p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.4C23, C4⋊C4.40D4, C8⋊C4.93C22, C4⋊1D4.23C22, C2.22(D4.8D4), C22.185C22≀C2, C42.C2.3C22, C42.2C22⋊13C2, C22.58C24⋊1C2, C42.29C22.4C2, (C2×C4).217(C2×D4), 2-Sylow(2A(2,4).C2), SmallGroup(128,390)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.4C23
G = < a,b,c,d,e | a4=b4=c2=1, d2=a2b2, e2=a2, ab=ba, cac=dad-1=a-1, eae-1=a-1b2, cbc=ebe-1=b-1, dbd-1=a2b-1, dcd-1=ac, ece-1=bc, de=ed >
Subgroups: 216 in 91 conjugacy classes, 30 normal (6 characteristic)
C1, C2, C2, C4, C22, C22, C8, C2×C4, C2×C4, D4, C23, C42, C42, C4⋊C4, C4⋊C4, C2×C8, C2×D4, C8⋊C4, D4⋊C4, C42.C2, C42.C2, C4⋊1D4, C42.2C22, C42.29C22, C22.58C24, C42.4C23
Quotients: C1, C2, C22, D4, C23, C2×D4, C22≀C2, D4.8D4, C42.4C23
Character table of C42.4C23
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 8A | 8B | 8C | 8D | 8E | 8F | |
| size | 1 | 1 | 1 | 1 | 16 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 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 | trivial |
| ρ2 | 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 | 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 | 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 | 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 | 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 | 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 | linear of order 2 |
| ρ9 | 2 | 2 | 2 | 2 | 0 | -2 | -2 | 2 | -2 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ10 | 2 | 2 | 2 | 2 | 0 | -2 | 2 | -2 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ11 | 2 | 2 | 2 | 2 | 0 | -2 | 2 | -2 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ12 | 2 | 2 | 2 | 2 | 0 | -2 | -2 | 2 | 2 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ13 | 2 | 2 | 2 | 2 | 0 | 2 | -2 | -2 | 0 | -2 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ14 | 2 | 2 | 2 | 2 | 0 | 2 | -2 | -2 | 0 | 2 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ15 | 4 | -4 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2i | 0 | -2i | 0 | 0 | 0 | complex lifted from D4.8D4 |
| ρ16 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2i | 0 | 0 | -2i | 0 | complex lifted from D4.8D4 |
| ρ17 | 4 | -4 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2i | 0 | 2i | 0 | 0 | 0 | complex lifted from D4.8D4 |
| ρ18 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2i | 0 | -2i | complex lifted from D4.8D4 |
| ρ19 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2i | 0 | 0 | 2i | 0 | complex lifted from D4.8D4 |
| ρ20 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2i | 0 | 2i | complex lifted from D4.8D4 |
►On 64 points
Generators in S64
(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)(49 50 51 52)(53 54 55 56)(57 58 59 60)(61 62 63 64)
