p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.282D4, C42.415C23, C4.542- 1+4, C4.Q16⋊8C2, C8⋊2Q8⋊14C2, (C2×C4).26Q16, C4.48(C2×Q16), (C4×C8).78C22, C4⋊C4.169C23, C4⋊C8.289C22, C4.30(C8⋊C22), (C2×C4).428C24, (C2×C8).167C23, C4.SD16⋊13C2, (C22×C4).511D4, C23.700(C2×D4), C4⋊Q8.312C22, C2.16(C22×Q16), C22.19(C2×Q16), C2.D8.37C22, (C4×Q8).109C22, (C2×Q8).162C23, Q8⋊C4.5C22, C22⋊C8.181C22, (C2×C42).889C22, C23.48D4.2C2, C22.688(C22×D4), C22⋊Q8.202C22, C42.12C4.36C2, (C22×C4).1093C23, C23.37C23.39C2, C2.76(C23.38C23), (C2×C4⋊Q8).55C2, (C2×C4).871(C2×D4), C2.61(C2×C8⋊C22), (C2×C4⋊C4).648C22, SmallGroup(128,1962)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.282D4
G = < a,b,c,d | a4=b4=1, c4=d2=b2, ab=ba, ac=ca, dad-1=a-1, cbc-1=a2b-1, dbd-1=a2b, dcd-1=a2c3 >
Subgroups: 316 in 180 conjugacy classes, 96 normal (26 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, Q8, C23, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C22×C4, C22×C4, C2×Q8, C2×Q8, C4×C8, C22⋊C8, Q8⋊C4, C4⋊C8, C2.D8, C2×C42, C2×C4⋊C4, C2×C4⋊C4, C42⋊C2, C4×Q8, C4×Q8, C22⋊Q8, C22⋊Q8, C42.C2, C4⋊Q8, C4⋊Q8, C4⋊Q8, C22×Q8, C42.12C4, C4.Q16, C23.48D4, C4.SD16, C8⋊2Q8, C2×C4⋊Q8, C23.37C23, C42.282D4
Quotients: C1, C2, C22, D4, C23, Q16, C2×D4, C24, C2×Q16, C8⋊C22, C22×D4, 2- 1+4, C23.38C23, C22×Q16, C2×C8⋊C22, C42.282D4
►On 64 points
(1 32 35 44)(2 25 36 45)(3 26 37 46)(4 27 38 47)(5 28 39 48)(6 29 40 41)(7 30 33 42)(8 31 34 43)(9 57 55 19)(10 58 56 20)(11 59 49 21)(12 60 50 22)(13 61 51 23)(14 62 52 24)(15 63 53 17)(16 64 54 18)
(1 37 5 33)(2 8 6 4)(3 39 7 35)(9 49 13 53)(10 16 14 12)(11 51 15 55)(17 57 21 61)(18 24 22 20)(19 59 23 63)(25 31 29 27)(26 48 30 44)(28 42 32 46)(34 40 38 36)(41 47 45 43)(50 56 54 52)(58 64 62 60)
(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 50 5 54)(2 15 6 11)(3 56 7 52)(4 13 8 9)(10 33 14 37)(12 39 16 35)(17 29 21 25)(18 44 22 48)(19 27 23 31)(20 42 24 46)(26 58 30 62)(28 64 32 60)(34 55 38 51)(36 53 40 49)(41 59 45 63)(43 57 47 61)
G:=sub<Sym(64)| (1,32,35,44)(2,25,36,45)(3,26,37,46)(4,27,38,47)(5,28,39,48)(6,29,40,41)(7,30,33,42)(8,31,34,43)(9,57,55,19)(10,58,56,20)(11,59,49,21)(12,60,50,22)(13,61,51,23)(14,62,52,24)(15,63,53,17)(16,64,54,18), (1,37,5,33)(2,8,6,4)(3,39,7,35)(9,49,13,53)(10,16,14,12)(11,51,15,55)(17,57,21,61)(18,24,22,20)(19,59,23,63)(25,31,29,27)(26,48,30,44)(28,42,32,46)(34,40,38,36)(41,47,45,43)(50,56,54,52)(58,64,62,60), (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), (1,50,5,54)(2,15,6,11)(3,56,7,52)(4,13,8,9)(10,33,14,37)(12,39,16,35)(17,29,21,25)(18,44,22,48)(19,27,23,31)(20,42,24,46)(26,58,30,62)(28,64,32,60)(34,55,38,51)(36,53,40,49)(41,59,45,63)(43,57,47,61)>;
