p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.199C23, Q8⋊C8⋊24C2, D4⋊C8.5C2, C4⋊C4.29D4, C4.Q16⋊5C2, (C2×D4).50D4, (C2×Q8).48D4, C4.35(C4○D8), (C4×C8).21C22, D4⋊2Q8.6C2, C4.SD16⋊3C2, C4⋊Q8.20C22, C4.10D8⋊11C2, C4⋊C8.165C22, (C4×D4).30C22, (C4×Q8).30C22, C4.62(C8.C22), C2.19(D4.8D4), C22.165C22≀C2, C2.18(D4.7D4), C22.50C24.1C2, (C2×C4).956(C2×D4), SmallGroup(128,370)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.199C23
G = < a,b,c,d,e | a4=b4=1, c2=a2, d2=b2, e2=a2b2, ab=ba, cac-1=dad-1=a-1, eae-1=ab2, cbc-1=dbd-1=ebe-1=b-1, dcd-1=ac, ece-1=bc, de=ed >
Subgroups: 208 in 97 conjugacy classes, 34 normal (32 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, D4, Q8, C23, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C4⋊C4, C2×C8, C22×C4, C2×D4, C2×Q8, C2×Q8, C4×C8, D4⋊C4, Q8⋊C4, C4⋊C8, C4.Q8, C2.D8, C42⋊C2, C4×D4, C4×Q8, C4×Q8, C22⋊Q8, C4.4D4, C42⋊2C2, C4⋊Q8, D4⋊C8, Q8⋊C8, C4.10D8, D4⋊2Q8, C4.Q16, C4.SD16, C22.50C24, C42.199C23
Quotients: C1, C2, C22, D4, C23, C2×D4, C22≀C2, C4○D8, C8.C22, D4.7D4, D4.8D4, C42.199C23
Character table of C42.199C23
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | |
| size | 1 | 1 | 1 | 1 | 8 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 16 | 4 | 4 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | linear of order 2 |
| ρ9 | 2 | 2 | 2 | 2 | -2 | -2 | -2 | 2 | 2 | -2 | 0 | 2 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ10 | 2 | 2 | 2 | 2 | 0 | -2 | -2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ11 | 2 | 2 | 2 | 2 | 0 | -2 | -2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ12 | 2 | 2 | 2 | 2 | 0 | 2 | 2 | -2 | -2 | -2 | 2 | 0 | 2 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ13 | 2 | 2 | 2 | 2 | 2 | -2 | -2 | 2 | 2 | -2 | 0 | -2 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ14 | 2 | 2 | 2 | 2 | 0 | 2 | 2 | -2 | -2 | -2 | -2 | 0 | -2 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ15 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | -2 | 2 | 0 | 0 | 2i | 0 | -2i | 0 | 0 | 0 | 0 | -√-2 | -√-2 | √-2 | √-2 | 0 | √2 | -√2 | 0 | complex lifted from C4○D8 |
| ρ16 | 2 | -2 | -2 | 2 | 0 | 2 | -2 | 0 | 0 | 0 | 2i | 0 | -2i | 0 | 0 | 0 | 0 | 0 | √2 | -√2 | -√2 | √2 | √-2 | 0 | 0 | -√-2 | complex lifted from C4○D8 |
| ρ17 | 2 | -2 | -2 | 2 | 0 | 2 | -2 | 0 | 0 | 0 | -2i | 0 | 2i | 0 | 0 | 0 | 0 | 0 | -√2 | √2 | √2 | -√2 | √-2 | 0 | 0 | -√-2 | complex lifted from C4○D8 |
| ρ18 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | -2 | 2 | 0 | 0 | -2i | 0 | 2i | 0 | 0 | 0 | 0 | √-2 | √-2 | -√-2 | -√-2 | 0 | √2 | -√2 | 0 | complex lifted from C4○D8 |
| ρ19 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | -2 | 2 | 0 | 0 | -2i | 0 | 2i | 0 | 0 | 0 | 0 | -√-2 | -√-2 | √-2 | √-2 | 0 | -√2 | √2 | 0 | complex lifted from C4○D8 |
