p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.213C23, Q8⋊C8⋊19C2, D4⋊C8.9C2, C4⋊C4.35D4, C4⋊2Q16⋊5C2, C8⋊3Q8⋊13C2, (C2×D4).55D4, (C2×Q8).53D4, C4.61(C4○D8), C4.10D8⋊3C2, C4⋊C8.16C22, D4⋊Q8.3C2, C4⋊Q8.33C22, C4.40(C8⋊C22), (C4×C8).246C22, (C4×D4).41C22, (C4×Q8).41C22, C2.25(D4⋊D4), C4.68(C8.C22), C22.179C22≀C2, C2.25(D4.7D4), C2.17(D4.10D4), C22.50C24.3C2, (C2×C4).970(C2×D4), SmallGroup(128,384)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.213C23
G = < a,b,c,d,e | a4=b4=1, c2=b2, d2=a2b2, e2=a2, 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: 216 in 100 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, Q16, 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, C2×Q16, D4⋊C8, Q8⋊C8, C4.10D8, C4⋊2Q16, D4⋊Q8, C8⋊3Q8, C22.50C24, C42.213C23
Quotients: C1, C2, C22, D4, C23, C2×D4, C22≀C2, C4○D8, C8⋊C22, C8.C22, D4⋊D4, D4.7D4, D4.10D4, C42.213C23
Character table of C42.213C23
| 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 | 0 | -2 | -2 | -2 | -2 | 0 | 0 | 0 | 2 | 0 | 2 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ10 | 2 | 2 | 2 | 2 | 0 | -2 | -2 | -2 | -2 | 0 | 0 | 0 | 2 | 0 | -2 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ11 | 2 | 2 | 2 | 2 | 0 | -2 | 2 | -2 | 2 | 0 | 2 | 0 | -2 | 2 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ12 | 2 | 2 | 2 | 2 | 0 | -2 | 2 | -2 | 2 | 0 | -2 | 0 | -2 | -2 | 0 | 2 | 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 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ14 | 2 | 2 | 2 | 2 | -2 | 2 | -2 | 2 | -2 | 2 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ15 | 2 | -2 | 2 | -2 | 0 | 0 | 2 | 0 | -2 | 0 | 2i | 0 | 0 | -2i | 0 | 0 | 0 | 0 | √-2 | -√-2 | -√-2 | √-2 | 0 | -√2 | 0 | √2 | complex lifted from C4○D8 |
| ρ16 | 2 | 2 | -2 | -2 | 0 | 2 | 0 | -2 | 0 | 2i | 0 | -2i | 0 | 0 | 0 | 0 | 0 | 0 | -√-2 | -√-2 | √-2 | √-2 | -√2 | 0 | √2 | 0 | complex lifted from C4○D8 |
| ρ17 | 2 | 2 | -2 | -2 | 0 | 2 | 0 | -2 | 0 | -2i | 0 | 2i | 0 | 0 | 0 | 0 | 0 | 0 | -√-2 | -√-2 | √-2 | √-2 | √2 | 0 | -√2 | 0 | complex lifted from C4○D8 |
| ρ18 | 2 | -2 | 2 | -2 | 0 | 0 | 2 | 0 | -2 | 0 | -2i | 0 | 0 | 2i | 0 | 0 | 0 | 0 | -√-2 | √-2 | √-2 | -√-2 | 0 | -√2 | 0 | √2 | complex lifted from C4○D8 |
| ρ19 | 2 | 2 | -2 | -2 | 0 | 2 | 0 | -2 | 0 | -2i | 0 | 2i | 0 | 0 | 0 | 0 | 0 | 0 | √-2 | √-2 | -√-2 | -√-2 | -√2 | 0 | √2 | 0 | complex lifted from C4○D8 |
| ρ20 | 2 | 2 | -2 | -2 | 0 | 2 | 0 | -2 | 0 | 2i | 0 | -2i | 0 | 0 | 0 | 0 | 0 | 0 | √-2 | √-2 | -√-2 | -√-2 | √2 | 0 | -√2 | 0 | complex lifted from C4○D8 |
| ρ21 | 2 | -2 | 2 | -2 | 0 | 0 | 2 | 0 | -2 | 0 | 2i | 0 | 0 | -2i | 0 | 0 | 0 | 0 | -√-2 | √-2 | √-2 | -√-2 | 0 | √2 | 0 | -√2 | complex lifted from C4○D8 |
| ρ22 | 2 | -2 | 2 | -2 | 0 | 0 | 2 | 0 | -2 | 0 | -2i | 0 | 0 | 2i | 0 | 0 | 0 | 0 | √-2 | -√-2 | -√-2 | √-2 | 0 | √2 | 0 | -√2 | complex lifted from C4○D8 |
| ρ23 | 4 | -4 | 4 | -4 | 0 | 0 | -4 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C8⋊C22 |
| ρ24 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | symplectic lifted from D4.10D4, Schur index 2 |
| ρ25 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | symplectic lifted from D4.10D4, Schur index 2 |
| ρ26 | 4 | 4 | -4 | -4 | 0 | -4 | 0 | 4 | 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 |
►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 18 15 5)(2 19 16 6)(3 20 13 7)(4 17 14 8)(9 62 56 59)(10 63 53 60)(11 64 54 57)(12 61 55 58)(21 31 33 26)(22 32 34 27)(23 29 35 28)(24 30 36 25)(37 48 49 44)(38 45 50 41)(39 46 51 42)(40 47 52 43)
(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 25 13 32)(2 28 14 31)(3 27 15 30)(4 26 16 29)(5 24 20 34)(6 23 17 33)(7 22 18 36)(8 21 19 35)(9 39 54 49)(10 38 55 52)(11 37 56 51)(12 40 53 50)(41 58 47 63)(42 57 48 62)(43 60 45 61)(44 59 46 64)
