C42.310C23 ...
34 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | ··· | 2I | 4A | ··· | 4X |
| 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 | 1 | 1 | 4 | 4 |
| type | + | + | + | + | + | + | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | 2+ 1+4 | 2- 1+4 |
| kernel | C23.33C23 | C2×C4⋊C4 | C42⋊C2 | C4×D4 | C4×Q8 | C2×C4○D4 | C4○D4 | C2 | C2 |
| # reps | 1 | 3 | 3 | 6 | 2 | 1 | 16 | 1 | 1 |
Matrix representation of C23.33C23 ►in GL5(𝔽5)
| 4 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 3 | 4 | 0 |
| 0 | 1 | 4 | 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 |
| 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 0 | 0 |
| 0 | 3 | 0 | 0 | 0 |
| 0 | 0 | 0 | 3 | 4 |
| 0 | 0 | 0 | 0 | 2 |
| 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 4 | 0 |
| 0 | 4 | 0 | 4 | 2 |
| 0 | 0 | 0 | 1 | 3 |
| 0 | 0 | 0 | 0 | 4 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 2 |
| 0 | 0 | 0 | 4 | 1 |
G:=sub<GL(5,GF(5))| [4,0,0,0,0,0,1,0,0,1,0,0,1,3,4,0,0,0,4,0,0,0,0,0,4],[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],[2,0,0,0,0,0,0,3,0,0,0,3,0,0,0,0,0,0,3,0,0,0,0,4,2],[4,0,0,0,0,0,0,4,0,0,0,4,0,0,0,0,4,4,1,0,0,0,2,3,4],[1,0,0,0,0,0,0,4,0,0,0,1,0,0,0,0,0,0,4,4,0,0,0,2,1] >;
C23.33C23 in GAP, Magma, Sage, TeX
C_2^3._{33}C_2^3 % in TeX
G:=Group("C2^3.33C2^3"); // GroupNames label
G:=SmallGroup(64,201);
// by ID
G=gap.SmallGroup(64,201);
# by ID
G:=PCGroup([6,-2,2,2,2,-2,2,192,217,188,579,69]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=e^2=1,d^2=c,f^2=b,e*a*e=a*b=b*a,a*c=c*a,a*d=d*a,a*f=f*a,b*c=c*b,f*d*f^-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*e=e*d>;
// generators/relations