metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C23.4D12, (C2×D4).6D6, C4.D4.S3, (C2×C12).14D4, C23.9(C4×S3), C6.D4⋊4C4, (C22×C6).13D4, C6.12(C23⋊C4), (C22×Dic3)⋊2C4, C3⋊2(C23.D4), C22.13(D6⋊C4), (C6×D4).171C22, C23.7D6.3C2, C23.23D6.4C2, C2.13(C23.6D6), (C2×C4).2(C3⋊D4), (C22×C6).4(C2×C4), (C2×C6).6(C22⋊C4), (C3×C4.D4).1C2, SmallGroup(192,35)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C23.4D12
G = < a,b,c,d,e | a2=b2=c2=1, d12=c, e2=ca=ac, ab=ba, dad-1=abc, ae=ea, dbd-1=ebe-1=bc=cb, cd=dc, ce=ec, ede-1=acd11 >
Subgroups: 240 in 68 conjugacy classes, 21 normal (all characteristic)
C1, C2, C2, C3, C4, C22, C22, C6, C6, C8, C2×C4, C2×C4, D4, C23, Dic3, C12, C2×C6, C2×C6, C22⋊C4, C4⋊C4, M4(2), C22×C4, C2×D4, C24, C2×Dic3, C2×C12, C3×D4, C22×C6, C23⋊C4, C4.D4, C22.D4, Dic3⋊C4, C6.D4, C6.D4, C3×M4(2), C22×Dic3, C6×D4, C23.D4, C23.7D6, C3×C4.D4, C23.23D6, C23.4D12
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, D6, C22⋊C4, C4×S3, D12, C3⋊D4, C23⋊C4, D6⋊C4, C23.D4, C23.6D6, C23.4D12
Character table of C23.4D12
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 6A | 6B | 6C | 6D | 8A | 8B | 12A | 12B | 24A | 24B | 24C | 24D | |
| size | 1 | 1 | 2 | 4 | 4 | 2 | 4 | 12 | 12 | 24 | 24 | 24 | 2 | 4 | 8 | 8 | 8 | 8 | 4 | 4 | 8 | 8 | 8 | 8 | |
| 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 | 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 | 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 | 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 | linear of order 2 |
| ρ5 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | -1 | -i | i | 1 | 1 | 1 | -1 | 1 | i | -i | -1 | -1 | -i | i | i | -i | linear of order 4 |
| ρ6 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | -1 | i | -i | 1 | 1 | 1 | -1 | 1 | -i | i | -1 | -1 | i | -i | -i | i | linear of order 4 |
| ρ7 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | 1 | 1 | i | -i | -1 | 1 | 1 | -1 | 1 | i | -i | -1 | -1 | -i | i | i | -i | linear of order 4 |
| ρ8 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | 1 | 1 | -i | i | -1 | 1 | 1 | -1 | 1 | -i | i | -1 | -1 | i | -i | -i | i | linear of order 4 |
| ρ9 | 2 | 2 | 2 | -2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | 2 | -2 | 0 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ10 | 2 | 2 | 2 | 2 | 2 | -1 | 2 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ11 | 2 | 2 | 2 | 2 | 2 | -1 | 2 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -2 | -2 | -1 | -1 | 1 | 1 | 1 | 1 | orthogonal lifted from D6 |
| ρ12 | 2 | 2 | 2 | -2 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | -2 | -2 | 0 | 0 | 2 | 2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ13 | 2 | 2 | 2 | -2 | 2 | -1 | -2 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 1 | 0 | 0 | 1 | 1 | -√3 | √3 | -√3 | √3 | orthogonal lifted from D12 |
| ρ14 | 2 | 2 | 2 | -2 | 2 | -1 | -2 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 1 | 0 | 0 | 1 | 1 | √3 | -√3 | √3 | -√3 | orthogonal lifted from D12 |
| ρ15 | 2 | 2 | 2 | 2 | -2 | -1 | -2 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | 1 | -1 | 2i | -2i | 1 | 1 | i | -i | -i | i | complex lifted from C4×S3 |