| ρ20 | 2 | -2 | -2 | 2 | 0 | 2 | -2 | 0 | 0 | 0 | -2i | 0 | 2i | 0 | 0 | 0 | 0 | 0 | √2 | -√2 | -√2 | √2 | -√-2 | 0 | 0 | √-2 | complex lifted from C4○D8 |
| ρ21 | 2 | -2 | -2 | 2 | 0 | 2 | -2 | 0 | 0 | 0 | 2i | 0 | -2i | 0 | 0 | 0 | 0 | 0 | -√2 | √2 | √2 | -√2 | -√-2 | 0 | 0 | √-2 | complex lifted from C4○D8 |
| ρ22 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | -2 | 2 | 0 | 0 | 2i | 0 | -2i | 0 | 0 | 0 | 0 | √-2 | √-2 | -√-2 | -√-2 | 0 | -√2 | √2 | 0 | complex lifted from C4○D8 |
| ρ23 | 4 | -4 | -4 | 4 | 0 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from C8.C22, Schur index 2 |
| ρ24 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from C8.C22, Schur index 2 |
| ρ25 | 4 | -4 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2i | 2i | -2i | 2i | 0 | 0 | 0 | 0 | complex lifted from D4.8D4 |
| ρ26 | 4 | -4 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2i | -2i | 2i | -2i | 0 | 0 | 0 | 0 | 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 17 15 5)(2 18 16 6)(3 19 13 7)(4 20 14 8)(9 62 57 54)(10 63 58 55)(11 64 59 56)(12 61 60 53)(21 28 29 35)(22 25 30 36)(23 26 31 33)(24 27 32 34)(37 44 45 49)(38 41 46 50)(39 42 47 51)(40 43 48 52)
(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 51 19 49)(18 50 20 52)(21 53 23 55)(22 56 24 54)(25 59 27 57)(26 58 28 60)(29 61 31 63)(30 64 32 62)
(1 29 15 21)(2 32 16 24)(3 31 13 23)(4 30 14 22)(5 35 17 28)(6 34 18 27)(7 33 19 26)(8 36 20 25)(9 49 57 44)(10 52 58 43)(11 51 59 42)(12 50 60 41)(37 62 45 54)(38 61 46 53)(39 64 47 56)(40 63 48 55)
(1 4 13 16)(2 15 14 3)(5 20 19 6)(7 18 17 8)(9 53 59 63)(10 62 60 56)(11 55 57 61)(12 64 58 54)(21 22 31 32)(23 24 29 30)(25 26 34 35)(27 28 36 33)(37 50 47 43)(38 42 48 49)(39 52 45 41)(40 44 46 51)
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), (1,17,15,5)(2,18,16,6)(3,19,13,7)(4,20,14,8)(9,62,57,54)(10,63,58,55)(11,64,59,56)(12,61,60,53)(21,28,29,35)(22,25,30,36)(23,26,31,33)(24,27,32,34)(37,44,45,49)(38,41,46,50)(39,42,47,51)(40,43,48,52), (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)(17,51,19,49)(18,50,20,52)(21,53,23,55)(22,56,24,54)(25,59,27,57)(26,58,28,60)(29,61,31,63)(30,64,32,62), (1,29,15,21)(2,32,16,24)(3,31,13,23)(4,30,14,22)(5,35,17,28)(6,34,18,27)(7,33,19,26)(8,36,20,25)(9,49,57,44)(10,52,58,43)(11,51,59,42)(12,50,60,41)(37,62,45,54)(38,61,46,53)(39,64,47,56)(40,63,48,55), (1,4,13,16)(2,15,14,3)(5,20,19,6)(7,18,17,8)(9,53,59,63)(10,62,60,56)(11,55,57,61)(12,64,58,54)(21,22,31,32)(23,24,29,30)(25,26,34,35)(27,28,36,33)(37,50,47,43)(38,42,48,49)(39,52,45,41)(40,44,46,51)>;
