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 [×3], C2, C4 [×4], C4 [×7], C22, C22 [×3], C8 [×3], C2×C4 [×3], C2×C4 [×8], D4 [×2], Q8 [×6], C23, C42, C42 [×3], C22⋊C4 [×5], C4⋊C4 [×2], C4⋊C4 [×2], C4⋊C4 [×6], C2×C8 [×3], C22×C4, C2×D4, C2×Q8, C2×Q8 [×2], C4×C8, D4⋊C4, Q8⋊C4 [×3], C4⋊C8 [×2], C4.Q8, C2.D8, C42⋊C2, C4×D4, C4×Q8, C4×Q8, C22⋊Q8, C4.4D4, C42⋊2C2 [×2], C4⋊Q8 [×2], D4⋊C8, Q8⋊C8, C4.10D8, D4⋊2Q8, C4.Q16, C4.SD16, C22.50C24, C42.199C23
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, C2×D4 [×3], C22≀C2, C4○D8 [×2], C8.C22 [×2], D4.7D4 [×2], 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 32 34 28)(22 29 35 25)(23 30 36 26)(24 31 33 27)(37 48 52 44)(38 45 49 41)(39 46 50 42)(40 47 51 43)
(1 43 3 41)(2 42 4 44)(5 40 7 38)(6 39 8 37)(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 57 23 59)(22 60 24 58)(25 53 27 55)(26 56 28 54)(29 61 31 63)(30 64 32 62)
(1 29 15 25)(2 32 16 28)(3 31 13 27)(4 30 14 26)(5 35 17 22)(6 34 18 21)(7 33 19 24)(8 36 20 23)(9 49 57 38)(10 52 58 37)(11 51 59 40)(12 50 60 39)(41 62 45 54)(42 61 46 53)(43 64 47 56)(44 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 36 33)(23 24 34 35)(25 26 31 32)(27 28 29 30)(37 41 50 47)(38 46 51 44)(39 43 52 45)(40 48 49 42)
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)(21,32,34,28)(22,29,35,25)(23,30,36,26)(24,31,33,27)(37,48,52,44)(38,45,49,41)(39,46,50,42)(40,47,51,43), (1,43,3,41)(2,42,4,44)(5,40,7,38)(6,39,8,37)(9,36,11,34)(10,35,12,33)(13,45,15,47)(14,48,16,46)(17,51,19,49)(18,50,20,52)(21,57,23,59)(22,60,24,58)(25,53,27,55)(26,56,28,54)(29,61,31,63)(30,64,32,62), (1,29,15,25)(2,32,16,28)(3,31,13,27)(4,30,14,26)(5,35,17,22)(6,34,18,21)(7,33,19,24)(8,36,20,23)(9,49,57,38)(10,52,58,37)(11,51,59,40)(12,50,60,39)(41,62,45,54)(42,61,46,53)(43,64,47,56)(44,63,48,55), (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)(21,22,36,33)(23,24,34,35)(25,26,31,32)(27,28,29,30)(37,41,50,47)(38,46,51,44)(39,43,52,45)(40,48,49,42)>;
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)(21,32,34,28)(22,29,35,25)(23,30,36,26)(24,31,33,27)(37,48,52,44)(38,45,49,41)(39,46,50,42)(40,47,51,43), (1,43,3,41)(2,42,4,44)(5,40,7,38)(6,39,8,37)(9,36,11,34)(10,35,12,33)(13,45,15,47)(14,48,16,46)(17,51,19,49)(18,50,20,52)(21,57,23,59)(22,60,24,58)(25,53,27,55)(26,56,28,54)(29,61,31,63)(30,64,32,62), (1,29,15,25)(2,32,16,28)(3,31,13,27)(4,30,14,26)(5,35,17,22)(6,34,18,21)(7,33,19,24)(8,36,20,23)(9,49,57,38)(10,52,58,37)(11,51,59,40)(12,50,60,39)(41,62,45,54)(42,61,46,53)(43,64,47,56)(44,63,48,55), (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)(21,22,36,33)(23,24,34,35)(25,26,31,32)(27,28,29,30)(37,41,50,47)(38,46,51,44)(39,43,52,45)(40,48,49,42) );
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),(21,32,34,28),(22,29,35,25),(23,30,36,26),(24,31,33,27),(37,48,52,44),(38,45,49,41),(39,46,50,42),(40,47,51,43)], [(1,43,3,41),(2,42,4,44),(5,40,7,38),(6,39,8,37),(9,36,11,34),(10,35,12,33),(13,45,15,47),(14,48,16,46),(17,51,19,49),(18,50,20,52),(21,57,23,59),(22,60,24,58),(25,53,27,55),(26,56,28,54),(29,61,31,63),(30,64,32,62)], [(1,29,15,25),(2,32,16,28),(3,31,13,27),(4,30,14,26),(5,35,17,22),(6,34,18,21),(7,33,19,24),(8,36,20,23),(9,49,57,38),(10,52,58,37),(11,51,59,40),(12,50,60,39),(41,62,45,54),(42,61,46,53),(43,64,47,56),(44,63,48,55)], [(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),(21,22,36,33),(23,24,34,35),(25,26,31,32),(27,28,29,30),(37,41,50,47),(38,46,51,44),(39,43,52,45),(40,48,49,42)])
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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