p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.2C23, C8⋊3D4⋊2C2, C4⋊C4.38D4, (C2×D4).28D4, (C2×Q8).28D4, C2.25(D4⋊4D4), C8⋊C4.91C22, C4⋊1D4.21C22, C4.4D4.7C22, C22.183C22≀C2, C2.20(D4.8D4), C42.C2.1C22, C42.C22⋊2C2, C42.2C22⋊1C2, C22.56C24⋊1C2, C42.29C22⋊15C2, (C2×C4).215(C2×D4), SmallGroup(128,388)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.2C23
G = < a,b,c,d,e | a4=b4=c2=e2=1, d2=a2, ab=ba, cac=dad-1=a-1, eae=a-1b2, cbc=ebe=b-1, dbd-1=a2b-1, dcd-1=ac, ece=bc, de=ed >
Subgroups: 328 in 115 conjugacy classes, 30 normal (12 characteristic)
C1, C2, C2, C2, C4, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, D8, SD16, C22×C4, C2×D4, C2×D4, C2×Q8, C8⋊C4, C8⋊C4, D4⋊C4, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C42.C2, C4⋊1D4, C2×D8, C2×SD16, C42.C22, C42.2C22, C42.29C22, C8⋊3D4, C22.56C24, C42.2C23
Quotients: C1, C2, C22, D4, C23, C2×D4, C22≀C2, D4⋊4D4, D4.8D4, C42.2C23
Character table of C42.2C23
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 8A | 8B | 8C | 8D | 8E | 8F | |
| size | 1 | 1 | 1 | 1 | 8 | 8 | 16 | 4 | 4 | 4 | 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 | 2 | 0 | 0 | -2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ10 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 2 | -2 | -2 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ11 | 2 | 2 | 2 | 2 | 0 | -2 | 0 | -2 | -2 | 2 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ12 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 2 | -2 | -2 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ13 | 2 | 2 | 2 | 2 | -2 | 0 | 0 | -2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ14 | 2 | 2 | 2 | 2 | 0 | 2 | 0 | -2 | -2 | 2 | 0 | -2 | 0 | 0 | 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 | 0 | 2 | 0 | 0 | -2 | 0 | orthogonal lifted from D4⋊4D4 |
| ρ16 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 2 | 0 | orthogonal lifted from D4⋊4D4 |
| ρ17 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 2 | 0 | 0 | orthogonal lifted from D4⋊4D4 |
| ρ18 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | -2 | 0 | 0 | orthogonal lifted from D4⋊4D4 |
| ρ19 | 4 | -4 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2i | 0 | 0 | -2i | complex lifted from D4.8D4 |
| ρ20 | 4 | -4 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2i | 0 | 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 42 47 50)(38 43 48 51)(39 44 45 52)(40 41 46 49)
(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 38)(39 40)(41 52)(42 51)(43 50)(44 49)(45 46)(47 48)(53 57)(54 60)(55 59)(56 58)
(1 39 3 37)(2 38 4 40)(5 42 7 44)(6 41 8 43)(9 36 11 34)(10 35 12 33)(13 45 15 47)(14 48 16 46)(17 49 19 51)(18 52 20 50)(21 53 23 55)(22 56 24 54)(25 57 27 59)(26 60 28 58)(29 61 31 63)(30 64 32 62)
(1 31)(2 27)(3 29)(4 25)(5 33)(6 22)(7 35)(8 24)(9 51)(10 42)(11 49)(12 44)(13 28)(14 30)(15 26)(16 32)(17 36)(18 21)(19 34)(20 23)(37 61)(38 59)(39 63)(40 57)(41 56)(43 54)(45 58)(46 62)(47 60)(48 64)(50 55)(52 53)
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)(37,42,47,50)(38,43,48,51)(39,44,45,52)(40,41,46,49), (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)(37,38)(39,40)(41,52)(42,51)(43,50)(44,49)(45,46)(47,48)(53,57)(54,60)(55,59)(56,58), (1,39,3,37)(2,38,4,40)(5,42,7,44)(6,41,8,43)(9,36,11,34)(10,35,12,33)(13,45,15,47)(14,48,16,46)(17,49,19,51)(18,52,20,50)(21,53,23,55)(22,56,24,54)(25,57,27,59)(26,60,28,58)(29,61,31,63)(30,64,32,62), (1,31)(2,27)(3,29)(4,25)(5,33)(6,22)(7,35)(8,24)(9,51)(10,42)(11,49)(12,44)(13,28)(14,30)(15,26)(16,32)(17,36)(18,21)(19,34)(20,23)(37,61)(38,59)(39,63)(40,57)(41,56)(43,54)(45,58)(46,62)(47,60)(48,64)(50,55)(52,53)>;