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), (1,17,15,5)(2,18,16,6)(3,19,13,7)(4,20,14,8)(9,62,57,54)(10,63,58,55)(11,64,59,56)(12,61,60,53)(21,28,29,35)(22,25,30,36)(23,26,31,33)(24,27,32,34)(37,44,45,49)(38,41,46,50)(39,42,47,51)(40,43,48,52), (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)(17,51,19,49)(18,50,20,52)(21,53,23,55)(22,56,24,54)(25,59,27,57)(26,58,28,60)(29,61,31,63)(30,64,32,62), (1,29,15,21)(2,32,16,24)(3,31,13,23)(4,30,14,22)(5,35,17,28)(6,34,18,27)(7,33,19,26)(8,36,20,25)(9,49,57,44)(10,52,58,43)(11,51,59,42)(12,50,60,41)(37,62,45,54)(38,61,46,53)(39,64,47,56)(40,63,48,55), (1,4,13,16)(2,15,14,3)(5,20,19,6)(7,18,17,8)(9,53,59,63)(10,62,60,56)(11,55,57,61)(12,64,58,54)(21,22,31,32)(23,24,29,30)(25,26,34,35)(27,28,36,33)(37,50,47,43)(38,42,48,49)(39,52,45,41)(40,44,46,51) );
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)], [(1,17,15,5),(2,18,16,6),(3,19,13,7),(4,20,14,8),(9,62,57,54),(10,63,58,55),(11,64,59,56),(12,61,60,53),(21,28,29,35),(22,25,30,36),(23,26,31,33),(24,27,32,34),(37,44,45,49),(38,41,46,50),(39,42,47,51),(40,43,48,52)], [(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),(17,51,19,49),(18,50,20,52),(21,53,23,55),(22,56,24,54),(25,59,27,57),(26,58,28,60),(29,61,31,63),(30,64,32,62)], [(1,29,15,21),(2,32,16,24),(3,31,13,23),(4,30,14,22),(5,35,17,28),(6,34,18,27),(7,33,19,26),(8,36,20,25),(9,49,57,44),(10,52,58,43),(11,51,59,42),(12,50,60,41),(37,62,45,54),(38,61,46,53),(39,64,47,56),(40,63,48,55)], [(1,4,13,16),(2,15,14,3),(5,20,19,6),(7,18,17,8),(9,53,59,63),(10,62,60,56),(11,55,57,61),(12,64,58,54),(21,22,31,32),(23,24,29,30),(25,26,34,35),(27,28,36,33),(37,50,47,43),(38,42,48,49),(39,52,45,41),(40,44,46,51)]])
Matrix representation of C42.199C23 ►in GL4(𝔽17) generated by
| 13 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 1 | 15 |
| 0 | 0 | 1 | 16 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 15 |
| 0 | 0 | 1 | 16 |
| 0 | 2 | 0 | 0 |
| 8 | 0 | 0 | 0 |
| 0 | 0 | 10 | 7 |
| 0 | 0 | 5 | 7 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 13 | 8 |
| 0 | 0 | 0 | 4 |
| 4 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 1 | 15 |
| 0 | 0 | 0 | 16 |
G:=sub<GL(4,GF(17))| [13,0,0,0,0,4,0,0,0,0,1,1,0,0,15,16],[1,0,0,0,0,1,0,0,0,0,1,1,0,0,15,16],[0,8,0,0,2,0,0,0,0,0,10,5,0,0,7,7],[0,1,0,0,1,0,0,0,0,0,13,0,0,0,8,4],[4,0,0,0,0,4,0,0,0,0,1,0,0,0,15,16] >;
C42.199C23 in GAP, Magma, Sage, TeX
C_4^2._{199}C_2^3 % in TeX
G:=Group("C4^2.199C2^3"); // GroupNames label
G:=SmallGroup(128,370);
// by ID
G=gap.SmallGroup(128,370);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,2,224,141,456,422,520,1123,570,521,136,2804,1411,718,172]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^4=1,c^2=a^2,d^2=b^2,e^2=a^2*b^2,a*b=b*a,c*a*c^-1=d*a*d^-1=a^-1,e*a*e^-1=a*b^2,c*b*c^-1=d*b*d^-1=e*b*e^-1=b^-1,d*c*d^-1=a*c,e*c*e^-1=b*c,d*e=e*d>;
// generators/relations
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