(1 20 13 5)(2 17 14 6)(3 18 15 7)(4 19 16 8)(9 64 54 59)(10 61 55 60)(11 62 56 57)(12 63 53 58)(21 29 35 26)(22 30 36 27)(23 31 33 28)(24 32 34 25)(37 48 51 42)(38 45 52 43)(39 46 49 44)(40 47 50 41)
(2 4)(5 20)(6 19)(7 18)(8 17)(9 61)(10 64)(11 63)(12 62)(14 16)(21 26)(22 25)(23 28)(24 27)(29 35)(30 34)(31 33)(32 36)(37 50)(38 49)(39 52)(40 51)(41 42)(43 44)(45 46)(47 48)(53 57)(54 60)(55 59)(56 58)
(1 45 15 41)(2 48 16 44)(3 47 13 43)(4 46 14 42)(5 50 18 38)(6 49 19 37)(7 52 20 40)(8 51 17 39)(9 36 56 24)(10 35 53 23)(11 34 54 22)(12 33 55 21)(25 62 30 59)(26 61 31 58)(27 64 32 57)(28 63 29 60)
(1 29 3 31)(2 25 4 27)(5 35 7 33)(6 24 8 22)(9 51 11 49)(10 40 12 38)(13 26 15 28)(14 32 16 30)(17 34 19 36)(18 23 20 21)(37 56 39 54)(41 63 43 61)(42 57 44 59)(45 60 47 58)(46 64 48 62)(50 53 52 55)
G:=sub<Sym(64)| (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)(49,50,51,52)(53,54,55,56)(57,58,59,60)(61,62,63,64), (1,20,13,5)(2,17,14,6)(3,18,15,7)(4,19,16,8)(9,64,54,59)(10,61,55,60)(11,62,56,57)(12,63,53,58)(21,29,35,26)(22,30,36,27)(23,31,33,28)(24,32,34,25)(37,48,51,42)(38,45,52,43)(39,46,49,44)(40,47,50,41), (2,4)(5,20)(6,19)(7,18)(8,17)(9,61)(10,64)(11,63)(12,62)(14,16)(21,26)(22,25)(23,28)(24,27)(29,35)(30,34)(31,33)(32,36)(37,50)(38,49)(39,52)(40,51)(41,42)(43,44)(45,46)(47,48)(53,57)(54,60)(55,59)(56,58), (1,45,15,41)(2,48,16,44)(3,47,13,43)(4,46,14,42)(5,50,18,38)(6,49,19,37)(7,52,20,40)(8,51,17,39)(9,36,56,24)(10,35,53,23)(11,34,54,22)(12,33,55,21)(25,62,30,59)(26,61,31,58)(27,64,32,57)(28,63,29,60), (1,29,3,31)(2,25,4,27)(5,35,7,33)(6,24,8,22)(9,51,11,49)(10,40,12,38)(13,26,15,28)(14,32,16,30)(17,34,19,36)(18,23,20,21)(37,56,39,54)(41,63,43,61)(42,57,44,59)(45,60,47,58)(46,64,48,62)(50,53,52,55)>;
G:=Group( (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)(49,50,51,52)(53,54,55,56)(57,58,59,60)(61,62,63,64), (1,20,13,5)(2,17,14,6)(3,18,15,7)(4,19,16,8)(9,64,54,59)(10,61,55,60)(11,62,56,57)(12,63,53,58)(21,29,35,26)(22,30,36,27)(23,31,33,28)(24,32,34,25)(37,48,51,42)(38,45,52,43)(39,46,49,44)(40,47,50,41), (2,4)(5,20)(6,19)(7,18)(8,17)(9,61)(10,64)(11,63)(12,62)(14,16)(21,26)(22,25)(23,28)(24,27)(29,35)(30,34)(31,33)(32,36)(37,50)(38,49)(39,52)(40,51)(41,42)(43,44)(45,46)(47,48)(53,57)(54,60)(55,59)(56,58), (1,45,15,41)(2,48,16,44)(3,47,13,43)(4,46,14,42)(5,50,18,38)(6,49,19,37)(7,52,20,40)(8,51,17,39)(9,36,56,24)(10,35,53,23)(11,34,54,22)(12,33,55,21)(25,62,30,59)(26,61,31,58)(27,64,32,57)(28,63,29,60), (1,29,3,31)(2,25,4,27)(5,35,7,33)(6,24,8,22)(9,51,11,49)(10,40,12,38)(13,26,15,28)(14,32,16,30)(17,34,19,36)(18,23,20,21)(37,56,39,54)(41,63,43,61)(42,57,44,59)(45,60,47,58)(46,64,48,62)(50,53,52,55) );
