p-group, metabelian, nilpotent (class 3), monomial
Aliases: C23.25D4, (C2×C8)⋊6C4, C4○(C4.Q8), C4○(C2.D8), C4.3(C2×Q8), C8.17(C2×C4), C4.Q8⋊13C2, C2.D8⋊13C2, C4.22(C4⋊C4), (C2×C4).20Q8, C2.2(C4○D8), (C2×C4).147D4, C4⋊C4.48C22, C22.9(C4⋊C4), (C2×C8).74C22, (C2×C4).69C23, (C22×C8).11C2, C4.26(C22×C4), C22.49(C2×D4), C42⋊C2.6C2, (C22×C4).115C22, C2.13(C2×C4⋊C4), (C2×C4)○(C2.D8), (C2×C4)○(C4.Q8), (C2×C4).74(C2×C4), SmallGroup(64,108)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C23.25D4
G = < a,b,c,d,e | a2=b2=c2=1, d4=c, e2=cb=bc, ab=ba, eae-1=ac=ca, ad=da, bd=db, be=eb, cd=dc, ce=ec, ede-1=d3 >
Subgroups: 81 in 57 conjugacy classes, 41 normal (13 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, C23, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, C4.Q8, C2.D8, C42⋊C2, C22×C8, C23.25D4
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C4⋊C4, C22×C4, C2×D4, C2×Q8, C2×C4⋊C4, C4○D8, C23.25D4
Character table of C23.25D4
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | |
| 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 | 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 | -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 | -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 | 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 | 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 | -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 | 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 | -1 | 1 | -1 | 1 | -1 | -1 | linear of order 2 |
| ρ9 | 1 | -1 | -1 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | -i | i | -i | -i | i | -i | i | i | 1 | -1 | -1 | -1 | 1 | 1 | 1 | -1 | linear of order 4 |
| ρ10 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | -i | i | i | i | -i | -i | i | -i | -1 | 1 | -1 | 1 | 1 | -1 | 1 | -1 | linear of order 4 |
| ρ11 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | i | i | -i | i | i | -i | -i | -i | 1 | -1 | 1 | -1 | -1 | 1 | -1 | 1 | linear of order 4 |
| ρ12 | 1 | -1 | -1 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | i | i | i | -i | -i | -i | -i | i | -1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | linear of order 4 |
| ρ13 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | -i | -i | i | -i | -i | i | i | i | 1 | -1 | 1 | -1 | -1 | 1 | -1 | 1 | linear of order 4 |
| ρ14 | 1 | -1 | -1 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | -i | -i | -i | i | i | i | i | -i | -1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | linear of order 4 |
| ρ15 | 1 | -1 | -1 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | i | -i | i | i | -i | i | -i | -i | 1 | -1 | -1 | -1 | 1 | 1 | 1 | -1 | linear of order 4 |
| ρ16 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | i | -i | -i | -i | i | i | -i | i | -1 | 1 | -1 | 1 | 1 | -1 | 1 | -1 | 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 | 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 | 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 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| 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 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ21 | 2 | 2 | -2 | -2 | 0 | 0 | 2i | 2i | -2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -√2 | √2 | √-2 | -√2 | -√-2 | √2 | √-2 | -√-2 | complex lifted from C4○D8 |
| ρ22 | 2 | -2 | 2 | -2 | 0 | 0 | 2i | -2i | -2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | √2 | √2 | √-2 | -√2 | √-2 | -√2 | -√-2 | -√-2 | complex lifted from C4○D8 |
| ρ23 | 2 | -2 | 2 | -2 | 0 | 0 | -2i | 2i | 2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -√2 | -√2 | √-2 | √2 | √-2 | √2 | -√-2 | -√-2 | complex lifted from C4○D8 |
| ρ24 | 2 | 2 | -2 | -2 | 0 | 0 | -2i | -2i | 2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | √2 | -√2 | √-2 | √2 | -√-2 | -√2 | √-2 | -√-2 | complex lifted from C4○D8 |
