p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.297D4, C4.52- (1+4), C42.431C23, C4.242+ (1+4), (C2×C4)⋊5Q16, C4.49(C2×Q16), C4.Q16⋊11C2, C4⋊2Q16⋊11C2, C22.4(C2×Q16), C4⋊C4.188C23, C4⋊C8.293C22, (C2×C4).447C24, (C2×C8).172C23, C8.18D4.7C2, C23.406(C2×D4), (C22×C4).525D4, C4⋊Q8.326C22, C2.17(C22×Q16), C2.D8.46C22, Q8⋊C4.9C22, (C2×Q16).30C22, (C2×Q8).177C23, (C4×Q8).124C22, (C2×C42).904C22, (C22×C8).154C22, C22.707(C22×D4), C22⋊Q8.214C22, C2.70(D8⋊C22), (C22×C4).1580C23, C23.37C23.43C2, C2.66(C22.31C24), (C2×C4⋊C8).43C2, (C2×C4).571(C2×D4), SmallGroup(128,1981)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 292 in 174 conjugacy classes, 94 normal (16 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×4], C4 [×11], C22, C22 [×2], C22 [×2], C8 [×4], C2×C4 [×2], C2×C4 [×8], C2×C4 [×11], Q8 [×12], C23, C42 [×4], C42 [×4], C22⋊C4 [×4], C4⋊C4 [×4], C4⋊C4 [×14], C2×C8 [×4], C2×C8 [×2], Q16 [×4], C22×C4 [×3], C2×Q8 [×4], C2×Q8 [×2], Q8⋊C4 [×8], C4⋊C8 [×4], C2.D8 [×4], C2×C42, C42⋊C2 [×2], C4×Q8 [×4], C4×Q8 [×2], C22⋊Q8 [×4], C22⋊Q8 [×2], C42.C2 [×2], C4⋊Q8 [×4], C22×C8 [×2], C2×Q16 [×4], C2×C4⋊C8, C4⋊2Q16 [×4], C8.18D4 [×4], C4.Q16 [×4], C23.37C23 [×2], C42.297D4
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], Q16 [×4], C2×D4 [×6], C24, C2×Q16 [×6], C22×D4, 2+ (1+4), 2- (1+4), C22.31C24, C22×Q16, D8⋊C22, C42.297D4
Generators and relations
G = < a,b,c,d | a4=b4=1, c4=d2=a2, ab=ba, ac=ca, dad-1=ab2, cbc-1=a2b-1, dbd-1=b-1, dcd-1=a2c3 >
►On 64 points
(1 7 5 3)(2 8 6 4)(9 34 13 38)(10 35 14 39)(11 36 15 40)(12 37 16 33)(17 23 21 19)(18 24 22 20)(25 48 29 44)(26 41 30 45)(27 42 31 46)(28 43 32 47)(49 55 53 51)(50 56 54 52)(57 63 61 59)(58 64 62 60)
(1 52 63 19)(2 24 64 49)(3 54 57 21)(4 18 58 51)(5 56 59 23)(6 20 60 53)(7 50 61 17)(8 22 62 55)(9 47 40 26)(10 31 33 44)(11 41 34 28)(12 25 35 46)(13 43 36 30)(14 27 37 48)(15 45 38 32)(16 29 39 42)
(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 25 5 29)(2 32 6 28)(3 31 7 27)(4 30 8 26)(9 51 13 55)(10 50 14 54)(11 49 15 53)(12 56 16 52)(17 37 21 33)(18 36 22 40)(19 35 23 39)(20 34 24 38)(41 64 45 60)(42 63 46 59)(43 62 47 58)(44 61 48 57)
G:=sub<Sym(64)| (1,7,5,3)(2,8,6,4)(9,34,13,38)(10,35,14,39)(11,36,15,40)(12,37,16,33)(17,23,21,19)(18,24,22,20)(25,48,29,44)(26,41,30,45)(27,42,31,46)(28,43,32,47)(49,55,53,51)(50,56,54,52)(57,63,61,59)(58,64,62,60), (1,52,63,19)(2,24,64,49)(3,54,57,21)(4,18,58,51)(5,56,59,23)(6,20,60,53)(7,50,61,17)(8,22,62,55)(9,47,40,26)(10,31,33,44)(11,41,34,28)(12,25,35,46)(13,43,36,30)(14,27,37,48)(15,45,38,32)(16,29,39,42), (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), (1,25,5,29)(2,32,6,28)(3,31,7,27)(4,30,8,26)(9,51,13,55)(10,50,14,54)(11,49,15,53)(12,56,16,52)(17,37,21,33)(18,36,22,40)(19,35,23,39)(20,34,24,38)(41,64,45,60)(42,63,46,59)(43,62,47,58)(44,61,48,57)>;