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)(37,42,47,50)(38,43,48,51)(39,44,45,52)(40,41,46,49), (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)(37,38)(39,40)(41,52)(42,51)(43,50)(44,49)(45,46)(47,48)(53,57)(54,60)(55,59)(56,58), (1,39,3,37)(2,38,4,40)(5,42,7,44)(6,41,8,43)(9,36,11,34)(10,35,12,33)(13,45,15,47)(14,48,16,46)(17,49,19,51)(18,52,20,50)(21,53,23,55)(22,56,24,54)(25,57,27,59)(26,60,28,58)(29,61,31,63)(30,64,32,62), (1,31)(2,27)(3,29)(4,25)(5,33)(6,22)(7,35)(8,24)(9,51)(10,42)(11,49)(12,44)(13,28)(14,30)(15,26)(16,32)(17,36)(18,21)(19,34)(20,23)(37,61)(38,59)(39,63)(40,57)(41,56)(43,54)(45,58)(46,62)(47,60)(48,64)(50,55)(52,53) );
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),(37,42,47,50),(38,43,48,51),(39,44,45,52),(40,41,46,49)], [(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),(37,38),(39,40),(41,52),(42,51),(43,50),(44,49),(45,46),(47,48),(53,57),(54,60),(55,59),(56,58)], [(1,39,3,37),(2,38,4,40),(5,42,7,44),(6,41,8,43),(9,36,11,34),(10,35,12,33),(13,45,15,47),(14,48,16,46),(17,49,19,51),(18,52,20,50),(21,53,23,55),(22,56,24,54),(25,57,27,59),(26,60,28,58),(29,61,31,63),(30,64,32,62)], [(1,31),(2,27),(3,29),(4,25),(5,33),(6,22),(7,35),(8,24),(9,51),(10,42),(11,49),(12,44),(13,28),(14,30),(15,26),(16,32),(17,36),(18,21),(19,34),(20,23),(37,61),(38,59),(39,63),(40,57),(41,56),(43,54),(45,58),(46,62),(47,60),(48,64),(50,55),(52,53)]])
Matrix representation of C42.2C23 ►in GL8(𝔽17)
| 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 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 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 | 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 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 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 | 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 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 16 |
| 9 | 9 | 9 | 8 | 0 | 0 | 0 | 0 |
| 9 | 8 | 8 | 8 | 0 | 0 | 0 | 0 |
| 9 | 8 | 9 | 9 | 0 | 0 | 0 | 0 |
| 8 | 8 | 9 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 8 | 2 | 8 |
| 0 | 0 | 0 | 0 | 9 | 15 | 9 | 15 |
| 0 | 0 | 0 | 0 | 2 | 8 | 15 | 9 |
| 0 | 0 | 0 | 0 | 9 | 15 | 8 | 2 |
| 8 | 8 | 15 | 15 | 0 | 0 | 0 | 0 |
| 8 | 9 | 15 | 2 | 0 | 0 | 0 | 0 |
| 2 | 2 | 9 | 9 | 0 | 0 | 0 | 0 |
| 2 | 15 | 9 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 9 | 9 | 8 |
| 0 | 0 | 0 | 0 | 9 | 9 | 8 | 9 |
| 0 | 0 | 0 | 0 | 9 | 8 | 8 | 8 |
| 0 | 0 | 0 | 0 | 8 | 9 | 8 | 8 |
G:=sub<GL(8,GF(17))| [0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0],[1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16],[9,9,9,8,0,0,0,0,9,8,8,8,0,0,0,0,9,8,9,9,0,0,0,0,8,8,9,8,0,0,0,0,0,0,0,0,2,9,2,9,0,0,0,0,8,15,8,15,0,0,0,0,2,9,15,8,0,0,0,0,8,15,9,2],[8,8,2,2,0,0,0,0,8,9,2,15,0,0,0,0,15,15,9,9,0,0,0,0,15,2,9,8,0,0,0,0,0,0,0,0,9,9,9,8,0,0,0,0,9,9,8,9,0,0,0,0,9,8,8,8,0,0,0,0,8,9,8,8] >;
C42.2C23 in GAP, Magma, Sage, TeX
C_4^2._2C_2^3
% in TeX
G:=Group("C4^2.2C2^3"); // GroupNames label
G:=SmallGroup(128,388);
// by ID
G=gap.SmallGroup(128,388);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,2,141,232,422,1123,570,521,136,3924,1411,998,242]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^4=c^2=e^2=1,d^2=a^2,a*b=b*a,c*a*c=d*a*d^-1=a^-1,e*a*e=a^-1*b^2,c*b*c=e*b*e=b^-1,d*b*d^-1=a^2*b^-1,d*c*d^-1=a*c,e*c*e=b*c,d*e=e*d>;
// generators/relations
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