p-group, metabelian, nilpotent (class 3), monomial
Aliases: C23.36D4, C4○D4⋊2C4, D4⋊4(C2×C4), Q8⋊4(C2×C4), C4.50(C2×D4), (C2×C4).123D4, C4.4(C22×C4), D4⋊C4⋊14C2, Q8⋊C4⋊14C2, C4⋊C4.44C22, C2.1(C8⋊C22), (C2×C8).43C22, (C2×C4).62C23, C22.44(C2×D4), C4.24(C22⋊C4), (C2×M4(2))⋊10C2, (C2×D4).48C22, C2.1(C8.C22), (C2×Q8).42C22, C22.4(C22⋊C4), (C22×C4).34C22, (C2×C4⋊C4)⋊10C2, (C2×C4).21(C2×C4), (C2×C4○D4).5C2, C2.20(C2×C22⋊C4), SmallGroup(64,98)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C23.36D4
G = < a,b,c,d,e | a2=b2=c2=1, d4=c, e2=b, ab=ba, dad-1=ac=ca, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede-1=bcd3 >
Subgroups: 137 in 81 conjugacy classes, 41 normal (17 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C23, C4⋊C4, C4⋊C4, C2×C8, M4(2), C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C4○D4, D4⋊C4, Q8⋊C4, C2×C4⋊C4, C2×M4(2), C2×C4○D4, C23.36D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, C2×C22⋊C4, C8⋊C22, C8.C22, C23.36D4
Character table of C23.36D4
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 8A | 8B | 8C | 8D | |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 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 | trivial |
| ρ2 | 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 | 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 | 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 | 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 | 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 | 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 | linear of order 2 |
| ρ9 | 1 | -1 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | -1 | 1 | -1 | -i | -1 | i | -i | i | 1 | -i | i | i | -i | linear of order 4 |
| ρ10 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | -i | 1 | i | i | -i | -1 | -i | -i | i | i | linear of order 4 |
| ρ11 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | -i | -1 | i | i | -i | 1 | i | i | -i | -i | linear of order 4 |
| ρ12 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | -i | 1 | i | -i | i | -1 | i | -i | -i | i | linear of order 4 |
| ρ13 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | i | 1 | -i | i | -i | -1 | -i | i | i | -i | linear of order 4 |
| ρ14 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | i | -1 | -i | -i | i | 1 | -i | -i | i | i | linear of order 4 |
| ρ15 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | i | 1 | -i | -i | i | -1 | i | i | -i | -i | linear of order 4 |
| ρ16 | 1 | -1 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | -1 | 1 | -1 | i | -1 | -i | i | -i | 1 | i | -i | -i | i | linear of order 4 |
| ρ17 | 2 | 2 | 2 | 2 | -2 | -2 | 0 | 0 | -2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ18 | 2 | -2 | 2 | -2 | -2 | 2 | 0 | 0 | -2 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ19 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | -2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ20 | 2 | -2 | 2 | -2 | 2 | -2 | 0 | 0 | -2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ21 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C8⋊C22 |
| ρ22 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from C8.C22, Schur index 2 |
►On 32 points
(1 19)(2 24)(3 21)(4 18)(5 23)(6 20)(7 17)(8 22)(9 25)(10 30)(11 27)(12 32)(13 29)(14 26)(15 31)(16 28)
(1 27)(2 28)(3 29)(4 30)(5 31)(6 32)(7 25)(8 26)(9 17)(10 18)(11 19)(12 20)(13 21)(14 22)(15 23)(16 24)
(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 26 27 8)(2 7 28 25)(3 32 29 6)(4 5 30 31)(9 24 17 16)(10 15 18 23)(11 22 19 14)(12 13 20 21)