| ρ16 | 2 | 2 | 2 | 2 | -2 | -1 | -2 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | 1 | -1 | -2i | 2i | 1 | 1 | -i | i | i | -i | complex lifted from C4×S3 |
| ρ17 | 2 | 2 | 2 | -2 | -2 | -1 | 2 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | 1 | 1 | 0 | 0 | -1 | -1 | -√-3 | -√-3 | √-3 | √-3 | complex lifted from C3⋊D4 |
| ρ18 | 2 | 2 | 2 | -2 | -2 | -1 | 2 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | 1 | 1 | 0 | 0 | -1 | -1 | √-3 | √-3 | -√-3 | -√-3 | complex lifted from C3⋊D4 |
| ρ19 | 4 | 4 | -4 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C23⋊C4 |
| ρ20 | 4 | 4 | -4 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | -2√-3 | 2√-3 | 0 | 0 | 0 | 0 | complex lifted from C23.6D6 |
| ρ21 | 4 | 4 | -4 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | 2√-3 | -2√-3 | 0 | 0 | 0 | 0 | complex lifted from C23.6D6 |
| ρ22 | 4 | -4 | 0 | 0 | 0 | 4 | 0 | -2i | 2i | 0 | 0 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C23.D4 |
| ρ23 | 4 | -4 | 0 | 0 | 0 | 4 | 0 | 2i | -2i | 0 | 0 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C23.D4 |
| ρ24 | 8 | -8 | 0 | 0 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic faithful, Schur index 2 |
►On 48 points
Generators in S48
(1 31)(3 45)(4 16)(5 35)(7 25)(8 20)(9 39)(11 29)(12 24)(13 43)(15 33)(17 47)(19 37)(21 27)(23 41)(26 38)(30 42)(34 46)
(1 43)(2 32)(3 45)(4 34)(5 47)(6 36)(7 25)(8 38)(9 27)(10 40)(11 29)(12 42)(13 31)(14 44)(15 33)(16 46)(17 35)(18 48)(19 37)(20 26)(21 39)(22 28)(23 41)(24 30)
(1 13)(2 14)(3 15)(4 16)(5 17)(6 18)(7 19)(8 20)(9 21)(10 22)(11 23)(12 24)(25 37)(26 38)(27 39)(28 40)(29 41)(30 42)(31 43)(32 44)(33 45)(34 46)(35 47)(36 48)
(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 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48)
(1 33 43 3)(2 32 14 44)(4 12)(5 29 47 23)(6 28 18 40)(7 21 37 39)(9 25 27 19)(10 48 22 36)(11 17 41 35)(13 45 31 15)(16 24)(26 38)(30 34)(42 46)
G:=sub<Sym(48)| (1,31)(3,45)(4,16)(5,35)(7,25)(8,20)(9,39)(11,29)(12,24)(13,43)(15,33)(17,47)(19,37)(21,27)(23,41)(26,38)(30,42)(34,46), (1,43)(2,32)(3,45)(4,34)(5,47)(6,36)(7,25)(8,38)(9,27)(10,40)(11,29)(12,42)(13,31)(14,44)(15,33)(16,46)(17,35)(18,48)(19,37)(20,26)(21,39)(22,28)(23,41)(24,30), (1,13)(2,14)(3,15)(4,16)(5,17)(6,18)(7,19)(8,20)(9,21)(10,22)(11,23)(12,24)(25,37)(26,38)(27,39)(28,40)(29,41)(30,42)(31,43)(32,44)(33,45)(34,46)(35,47)(36,48), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48), (1,33,43,3)(2,32,14,44)(4,12)(5,29,47,23)(6,28,18,40)(7,21,37,39)(9,25,27,19)(10,48,22,36)(11,17,41,35)(13,45,31,15)(16,24)(26,38)(30,34)(42,46)>;
G:=Group( (1,31)(3,45)(4,16)(5,35)(7,25)(8,20)(9,39)(11,29)(12,24)(13,43)(15,33)(17,47)(19,37)(21,27)(23,41)(26,38)(30,42)(34,46), (1,43)(2,32)(3,45)(4,34)(5,47)(6,36)(7,25)(8,38)(9,27)(10,40)(11,29)(12,42)(13,31)(14,44)(15,33)(16,46)(17,35)(18,48)(19,37)(20,26)(21,39)(22,28)(23,41)(24,30), (1,13)(2,14)(3,15)(4,16)(5,17)(6,18)(7,19)(8,20)(9,21)(10,22)(11,23)(12,24)(25,37)(26,38)(27,39)(28,40)(29,41)(30,42)(31,43)(32,44)(33,45)(34,46)(35,47)(36,48), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48), (1,33,43,3)(2,32,14,44)(4,12)(5,29,47,23)(6,28,18,40)(7,21,37,39)(9,25,27,19)(10,48,22,36)(11,17,41,35)(13,45,31,15)(16,24)(26,38)(30,34)(42,46) );