G:=Group( (1,32,35,44)(2,25,36,45)(3,26,37,46)(4,27,38,47)(5,28,39,48)(6,29,40,41)(7,30,33,42)(8,31,34,43)(9,57,55,19)(10,58,56,20)(11,59,49,21)(12,60,50,22)(13,61,51,23)(14,62,52,24)(15,63,53,17)(16,64,54,18), (1,37,5,33)(2,8,6,4)(3,39,7,35)(9,49,13,53)(10,16,14,12)(11,51,15,55)(17,57,21,61)(18,24,22,20)(19,59,23,63)(25,31,29,27)(26,48,30,44)(28,42,32,46)(34,40,38,36)(41,47,45,43)(50,56,54,52)(58,64,62,60), (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), (1,50,5,54)(2,15,6,11)(3,56,7,52)(4,13,8,9)(10,33,14,37)(12,39,16,35)(17,29,21,25)(18,44,22,48)(19,27,23,31)(20,42,24,46)(26,58,30,62)(28,64,32,60)(34,55,38,51)(36,53,40,49)(41,59,45,63)(43,57,47,61) );
G=PermutationGroup([[(1,32,35,44),(2,25,36,45),(3,26,37,46),(4,27,38,47),(5,28,39,48),(6,29,40,41),(7,30,33,42),(8,31,34,43),(9,57,55,19),(10,58,56,20),(11,59,49,21),(12,60,50,22),(13,61,51,23),(14,62,52,24),(15,63,53,17),(16,64,54,18)], [(1,37,5,33),(2,8,6,4),(3,39,7,35),(9,49,13,53),(10,16,14,12),(11,51,15,55),(17,57,21,61),(18,24,22,20),(19,59,23,63),(25,31,29,27),(26,48,30,44),(28,42,32,46),(34,40,38,36),(41,47,45,43),(50,56,54,52),(58,64,62,60)], [(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)], [(1,50,5,54),(2,15,6,11),(3,56,7,52),(4,13,8,9),(10,33,14,37),(12,39,16,35),(17,29,21,25),(18,44,22,48),(19,27,23,31),(20,42,24,46),(26,58,30,62),(28,64,32,60),(34,55,38,51),(36,53,40,49),(41,59,45,63),(43,57,47,61)]])
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | ··· | 4H | 4I | 4J | 4K | ··· | 4R | 8A | ··· | 8H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | ··· | 2 | 4 | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | - | + | - |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | Q16 | C8⋊C22 | 2- 1+4 |
| kernel | C42.282D4 | C42.12C4 | C4.Q16 | C23.48D4 | C4.SD16 | C8⋊2Q8 | C2×C4⋊Q8 | C23.37C23 | C42 | C22×C4 | C2×C4 | C4 | C4 |
| # reps | 1 | 1 | 4 | 4 | 2 | 2 | 1 | 1 | 2 | 2 | 8 | 2 | 2 |
Matrix representation of C42.282D4 ►in GL6(𝔽17)
| 0 | 1 | 0 | 0 | 0 | 0 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 15 | 0 | 0 |
| 0 | 0 | 1 | 16 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 1 |
| 0 | 0 | 16 | 1 | 16 | 0 |
| 3 | 14 | 0 | 0 | 0 | 0 |
| 3 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 2 | 0 |
| 0 | 0 | 1 | 0 | 1 | 1 |
| 0 | 0 | 0 | 16 | 16 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 12 | 5 | 0 | 0 | 0 | 0 |
| 5 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 15 | 0 |
| 0 | 0 | 0 | 0 | 16 | 1 |
| 0 | 0 | 1 | 0 | 1 | 0 |
| 0 | 0 | 1 | 16 | 1 | 0 |
G:=sub<GL(6,GF(17))| [0,16,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,16,0,0,0,0,0,0,0,1,1,0,16,0,0,15,16,1,1,0,0,0,0,0,16,0,0,0,0,1,0],[3,3,0,0,0,0,14,3,0,0,0,0,0,0,1,1,0,0,0,0,0,0,16,0,0,0,2,1,16,16,0,0,0,1,0,0],[12,5,0,0,0,0,5,5,0,0,0,0,0,0,16,0,1,1,0,0,0,0,0,16,0,0,15,16,1,1,0,0,0,1,0,0] >;
C42.282D4 in GAP, Magma, Sage, TeX
C_4^2._{282}D_4 % in TeX
G:=Group("C4^2.282D4"); // GroupNames label
G:=SmallGroup(128,1962);
// by ID
G=gap.SmallGroup(128,1962);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,120,758,219,436,675,4037,1027,124]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^4=d^2=b^2,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,c*b*c^-1=a^2*b^-1,d*b*d^-1=a^2*b,d*c*d^-1=a^2*c^3>;
// generators/relations