p-group, metabelian, nilpotent (class 3), monomial
Aliases: C23.24D4, C4○D4⋊1C4, (C22×C8)⋊4C2, D4.5(C2×C4), C4.49(C2×D4), Q8.5(C2×C4), C4○(D4⋊C4), C4○(Q8⋊C4), C2.1(C4○D8), (C2×C4).145D4, C4.3(C22×C4), D4⋊C4⋊20C2, C42⋊C2⋊2C2, Q8⋊C4⋊20C2, C4⋊C4.43C22, (C2×C8).59C22, (C2×C4).61C23, C22.43(C2×D4), C4.33(C22⋊C4), (C2×D4).47C22, (C2×Q8).41C22, C22.3(C22⋊C4), (C22×C4).109C22, (C2×C4).45(C2×C4), (C2×C4○D4).4C2, (C2×C4)○(D4⋊C4), (C2×C4)○(Q8⋊C4), C2.19(C2×C22⋊C4), SmallGroup(64,97)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C23.24D4
G = < a,b,c,d,e | a2=b2=c2=1, d4=c, e2=b, ab=ba, eae-1=ac=ca, ad=da, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede-1=bcd3 >
Subgroups: 129 in 79 conjugacy classes, 41 normal (15 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C23, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C4○D4, D4⋊C4, Q8⋊C4, C42⋊C2, C22×C8, C2×C4○D4, C23.24D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, C2×C22⋊C4, C4○D8, C23.24D4
Character table of C23.24D4
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 1 | 1 | 1 | 1 | 2 | 2 | 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 | 1 | -1 | -i | 1 | -1 | -i | i | i | i | i | -i | -i | -i | i | i | -i | linear of order 4 |
| ρ10 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | i | -1 | 1 | i | -i | -i | i | i | -i | -i | -i | i | i | -i | linear of order 4 |
| ρ11 | 1 | -1 | -1 | 1 | -1 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | 1 | -1 | -i | -1 | 1 | i | i | -i | -i | i | -i | i | i | -i | i | -i | linear of order 4 |
| ρ12 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | i | 1 | -1 | -i | -i | i | -i | i | -i | i | i | -i | i | -i | linear of order 4 |
| ρ13 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | -i | 1 | -1 | i | i | -i | i | -i | i | -i | -i | i | -i | i | linear of order 4 |
| ρ14 | 1 | -1 | -1 | 1 | -1 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | 1 | -1 | i | -1 | 1 | -i | -i | i | i | -i | i | -i | -i | i | -i | i | linear of order 4 |
| ρ15 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | -i | -1 | 1 | -i | i | i | -i | -i | i | i | i | -i | -i | i | linear of order 4 |
| ρ16 | 1 | -1 | -1 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | i | 1 | -1 | i | -i | -i | -i | -i | i | i | i | -i | -i | i | linear of order 4 |
| ρ17 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | -2 | -2 | -2 | -2 | -2 | -2 | 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 | 0 | 0 | 2 | -2 | 2 | -2 | -2 | 2 | 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 | 0 | 0 | -2 | 2 | -2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 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 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ21 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 2i | -2i | -2i | 2i | 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 | 0 | 0 | 2i | 2i | -2i | -2i | 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 | 0 | 0 | -2i | 2i | 2i | -2i | 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 | 0 | 0 | -2i | -2i | 2i | 2i | 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 | 0 | 0 | 2i | -2i | -2i | 2i | 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 | 0 | 0 | 2i | 2i | -2i | -2i | 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 | 0 | 0 | -2i | 2i | 2i | -2i | 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 | 0 | 0 | -2i | -2i | 2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | √-2 | √2 | -√2 | √-2 | -√-2 | -√-2 | -√2 | √2 | complex lifted from C4○D8 |
►On 32 points
(1 14)(2 15)(3 16)(4 9)(5 10)(6 11)(7 12)(8 13)(17 25)(18 26)(19 27)(20 28)(21 29)(22 30)(23 31)(24 32)
