p-group, metabelian, nilpotent (class 2), monomial
Aliases: C23.34D4, C24.25C22, C23.57C23, (C22×C4)⋊7C4, (C23×C4).3C2, C23.26(C2×C4), C22.31(C2×D4), C2.C42⋊2C2, C2.6(C42⋊C2), C22.16(C4○D4), (C22×C4).88C22, C22.30(C22×C4), C22.15(C22⋊C4), C2.1(C22.D4), (C2×C4).52(C2×C4), C2.6(C2×C22⋊C4), (C2×C22⋊C4).4C2, SmallGroup(64,62)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C23.34D4
G = < a,b,c,d,e | a2=b2=c2=d4=1, e2=cb=bc, dad-1=eae-1=ab=ba, ac=ca, bd=db, be=eb, cd=dc, ce=ec, ede-1=cd-1 >
Subgroups: 177 in 109 conjugacy classes, 49 normal (7 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C2×C4, C2×C4, C23, C23, C23, C22⋊C4, C22×C4, C22×C4, C24, C2.C42, C2×C22⋊C4, C23×C4, C23.34D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, C4○D4, C2×C22⋊C4, C42⋊C2, C22.D4, C23.34D4
Character table of C23.34D4
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 2K | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 4P | |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 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 | 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 | -1 | -1 | 1 | 1 | -1 | 1 | 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 | 1 | -1 | 1 | -1 | 1 | -1 | 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 | -1 | -1 | 1 | 1 | -1 | 1 | -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 | 1 | -1 | 1 | -1 | 1 | -1 | -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 | 1 | 1 | -1 | -1 | 1 | -1 | -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 | -1 | 1 | -1 | 1 | -1 | 1 | -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 | 1 | 1 | -1 | -1 | 1 | -1 | 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 | -1 | 1 | -1 | 1 | -1 | 1 | i | -i | i | -i | i | -i | i | -i | 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 | orthogonal lifted from D4 |
| 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 | orthogonal lifted from D4 |
| ρ21 | 2 | -2 | -2 | 2 | -2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 2i | 0 | 0 | -2i | -2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ22 | 2 | 2 | -2 | -2 | -2 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 2i | 0 | 0 | 2i | -2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ23 | 2 | 2 | 2 | -2 | -2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 2i | 2i | 0 | -2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ24 | 2 | 2 | 2 | -2 | -2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | -2i | -2i | 0 | 2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ25 | 2 | 2 | -2 | -2 | -2 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2i | 0 | 0 | -2i | 2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ26 | 2 | -2 | -2 | 2 | -2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -2i | 0 | 0 | 2i | 2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ27 | 2 | -2 | 2 | 2 | -2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | -2i | 2i | 0 | -2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ28 | 2 | -2 | 2 | 2 | -2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 2i | -2i | 0 | 2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
►On 32 points
(2 12)(4 10)(5 22)(7 24)(14 26)(16 28)(18 32)(20 30)
(1 11)(2 12)(3 9)(4 10)(5 22)(6 23)(7 24)(8 21)(13 25)(14 26)(15 27)(16 28)(17 31)(18 32)(19 29)(20 30)
(1 25)(2 26)(3 27)(4 28)(5 18)(6 19)(7 20)(8 17)(9 15)(10 16)(11 13)(12 14)(21 31)(22 32)(23 29)(24 30)
(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 18 13 22)(2 8 14 31)(3 20 15 24)(4 6 16 29)(5 11 32 25)(7 9 30 27)(10 23 28 19)(12 21 26 17)
