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×C8).167C23, (C2×C4).428C24, C4.SD16⋊13C2, C23.700(C2×D4), (C22×C4).511D4, C4⋊Q8.312C22, C22.19(C2×Q16), C2.16(C22×Q16), C2.D8.37C22, (C4×Q8).109C22, Q8⋊C4.5C22, (C2×Q8).162C23, 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
Subgroups: 316 in 180 conjugacy classes, 96 normal (26 characteristic)
C1, C2 [×3], C2 [×2], C4 [×8], C4 [×10], C22, C22 [×2], C22 [×2], C8 [×4], C2×C4 [×6], C2×C4 [×4], C2×C4 [×16], Q8 [×14], C23, C42 [×4], C42 [×2], C22⋊C4 [×2], C4⋊C4 [×6], C4⋊C4 [×13], C2×C8 [×4], C22×C4 [×3], C22×C4 [×2], C2×Q8 [×2], C2×Q8 [×9], C4×C8 [×2], C22⋊C8 [×2], Q8⋊C4 [×8], C4⋊C8 [×2], C2.D8 [×8], C2×C42, C2×C4⋊C4 [×2], C2×C4⋊C4, C42⋊C2, C4×Q8 [×2], C4×Q8, C22⋊Q8 [×2], C22⋊Q8, C42.C2, C4⋊Q8 [×2], C4⋊Q8 [×4], C4⋊Q8 [×2], C22×Q8, C42.12C4, C4.Q16 [×4], C23.48D4 [×4], C4.SD16 [×2], C8⋊2Q8 [×2], C2×C4⋊Q8, C23.37C23, C42.282D4
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], Q16 [×4], C2×D4 [×6], C24, C2×Q16 [×6], C8⋊C22 [×2], C22×D4, 2- (1+4) [×2], C23.38C23, C22×Q16, C2×C8⋊C22, C42.282D4
Generators and relations
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 >
►On 64 points
(1 28 37 44)(2 29 38 45)(3 30 39 46)(4 31 40 47)(5 32 33 48)(6 25 34 41)(7 26 35 42)(8 27 36 43)(9 57 55 17)(10 58 56 18)(11 59 49 19)(12 60 50 20)(13 61 51 21)(14 62 52 22)(15 63 53 23)(16 64 54 24)
(1 39 5 35)(2 8 6 4)(3 33 7 37)(9 49 13 53)(10 16 14 12)(11 51 15 55)(17 59 21 63)(18 24 22 20)(19 61 23 57)(25 31 29 27)(26 44 30 48)(28 46 32 42)(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 35 14 39)(12 33 16 37)(17 31 21 27)(18 42 22 46)(19 29 23 25)(20 48 24 44)(26 62 30 58)(28 60 32 64)(34 49 38 53)(36 55 40 51)(41 59 45 63)(43 57 47 61)
G:=sub<Sym(64)| (1,28,37,44)(2,29,38,45)(3,30,39,46)(4,31,40,47)(5,32,33,48)(6,25,34,41)(7,26,35,42)(8,27,36,43)(9,57,55,17)(10,58,56,18)(11,59,49,19)(12,60,50,20)(13,61,51,21)(14,62,52,22)(15,63,53,23)(16,64,54,24), (1,39,5,35)(2,8,6,4)(3,33,7,37)(9,49,13,53)(10,16,14,12)(11,51,15,55)(17,59,21,63)(18,24,22,20)(19,61,23,57)(25,31,29,27)(26,44,30,48)(28,46,32,42)(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)(10,35,14,39)(12,33,16,37)(17,31,21,27)(18,42,22,46)(19,29,23,25)(20,48,24,44)(26,62,30,58)(28,60,32,64)(34,49,38,53)(36,55,40,51)(41,59,45,63)(43,57,47,61)>;
G:=Group( (1,28,37,44)(2,29,38,45)(3,30,39,46)(4,31,40,47)(5,32,33,48)(6,25,34,41)(7,26,35,42)(8,27,36,43)(9,57,55,17)(10,58,56,18)(11,59,49,19)(12,60,50,20)(13,61,51,21)(14,62,52,22)(15,63,53,23)(16,64,54,24), (1,39,5,35)(2,8,6,4)(3,33,7,37)(9,49,13,53)(10,16,14,12)(11,51,15,55)(17,59,21,63)(18,24,22,20)(19,61,23,57)(25,31,29,27)(26,44,30,48)(28,46,32,42)(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)(10,35,14,39)(12,33,16,37)(17,31,21,27)(18,42,22,46)(19,29,23,25)(20,48,24,44)(26,62,30,58)(28,60,32,64)(34,49,38,53)(36,55,40,51)(41,59,45,63)(43,57,47,61) );
G=PermutationGroup([(1,28,37,44),(2,29,38,45),(3,30,39,46),(4,31,40,47),(5,32,33,48),(6,25,34,41),(7,26,35,42),(8,27,36,43),(9,57,55,17),(10,58,56,18),(11,59,49,19),(12,60,50,20),(13,61,51,21),(14,62,52,22),(15,63,53,23),(16,64,54,24)], [(1,39,5,35),(2,8,6,4),(3,33,7,37),(9,49,13,53),(10,16,14,12),(11,51,15,55),(17,59,21,63),(18,24,22,20),(19,61,23,57),(25,31,29,27),(26,44,30,48),(28,46,32,42),(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),(10,35,14,39),(12,33,16,37),(17,31,21,27),(18,42,22,46),(19,29,23,25),(20,48,24,44),(26,62,30,58),(28,60,32,64),(34,49,38,53),(36,55,40,51),(41,59,45,63),(43,57,47,61)])
Matrix representation ►G ⊆ 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] >;
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 |
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