G:=Group( (1,7,5,3)(2,8,6,4)(9,34,13,38)(10,35,14,39)(11,36,15,40)(12,37,16,33)(17,23,21,19)(18,24,22,20)(25,48,29,44)(26,41,30,45)(27,42,31,46)(28,43,32,47)(49,55,53,51)(50,56,54,52)(57,63,61,59)(58,64,62,60), (1,52,63,19)(2,24,64,49)(3,54,57,21)(4,18,58,51)(5,56,59,23)(6,20,60,53)(7,50,61,17)(8,22,62,55)(9,47,40,26)(10,31,33,44)(11,41,34,28)(12,25,35,46)(13,43,36,30)(14,27,37,48)(15,45,38,32)(16,29,39,42), (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), (1,25,5,29)(2,32,6,28)(3,31,7,27)(4,30,8,26)(9,51,13,55)(10,50,14,54)(11,49,15,53)(12,56,16,52)(17,37,21,33)(18,36,22,40)(19,35,23,39)(20,34,24,38)(41,64,45,60)(42,63,46,59)(43,62,47,58)(44,61,48,57) );
G=PermutationGroup([(1,7,5,3),(2,8,6,4),(9,34,13,38),(10,35,14,39),(11,36,15,40),(12,37,16,33),(17,23,21,19),(18,24,22,20),(25,48,29,44),(26,41,30,45),(27,42,31,46),(28,43,32,47),(49,55,53,51),(50,56,54,52),(57,63,61,59),(58,64,62,60)], [(1,52,63,19),(2,24,64,49),(3,54,57,21),(4,18,58,51),(5,56,59,23),(6,20,60,53),(7,50,61,17),(8,22,62,55),(9,47,40,26),(10,31,33,44),(11,41,34,28),(12,25,35,46),(13,43,36,30),(14,27,37,48),(15,45,38,32),(16,29,39,42)], [(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)], [(1,25,5,29),(2,32,6,28),(3,31,7,27),(4,30,8,26),(9,51,13,55),(10,50,14,54),(11,49,15,53),(12,56,16,52),(17,37,21,33),(18,36,22,40),(19,35,23,39),(20,34,24,38),(41,64,45,60),(42,63,46,59),(43,62,47,58),(44,61,48,57)])
Matrix representation ►G ⊆ GL6(𝔽17)
| 4 | 0 | 0 | 0 | 0 | 0 |
| 13 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 13 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 13 | 0 | 0 |
| 0 | 0 | 8 | 13 | 0 | 0 |
| 0 | 0 | 0 | 0 | 13 | 4 |
| 0 | 0 | 0 | 0 | 9 | 4 |
| 15 | 0 | 0 | 0 | 0 | 0 |
| 5 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 0 | 0 | 0 |
| 0 | 0 | 16 | 9 | 0 | 0 |
| 0 | 0 | 0 | 0 | 15 | 0 |
| 0 | 0 | 0 | 0 | 13 | 2 |
| 9 | 1 | 0 | 0 | 0 | 0 |
| 3 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 0 | 4 | 15 |
| 0 | 0 | 8 | 0 | 0 | 0 |
| 0 | 0 | 16 | 9 | 0 | 0 |
G:=sub<GL(6,GF(17))| [4,13,0,0,0,0,0,13,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[4,13,0,0,0,0,0,13,0,0,0,0,0,0,4,8,0,0,0,0,13,13,0,0,0,0,0,0,13,9,0,0,0,0,4,4],[15,5,0,0,0,0,0,8,0,0,0,0,0,0,8,16,0,0,0,0,0,9,0,0,0,0,0,0,15,13,0,0,0,0,0,2],[9,3,0,0,0,0,1,8,0,0,0,0,0,0,0,0,8,16,0,0,0,0,0,9,0,0,2,4,0,0,0,0,0,15,0,0] >;
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 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | Q16 | 2+ (1+4) | 2- (1+4) | D8⋊C22 |
| kernel | C42.297D4 | C2×C4⋊C8 | C4⋊2Q16 | C8.18D4 | C4.Q16 | C23.37C23 | C42 | C22×C4 | C2×C4 | C4 | C4 | C2 |
| # reps | 1 | 1 | 4 | 4 | 4 | 2 | 2 | 2 | 8 | 1 | 1 | 2 |
In GAP, Magma, Sage, TeX
C_4^2._{297}D_4 % in TeX
G:=Group("C4^2.297D4"); // GroupNames label
G:=SmallGroup(128,1981);
// by ID
G=gap.SmallGroup(128,1981);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,758,219,352,675,80,4037,1027,124]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^4=d^2=a^2,a*b=b*a,a*c=c*a,d*a*d^-1=a*b^2,c*b*c^-1=a^2*b^-1,d*b*d^-1=b^-1,d*c*d^-1=a^2*c^3>;
// generators/relations