G:=sub<Sym(32)| (1,19)(2,24)(3,21)(4,18)(5,23)(6,20)(7,17)(8,22)(9,25)(10,30)(11,27)(12,32)(13,29)(14,26)(15,31)(16,28), (1,27)(2,28)(3,29)(4,30)(5,31)(6,32)(7,25)(8,26)(9,17)(10,18)(11,19)(12,20)(13,21)(14,22)(15,23)(16,24), (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), (1,26,27,8)(2,7,28,25)(3,32,29,6)(4,5,30,31)(9,24,17,16)(10,15,18,23)(11,22,19,14)(12,13,20,21)>;
G:=Group( (1,19)(2,24)(3,21)(4,18)(5,23)(6,20)(7,17)(8,22)(9,25)(10,30)(11,27)(12,32)(13,29)(14,26)(15,31)(16,28), (1,27)(2,28)(3,29)(4,30)(5,31)(6,32)(7,25)(8,26)(9,17)(10,18)(11,19)(12,20)(13,21)(14,22)(15,23)(16,24), (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), (1,26,27,8)(2,7,28,25)(3,32,29,6)(4,5,30,31)(9,24,17,16)(10,15,18,23)(11,22,19,14)(12,13,20,21) );
G=PermutationGroup([[(1,19),(2,24),(3,21),(4,18),(5,23),(6,20),(7,17),(8,22),(9,25),(10,30),(11,27),(12,32),(13,29),(14,26),(15,31),(16,28)], [(1,27),(2,28),(3,29),(4,30),(5,31),(6,32),(7,25),(8,26),(9,17),(10,18),(11,19),(12,20),(13,21),(14,22),(15,23),(16,24)], [(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)], [(1,26,27,8),(2,7,28,25),(3,32,29,6),(4,5,30,31),(9,24,17,16),(10,15,18,23),(11,22,19,14),(12,13,20,21)]])
C23.36D4 is a maximal subgroup of
2+ 1+4⋊3C4 2- 1+4⋊2C4 C42⋊11D4 C42⋊12D4 C24.98D4 2+ 1+4⋊5C4 2- 1+4⋊4C4 C4×C8.C22 C24.105D4 C24.106D4 (C2×D4)⋊21D4 (C2×Q8)⋊17D4 C42.211D4 C42.212D4 M4(2)⋊14D4 M4(2)⋊15D4 (C2×C8)⋊13D4 (C2×C8)⋊14D4 C42.448D4 C42.449D4 C24.183D4 C24.116D4 (C2×D4).303D4 (C2×D4).304D4 C42.243D4 C42.244D4 (C2×C4).S4
C4○D4p⋊C4: C42.280C23 C42.281C23 C4○D12⋊C4 C23.54D12 C4○D20⋊9C4 C23.49D20 C4○D4⋊F5 C4○D28⋊C4 ...
D4p⋊(C2×C4): C4×C8⋊C22 Q8⋊7(C4×S3) Q8⋊(C4×D5) Q8⋊(C4×D7) ...
(C2×C4p).D4: C4○D4.4Q8 C4○D4.5Q8 M4(2).48D4 M4(2).49D4 M4(2).10D4 M4(2).11D4 (C2×C8).55D4 (C2×C8).165D4 ...
C4⋊C4.D2p: C4≀C2⋊C4 C42⋊9(C2×C4) M4(2)⋊6D4 M4(2).7D4 C42.18C23 C42.19C23 C42.22C23 C42.23C23 ...
C23.36D4 is a maximal quotient of
C42.397D4 C42.47D4 D4⋊4M4(2) C42.52D4 C24.150D4 C42.57D4 C42.58D4 C24.58D4 C42.79D4 C42.80D4 C42.81D4 C42.84D4 C42.86D4 C42.87D4 C42.88D4 C24.152D4 D4⋊C42 Q8⋊C42 C24.65D4 C42.99D4 C24.69D4 C24.76D4 C42.123D4 C42.124D4 C4○D4⋊F5
C23.D4p: C23.38D8 C23.54D12 C23.49D20 C23.49D28 ...
(C2×C4).D4p: C42.78D4 C42.98D4 C4○D12⋊C4 C4○D20⋊9C4 C4○D28⋊C4 ...
(C2×C4p).D4: C24.75D4 C42.110D4 C4○D4⋊3Dic3 C4○D4⋊Dic5 C4○D4⋊Dic7 ...
C4⋊C4.D2p: C24.160D4 C42.119D4 D4⋊(C4×S3) Q8⋊7(C4×S3) D4⋊(C4×D5) Q8⋊(C4×D5) D4⋊(C4×D7) Q8⋊(C4×D7) ...
Matrix representation of C23.36D4 ►in GL6(𝔽17)
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 13 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 4 | 4 | 4 | 8 |
| 0 | 0 | 13 | 0 | 13 | 13 |
| 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 |
| 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 | 15 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 1 | 1 | 1 | 2 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 16 |
| 16 | 15 | 0 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 16 | 16 | 16 | 15 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,4,4,13,0,0,13,0,4,0,0,0,0,0,4,13,0,0,0,0,8,13],[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],[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],[16,0,0,0,0,0,15,1,0,0,0,0,0,0,0,1,0,0,0,0,0,1,16,0,0,0,1,1,0,16,0,0,0,2,0,16],[16,1,0,0,0,0,15,1,0,0,0,0,0,0,0,16,1,0,0,0,0,16,0,0,0,0,1,16,0,0,0,0,0,15,0,1] >;
C23.36D4 in GAP, Magma, Sage, TeX
C_2^3._{36}D_4 % in TeX
G:=Group("C2^3.36D4"); // GroupNames label
G:=SmallGroup(64,98);
// by ID
G=gap.SmallGroup(64,98);
# by ID
G:=PCGroup([6,-2,2,2,-2,2,-2,96,121,332,963,489,117]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=1,d^4=c,e^2=b,a*b=b*a,d*a*d^-1=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=b*c*d^3>;
// generators/relations
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