G=PermutationGroup([[(1,31),(3,45),(4,16),(5,35),(7,25),(8,20),(9,39),(11,29),(12,24),(13,43),(15,33),(17,47),(19,37),(21,27),(23,41),(26,38),(30,42),(34,46)], [(1,43),(2,32),(3,45),(4,34),(5,47),(6,36),(7,25),(8,38),(9,27),(10,40),(11,29),(12,42),(13,31),(14,44),(15,33),(16,46),(17,35),(18,48),(19,37),(20,26),(21,39),(22,28),(23,41),(24,30)], [(1,13),(2,14),(3,15),(4,16),(5,17),(6,18),(7,19),(8,20),(9,21),(10,22),(11,23),(12,24),(25,37),(26,38),(27,39),(28,40),(29,41),(30,42),(31,43),(32,44),(33,45),(34,46),(35,47),(36,48)], [(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24),(25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48)], [(1,33,43,3),(2,32,14,44),(4,12),(5,29,47,23),(6,28,18,40),(7,21,37,39),(9,25,27,19),(10,48,22,36),(11,17,41,35),(13,45,31,15),(16,24),(26,38),(30,34),(42,46)]])
Matrix representation of C23.4D12 ►in GL8(𝔽73)
| 72 | 0 | 72 | 71 | 0 | 0 | 0 | 0 |
| 0 | 72 | 1 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 71 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 72 |
| 72 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 71 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 71 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 72 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 72 |
| 60 | 43 | 9 | 8 | 0 | 0 | 0 | 0 |
| 30 | 30 | 1 | 9 | 0 | 0 | 0 | 0 |
| 0 | 60 | 43 | 30 | 0 | 0 | 0 | 0 |
| 13 | 0 | 43 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 46 | 27 | 0 | 46 |
| 0 | 0 | 0 | 0 | 46 | 27 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 46 | 27 | 46 |
| 0 | 0 | 0 | 0 | 0 | 0 | 27 | 46 |
| 43 | 60 | 65 | 1 | 0 | 0 | 0 | 0 |
| 30 | 30 | 1 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 72 | 2 |
| 0 | 0 | 0 | 0 | 0 | 0 | 72 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 72 | 0 | 0 |
G:=sub<GL(8,GF(73))| [72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,72,1,1,0,0,0,0,0,71,72,0,1,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,2,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,71,72],[72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,71,72,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,71,72],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72],[60,30,0,13,0,0,0,0,43,30,60,0,0,0,0,0,9,1,43,43,0,0,0,0,8,9,30,13,0,0,0,0,0,0,0,0,46,46,0,0,0,0,0,0,27,27,46,0,0,0,0,0,0,0,27,27,0,0,0,0,46,0,46,46],[43,30,0,0,0,0,0,0,60,30,0,0,0,0,0,0,65,1,1,72,0,0,0,0,1,9,0,72,0,0,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,72,0,0,0,0,72,72,0,0,0,0,0,0,2,1,0,0] >;
C23.4D12 in GAP, Magma, Sage, TeX
C_2^3._4D_{12} % in TeX
G:=Group("C2^3.4D12"); // GroupNames label
G:=SmallGroup(192,35);
// by ID
G=gap.SmallGroup(192,35);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,141,36,422,184,346,297,851,6278]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=1,d^12=c,e^2=c*a=a*c,a*b=b*a,d*a*d^-1=a*b*c,a*e=e*a,d*b*d^-1=e*b*e^-1=b*c=c*b,c*d=d*c,c*e=e*c,e*d*e^-1=a*c*d^11>;
// generators/relations
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