G=PermutationGroup([[(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),(49,50,51,52),(53,54,55,56),(57,58,59,60),(61,62,63,64)], [(1,20,13,5),(2,17,14,6),(3,18,15,7),(4,19,16,8),(9,64,54,59),(10,61,55,60),(11,62,56,57),(12,63,53,58),(21,29,35,26),(22,30,36,27),(23,31,33,28),(24,32,34,25),(37,48,51,42),(38,45,52,43),(39,46,49,44),(40,47,50,41)], [(2,4),(5,20),(6,19),(7,18),(8,17),(9,61),(10,64),(11,63),(12,62),(14,16),(21,26),(22,25),(23,28),(24,27),(29,35),(30,34),(31,33),(32,36),(37,50),(38,49),(39,52),(40,51),(41,42),(43,44),(45,46),(47,48),(53,57),(54,60),(55,59),(56,58)], [(1,45,15,41),(2,48,16,44),(3,47,13,43),(4,46,14,42),(5,50,18,38),(6,49,19,37),(7,52,20,40),(8,51,17,39),(9,36,56,24),(10,35,53,23),(11,34,54,22),(12,33,55,21),(25,62,30,59),(26,61,31,58),(27,64,32,57),(28,63,29,60)], [(1,29,3,31),(2,25,4,27),(5,35,7,33),(6,24,8,22),(9,51,11,49),(10,40,12,38),(13,26,15,28),(14,32,16,30),(17,34,19,36),(18,23,20,21),(37,56,39,54),(41,63,43,61),(42,57,44,59),(45,60,47,58),(46,64,48,62),(50,53,52,55)]])
Matrix representation of C42.4C23 ►in GL8(𝔽17)
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 16 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 16 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 16 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 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 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 16 |
| 5 | 4 | 5 | 13 | 0 | 0 | 0 | 0 |
| 13 | 5 | 4 | 5 | 0 | 0 | 0 | 0 |
| 5 | 13 | 12 | 13 | 0 | 0 | 0 | 0 |
| 4 | 5 | 4 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 15 | 15 | 15 | 2 |
| 0 | 0 | 0 | 0 | 15 | 2 | 2 | 2 |
| 0 | 0 | 0 | 0 | 15 | 2 | 15 | 15 |
| 0 | 0 | 0 | 0 | 2 | 2 | 15 | 2 |
| 3 | 3 | 16 | 1 | 0 | 0 | 0 | 0 |
| 3 | 14 | 1 | 1 | 0 | 0 | 0 | 0 |
| 1 | 16 | 3 | 3 | 0 | 0 | 0 | 0 |
| 16 | 16 | 3 | 14 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 1 | 3 | 3 |
| 0 | 0 | 0 | 0 | 1 | 16 | 3 | 14 |
| 0 | 0 | 0 | 0 | 14 | 14 | 16 | 16 |
| 0 | 0 | 0 | 0 | 14 | 3 | 16 | 1 |
G:=sub<GL(8,GF(17))| [0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0],[1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16],[5,13,5,4,0,0,0,0,4,5,13,5,0,0,0,0,5,4,12,4,0,0,0,0,13,5,13,12,0,0,0,0,0,0,0,0,15,15,15,2,0,0,0,0,15,2,2,2,0,0,0,0,15,2,15,15,0,0,0,0,2,2,15,2],[3,3,1,16,0,0,0,0,3,14,16,16,0,0,0,0,16,1,3,3,0,0,0,0,1,1,3,14,0,0,0,0,0,0,0,0,1,1,14,14,0,0,0,0,1,16,14,3,0,0,0,0,3,3,16,16,0,0,0,0,3,14,16,1] >;
C42.4C23 in GAP, Magma, Sage, TeX
C_4^2._4C_2^3
% in TeX
G:=Group("C4^2.4C2^3"); // GroupNames label
G:=SmallGroup(128,390);
// by ID
G=gap.SmallGroup(128,390);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,2,141,456,422,184,1123,570,521,136,3924,1411,998,242]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^4=c^2=1,d^2=a^2*b^2,e^2=a^2,a*b=b*a,c*a*c=d*a*d^-1=a^-1,e*a*e^-1=a^-1*b^2,c*b*c=e*b*e^-1=b^-1,d*b*d^-1=a^2*b^-1,d*c*d^-1=a*c,e*c*e^-1=b*c,d*e=e*d>;
// generators/relations
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