| ρ25 | 2 | 2 | -2 | -2 | 0 | 0 | 2i | 2i | -2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | √2 | -√2 | -√-2 | √2 | √-2 | -√2 | -√-2 | √-2 | complex lifted from C4○D8 |
| ρ26 | 2 | -2 | 2 | -2 | 0 | 0 | 2i | -2i | -2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -√2 | -√2 | -√-2 | √2 | -√-2 | √2 | √-2 | √-2 | complex lifted from C4○D8 |
| ρ27 | 2 | -2 | 2 | -2 | 0 | 0 | -2i | 2i | 2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | √2 | √2 | -√-2 | -√2 | -√-2 | -√2 | √-2 | √-2 | complex lifted from C4○D8 |
| ρ28 | 2 | 2 | -2 | -2 | 0 | 0 | -2i | -2i | 2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -√2 | √2 | -√-2 | -√2 | √-2 | √2 | -√-2 | √-2 | complex lifted from C4○D8 |
►On 32 points
(9 13)(10 14)(11 15)(12 16)(25 29)(26 30)(27 31)(28 32)
(1 19)(2 20)(3 21)(4 22)(5 23)(6 24)(7 17)(8 18)(9 30)(10 31)(11 32)(12 25)(13 26)(14 27)(15 28)(16 29)
(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 14 23 31)(2 9 24 26)(3 12 17 29)(4 15 18 32)(5 10 19 27)(6 13 20 30)(7 16 21 25)(8 11 22 28)
G:=sub<Sym(32)| (9,13)(10,14)(11,15)(12,16)(25,29)(26,30)(27,31)(28,32), (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,17)(8,18)(9,30)(10,31)(11,32)(12,25)(13,26)(14,27)(15,28)(16,29), (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), (1,14,23,31)(2,9,24,26)(3,12,17,29)(4,15,18,32)(5,10,19,27)(6,13,20,30)(7,16,21,25)(8,11,22,28)>;
G:=Group( (9,13)(10,14)(11,15)(12,16)(25,29)(26,30)(27,31)(28,32), (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,17)(8,18)(9,30)(10,31)(11,32)(12,25)(13,26)(14,27)(15,28)(16,29), (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), (1,14,23,31)(2,9,24,26)(3,12,17,29)(4,15,18,32)(5,10,19,27)(6,13,20,30)(7,16,21,25)(8,11,22,28) );
G=PermutationGroup([[(9,13),(10,14),(11,15),(12,16),(25,29),(26,30),(27,31),(28,32)], [(1,19),(2,20),(3,21),(4,22),(5,23),(6,24),(7,17),(8,18),(9,30),(10,31),(11,32),(12,25),(13,26),(14,27),(15,28),(16,29)], [(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)], [(1,14,23,31),(2,9,24,26),(3,12,17,29),(4,15,18,32),(5,10,19,27),(6,13,20,30),(7,16,21,25),(8,11,22,28)]])
C23.25D4 is a maximal subgroup of
(C2×D4).24Q8 C8○D4⋊C4 M4(2).3Q8 M4(2).30D4 C24.100D4 C4○D4.7Q8 C4○D4.8Q8 C4×C4○D8 C42.383D4 C24.144D4 C24.110D4 (C2×C8)⋊11D4 (C2×C8)⋊12D4 C8.D4⋊C2 C42.447D4 C24.115D4 (C2×D4).301D4 (C2×D4).302D4 C42.364D4 C42.252D4
C23.D4p: C23.9D8 C8.C42 C23.24D8 C23.40D8 C23.41D8 C23.13D8 C23.25D8 M5(2)⋊1C4 ...
C4p.(C4⋊C4): C8.9C42 C8.4C42 C8.14C42 C8.5C42 C8.(C4⋊C4) C42.62Q8 C42.28Q8 M5(2)⋊3C4 ...
C4⋊C4.D2p: C42.277C23 C42.278C23 C42.279C23 C42.20C23 C42.21C23 M4(2)⋊3Q8 M4(2)⋊4Q8 C42.485C23 ...
C23.25D4 is a maximal quotient of
C42.42Q8 C42.43Q8 C42.Q8 C24.132D4 C4×C4.Q8 C4×C2.D8 C24.133D4 C42.56Q8
C23.D4p: C23.22D8 C23.27D12 C23.22D20 C23.22D28 ...
C4p.(C4⋊C4): C42.60Q8 C4⋊C4.234D6 C20.76(C4⋊C4) (C2×C8)⋊6F5 C28.45(C4⋊C4) ...
C4⋊C4.D2p: C24.71D4 C42.31Q8 (S3×C8)⋊C4 C8.27(C4×S3) (C8×D5)⋊C4 C8.27(C4×D5) (C8×D7)⋊C4 C8.27(C4×D7) ...
Matrix representation of C23.25D4 ►in GL3(𝔽17) generated by
| 16 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 4 | 16 |
| 16 | 0 | 0 |
| 0 | 16 | 0 |
| 0 | 0 | 16 |
| 1 | 0 | 0 |
| 0 | 16 | 0 |
| 0 | 0 | 16 |
| 16 | 0 | 0 |
| 0 | 2 | 0 |
| 0 | 5 | 8 |
| 4 | 0 | 0 |
| 0 | 15 | 1 |
| 0 | 14 | 2 |
G:=sub<GL(3,GF(17))| [16,0,0,0,1,4,0,0,16],[16,0,0,0,16,0,0,0,16],[1,0,0,0,16,0,0,0,16],[16,0,0,0,2,5,0,0,8],[4,0,0,0,15,14,0,1,2] >;
C23.25D4 in GAP, Magma, Sage, TeX
C_2^3._{25}D_4 % in TeX
G:=Group("C2^3.25D4"); // GroupNames label
G:=SmallGroup(64,108);
// by ID
G=gap.SmallGroup(64,108);
# by ID
G:=PCGroup([6,-2,2,2,-2,2,-2,96,121,55,158,963,117]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=1,d^4=c,e^2=c*b=b*c,a*b=b*a,e*a*e^-1=a*c=c*a,a*d=d*a,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^3>;
// generators/relations
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