G:=sub<Sym(32)| (2,12)(4,10)(5,22)(7,24)(14,26)(16,28)(18,32)(20,30), (1,11)(2,12)(3,9)(4,10)(5,22)(6,23)(7,24)(8,21)(13,25)(14,26)(15,27)(16,28)(17,31)(18,32)(19,29)(20,30), (1,25)(2,26)(3,27)(4,28)(5,18)(6,19)(7,20)(8,17)(9,15)(10,16)(11,13)(12,14)(21,31)(22,32)(23,29)(24,30), (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), (1,18,13,22)(2,8,14,31)(3,20,15,24)(4,6,16,29)(5,11,32,25)(7,9,30,27)(10,23,28,19)(12,21,26,17)>;
G:=Group( (2,12)(4,10)(5,22)(7,24)(14,26)(16,28)(18,32)(20,30), (1,11)(2,12)(3,9)(4,10)(5,22)(6,23)(7,24)(8,21)(13,25)(14,26)(15,27)(16,28)(17,31)(18,32)(19,29)(20,30), (1,25)(2,26)(3,27)(4,28)(5,18)(6,19)(7,20)(8,17)(9,15)(10,16)(11,13)(12,14)(21,31)(22,32)(23,29)(24,30), (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), (1,18,13,22)(2,8,14,31)(3,20,15,24)(4,6,16,29)(5,11,32,25)(7,9,30,27)(10,23,28,19)(12,21,26,17) );
G=PermutationGroup([[(2,12),(4,10),(5,22),(7,24),(14,26),(16,28),(18,32),(20,30)], [(1,11),(2,12),(3,9),(4,10),(5,22),(6,23),(7,24),(8,21),(13,25),(14,26),(15,27),(16,28),(17,31),(18,32),(19,29),(20,30)], [(1,25),(2,26),(3,27),(4,28),(5,18),(6,19),(7,20),(8,17),(9,15),(10,16),(11,13),(12,14),(21,31),(22,32),(23,29),(24,30)], [(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)], [(1,18,13,22),(2,8,14,31),(3,20,15,24),(4,6,16,29),(5,11,32,25),(7,9,30,27),(10,23,28,19),(12,21,26,17)]])
C23.34D4 is a maximal subgroup of
C23.165C24 C23.195C24 C24.547C23 C23.201C24 C23.214C24 C23.215C24 C24.549C23 C23.224C24 C23.225C24 C23.226C24 C23.235C24 C24.212C23 C23.259C24 C24.227C23 C23.304C24 C24.563C23 C24.254C23 C24.567C23 C23.344C24 C24.271C23 C24.278C23 C23.359C24 C24.286C23 C23.368C24 C23.388C24 C24.577C23 C24.309C23 C23.416C24 C23.426C24 C24.315C23 C23.434C24 C23.472C24 C23.473C24 C24.339C23 C24.340C23 C23.500C24 C24.355C23 C23.508C24 C24⋊9D4 C23.535C24 C24.379C23 C23.567C24 C24.393C23 C24.394C23 C24.395C23 C23.589C24 C24.405C23 C24.406C23 C23.600C24 C24.407C23 C23.637C24 C24.426C23 C24.427C23 C23.640C24 C23.643C24 C24.430C23 C23.645C24 C24.432C23 C23.649C24 C24.435C23 C23.651C24 C23.652C24 C23.660C24 C24.440C23 C23.664C24 C24.443C23 C23.671C24 C23.715C24 C23.741C24 (C22×C4)⋊7F5
C24.D2p: C24.46D4 C24.56D4 C24.57D4 C24.59D4 C24.26D4 C24.31D4 C24.90D4 C24.95D4 ...
(C22×C4).D2p: C23.8C42 C23.15M4(2) C24.165C23 C24.169C23 C24.174C23 C24.180C23 C25.85C22 C4×C22.D4 ...
C23.34D4 is a maximal quotient of
(C2×C42)⋊C4 C24.C23 C24.6(C2×C4) (C2×Q8).211D4 (C22×C4)⋊7F5
C24.D2p: C24.17Q8 C24.52D4 C24.68D4 C23.36D8 C24.157D4 C24.69D4 C24.70D4 C24.56D6 ...
(C22×C4).D2p: C24.624C23 C24.635C23 C23.32M4(2) C24.53(C2×C4) C24.169C23 (C22×C4).275D4 (C22×C4).276D4 C22.58(S3×D4) ...
Matrix representation of C23.34D4 ►in GL5(𝔽5)
| 4 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 | 4 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 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 | 1 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 1 | 3 |
| 0 | 0 | 0 | 0 | 4 |
| 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 2 | 1 |
| 0 | 0 | 0 | 2 | 3 |
G:=sub<GL(5,GF(5))| [4,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,1,0,0,0,0,4],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,4,0,0,0,0,0,4],[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],[2,0,0,0,0,0,1,0,0,0,0,0,4,0,0,0,0,0,1,0,0,0,0,3,4],[2,0,0,0,0,0,0,1,0,0,0,4,0,0,0,0,0,0,2,2,0,0,0,1,3] >;
C23.34D4 in GAP, Magma, Sage, TeX
C_2^3._{34}D_4 % in TeX
G:=Group("C2^3.34D4"); // GroupNames label
G:=SmallGroup(64,62);
// by ID
G=gap.SmallGroup(64,62);
# by ID
G:=PCGroup([6,-2,2,2,-2,2,2,192,121,362,50]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^4=1,e^2=c*b=b*c,d*a*d^-1=e*a*e^-1=a*b=b*a,a*c=c*a,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=c*d^-1>;
// generators/relations
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