(1 29 3 31)(2 25 4 27)(5 35 7 33)(6 24 8 22)(9 46 11 48)(10 43 12 41)(13 26 15 28)(14 32 16 30)(17 34 19 36)(18 23 20 21)(37 62 39 64)(38 60 40 58)(42 54 44 56)(45 53 47 55)(49 59 51 57)(50 63 52 61)
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), (1,18,15,5)(2,19,16,6)(3,20,13,7)(4,17,14,8)(9,62,56,59)(10,63,53,60)(11,64,54,57)(12,61,55,58)(21,31,33,26)(22,32,34,27)(23,29,35,28)(24,30,36,25)(37,48,49,44)(38,45,50,41)(39,46,51,42)(40,47,52,43), (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), (1,25,13,32)(2,28,14,31)(3,27,15,30)(4,26,16,29)(5,24,20,34)(6,23,17,33)(7,22,18,36)(8,21,19,35)(9,39,54,49)(10,38,55,52)(11,37,56,51)(12,40,53,50)(41,58,47,63)(42,57,48,62)(43,60,45,61)(44,59,46,64), (1,29,3,31)(2,25,4,27)(5,35,7,33)(6,24,8,22)(9,46,11,48)(10,43,12,41)(13,26,15,28)(14,32,16,30)(17,34,19,36)(18,23,20,21)(37,62,39,64)(38,60,40,58)(42,54,44,56)(45,53,47,55)(49,59,51,57)(50,63,52,61)>;
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), (1,18,15,5)(2,19,16,6)(3,20,13,7)(4,17,14,8)(9,62,56,59)(10,63,53,60)(11,64,54,57)(12,61,55,58)(21,31,33,26)(22,32,34,27)(23,29,35,28)(24,30,36,25)(37,48,49,44)(38,45,50,41)(39,46,51,42)(40,47,52,43), (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), (1,25,13,32)(2,28,14,31)(3,27,15,30)(4,26,16,29)(5,24,20,34)(6,23,17,33)(7,22,18,36)(8,21,19,35)(9,39,54,49)(10,38,55,52)(11,37,56,51)(12,40,53,50)(41,58,47,63)(42,57,48,62)(43,60,45,61)(44,59,46,64), (1,29,3,31)(2,25,4,27)(5,35,7,33)(6,24,8,22)(9,46,11,48)(10,43,12,41)(13,26,15,28)(14,32,16,30)(17,34,19,36)(18,23,20,21)(37,62,39,64)(38,60,40,58)(42,54,44,56)(45,53,47,55)(49,59,51,57)(50,63,52,61) );
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)], [(1,18,15,5),(2,19,16,6),(3,20,13,7),(4,17,14,8),(9,62,56,59),(10,63,53,60),(11,64,54,57),(12,61,55,58),(21,31,33,26),(22,32,34,27),(23,29,35,28),(24,30,36,25),(37,48,49,44),(38,45,50,41),(39,46,51,42),(40,47,52,43)], [(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)], [(1,25,13,32),(2,28,14,31),(3,27,15,30),(4,26,16,29),(5,24,20,34),(6,23,17,33),(7,22,18,36),(8,21,19,35),(9,39,54,49),(10,38,55,52),(11,37,56,51),(12,40,53,50),(41,58,47,63),(42,57,48,62),(43,60,45,61),(44,59,46,64)], [(1,29,3,31),(2,25,4,27),(5,35,7,33),(6,24,8,22),(9,46,11,48),(10,43,12,41),(13,26,15,28),(14,32,16,30),(17,34,19,36),(18,23,20,21),(37,62,39,64),(38,60,40,58),(42,54,44,56),(45,53,47,55),(49,59,51,57),(50,63,52,61)]])
Matrix representation of C42.213C23 ►in GL4(𝔽17) generated by
| 16 | 15 | 0 | 0 |
| 1 | 1 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 9 | 13 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 9 | 13 |
| 0 | 6 | 0 | 0 |
| 3 | 0 | 0 | 0 |
| 0 | 0 | 13 | 13 |
| 0 | 0 | 0 | 4 |
| 13 | 9 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 9 | 9 |
| 0 | 0 | 10 | 8 |
| 13 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 |
| 0 | 0 | 15 | 15 |
| 0 | 0 | 11 | 2 |
G:=sub<GL(4,GF(17))| [16,1,0,0,15,1,0,0,0,0,4,9,0,0,0,13],[1,0,0,0,0,1,0,0,0,0,4,9,0,0,0,13],[0,3,0,0,6,0,0,0,0,0,13,0,0,0,13,4],[13,0,0,0,9,4,0,0,0,0,9,10,0,0,9,8],[13,0,0,0,0,13,0,0,0,0,15,11,0,0,15,2] >;
C42.213C23 in GAP, Magma, Sage, TeX
C_4^2._{213}C_2^3 % in TeX
G:=Group("C4^2.213C2^3"); // GroupNames label
G:=SmallGroup(128,384);
// by ID
G=gap.SmallGroup(128,384);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,2,448,141,680,422,184,1123,570,521,136,2804,1411,718,172]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^4=1,c^2=b^2,d^2=a^2*b^2,e^2=a^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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