(1 25)(2 26)(3 27)(4 28)(5 29)(6 30)(7 31)(8 32)(9 20)(10 21)(11 22)(12 23)(13 24)(14 17)(15 18)(16 19)
(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 32 25 8)(2 7 26 31)(3 30 27 6)(4 5 28 29)(9 14 20 17)(10 24 21 13)(11 12 22 23)(15 16 18 19)
G:=sub<Sym(32)| (1,14)(2,15)(3,16)(4,9)(5,10)(6,11)(7,12)(8,13)(17,25)(18,26)(19,27)(20,28)(21,29)(22,30)(23,31)(24,32), (1,25)(2,26)(3,27)(4,28)(5,29)(6,30)(7,31)(8,32)(9,20)(10,21)(11,22)(12,23)(13,24)(14,17)(15,18)(16,19), (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), (1,32,25,8)(2,7,26,31)(3,30,27,6)(4,5,28,29)(9,14,20,17)(10,24,21,13)(11,12,22,23)(15,16,18,19)>;
G:=Group( (1,14)(2,15)(3,16)(4,9)(5,10)(6,11)(7,12)(8,13)(17,25)(18,26)(19,27)(20,28)(21,29)(22,30)(23,31)(24,32), (1,25)(2,26)(3,27)(4,28)(5,29)(6,30)(7,31)(8,32)(9,20)(10,21)(11,22)(12,23)(13,24)(14,17)(15,18)(16,19), (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), (1,32,25,8)(2,7,26,31)(3,30,27,6)(4,5,28,29)(9,14,20,17)(10,24,21,13)(11,12,22,23)(15,16,18,19) );
G=PermutationGroup([[(1,14),(2,15),(3,16),(4,9),(5,10),(6,11),(7,12),(8,13),(17,25),(18,26),(19,27),(20,28),(21,29),(22,30),(23,31),(24,32)], [(1,25),(2,26),(3,27),(4,28),(5,29),(6,30),(7,31),(8,32),(9,20),(10,21),(11,22),(12,23),(13,24),(14,17),(15,18),(16,19)], [(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)], [(1,32,25,8),(2,7,26,31),(3,30,27,6),(4,5,28,29),(9,14,20,17),(10,24,21,13),(11,12,22,23),(15,16,18,19)]])
C23.24D4 is a maximal subgroup of
Q8.C42 D4.3C42 2+ 1+4⋊4C4 M4(2).42D4 M4(2).44D4 M4(2).24D4 C42.428D4 C42.107D4 C22⋊C4.7D4 C42.9D4 C24.98D4 2+ 1+4⋊5C4 2- 1+4⋊4C4 C4×C4○D8 C42.275C23 C42.276C23 C24.103D4 C4○D4⋊D4 D4.(C2×D4) (C2×Q8)⋊16D4 Q8.(C2×D4) C42.443D4 C42.14C23 C42.15C23 C42.16C23 C42.17C23 C42.447D4 C42.22C23 C42.23C23 C24.115D4 (C2×D4).303D4 (C2×D4).304D4 C42.355D4 C42.239D4 C42.366C23 C42.367C23 C42.461C23 C42.462C23 C42.465C23 C42.466C23 C42.467C23 C42.468C23 C42.469C23 C42.470C23 C42.42C23 C42.44C23 C42.46C23 C42.48C23 C42.50C23 C42.52C23 C42.54C23 C42.56C23 C4.A4⋊C4
C4○D4p⋊C4: C42.383D4 C4.(C2×D12) C23.28D12 C4○D20⋊10C4 C23.23D20 C4○D20⋊C4 C4.(C2×D28) C23.23D28 ...
C4p.(C2×D4): C24.144D4 C24.110D4 M4(2)⋊16D4 M4(2)⋊17D4 D4⋊2S3⋊C4 C4⋊C4.150D6 C4○D4⋊4Dic3 D4⋊2D5⋊C4 ...
C23.24D4 is a maximal quotient of
C42.455D4 C42.373D4 C42.374D4 C42.305D4 C42.375D4 C24.53D4 C24.59D4 C42.63D4 C42.410D4 C42.411D4 C42.412D4 C42.80D4 C42.81D4 C42.417D4 C42.418D4 C24.132D4 C4×D4⋊C4 C4×Q8⋊C4 C24.65D4 C42.100D4 C42.101D4 C24.69D4 C24.74D4 C42.123D4 C42.437D4 C4○D20⋊C4
C23.D4p: C23.23D8 C23.28D12 C23.23D20 C23.23D28 ...
C4.(C2×D4p): C42.409D4 C4.(C2×D12) C4○D20⋊10C4 C4.(C2×D28) ...
(C2×C4p).D4: C24.135D4 C42.433D4 C4○D4⋊4Dic3 C20.(C2×D4) C28.(C2×D4) ...
C4⋊C4.D2p: C24.73D4 C42.119D4 D4⋊2S3⋊C4 C4⋊C4.150D6 D4⋊2D5⋊C4 Q8⋊2D5⋊C4 D4⋊2D7⋊C4 Q8⋊2D7⋊C4 ...
Matrix representation of C23.24D4 ►in GL4(𝔽17) generated by
| 16 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 4 |
| 0 | 0 | 13 | 0 |
| 16 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 12 | 5 |
| 0 | 0 | 12 | 12 |
| 0 | 1 | 0 | 0 |
| 16 | 0 | 0 | 0 |
| 0 | 0 | 12 | 5 |
| 0 | 0 | 5 | 5 |
G:=sub<GL(4,GF(17))| [16,0,0,0,0,16,0,0,0,0,0,13,0,0,4,0],[16,0,0,0,0,16,0,0,0,0,16,0,0,0,0,16],[1,0,0,0,0,1,0,0,0,0,16,0,0,0,0,16],[0,1,0,0,1,0,0,0,0,0,12,12,0,0,5,12],[0,16,0,0,1,0,0,0,0,0,12,5,0,0,5,5] >;
C23.24D4 in GAP, Magma, Sage, TeX
C_2^3._{24}D_4 % in TeX
G:=Group("C2^3.24D4"); // GroupNames label
G:=SmallGroup(64,97);
// by ID
G=gap.SmallGroup(64,97);
# by ID
G:=PCGroup([6,-2,2,2,-2,2,-2,96,121,158,963,489,117]);
// 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,e*a*e^-1=a*c=c*a,a*d=d*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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