direct product, p-group, metabelian, nilpotent (class 3), monomial
Aliases: C2×C22⋊Q16, C23⋊5Q16, C24.176D4, C4⋊C4.2C23, Q8.37(C2×D4), C22⋊3(C2×Q16), C4.36C22≀C2, (C2×Q8).227D4, (C22×Q16)⋊5C2, C4.36(C22×D4), C2.4(C22×Q16), (C2×C4).218C24, (C2×C8).126C23, (C2×Q16)⋊36C22, C23.849(C2×D4), (C22×C4).418D4, (Q8×C23).12C2, (C2×Q8).16C23, Q8⋊C4⋊63C22, C22.115C22≀C2, C22⋊C8.170C22, (C23×C4).538C22, (C22×C8).133C22, (C22×C4).956C23, C22.478(C22×D4), C22⋊Q8.149C22, C22.99(C8.C22), (C22×Q8).464C22, C2.8(C2×C8.C22), (C2×Q8⋊C4)⋊21C2, C2.36(C2×C22≀C2), (C2×C4).1092(C2×D4), (C2×C22⋊C8).30C2, (C2×C22⋊Q8).51C2, (C2×C4⋊C4).579C22, SmallGroup(128,1731)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 636 in 370 conjugacy classes, 124 normal (18 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C4 [×4], C4 [×14], C22, C22 [×10], C22 [×12], C8 [×4], C2×C4 [×2], C2×C4 [×6], C2×C4 [×46], Q8 [×8], Q8 [×34], C23, C23 [×6], C23 [×4], C22⋊C4 [×4], C4⋊C4 [×2], C4⋊C4 [×5], C2×C8 [×4], C2×C8 [×4], Q16 [×16], C22×C4 [×2], C22×C4 [×4], C22×C4 [×20], C2×Q8 [×14], C2×Q8 [×55], C24, C22⋊C8 [×4], Q8⋊C4 [×8], C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C22⋊Q8 [×4], C22⋊Q8 [×2], C22×C8 [×2], C2×Q16 [×8], C2×Q16 [×8], C23×C4, C23×C4, C22×Q8, C22×Q8 [×6], C22×Q8 [×11], C2×C22⋊C8, C2×Q8⋊C4 [×2], C22⋊Q16 [×8], C2×C22⋊Q8, C22×Q16 [×2], Q8×C23, C2×C22⋊Q16
Quotients:
C1, C2 [×15], C22 [×35], D4 [×12], C23 [×15], Q16 [×4], C2×D4 [×18], C24, C22≀C2 [×4], C2×Q16 [×6], C8.C22 [×2], C22×D4 [×3], C22⋊Q16 [×4], C2×C22≀C2, C22×Q16, C2×C8.C22, C2×C22⋊Q16
Generators and relations
G = < a,b,c,d,e | a2=b2=c2=d8=1, e2=d4, ab=ba, ac=ca, ad=da, ae=ea, dbd-1=bc=cb, be=eb, cd=dc, ce=ec, ede-1=d-1 >
►On 64 points
(1 52)(2 53)(3 54)(4 55)(5 56)(6 49)(7 50)(8 51)(9 28)(10 29)(11 30)(12 31)(13 32)(14 25)(15 26)(16 27)(17 62)(18 63)(19 64)(20 57)(21 58)(22 59)(23 60)(24 61)(33 42)(34 43)(35 44)(36 45)(37 46)(38 47)(39 48)(40 41)
(1 5)(2 64)(3 7)(4 58)(6 60)(8 62)(9 13)(10 33)(11 15)(12 35)(14 37)(16 39)(17 51)(18 22)(19 53)(20 24)(21 55)(23 49)(25 46)(26 30)(27 48)(28 32)(29 42)(31 44)(34 38)(36 40)(41 45)(43 47)(50 54)(52 56)(57 61)(59 63)
(1 59)(2 60)(3 61)(4 62)(5 63)(6 64)(7 57)(8 58)(9 36)(10 37)(11 38)(12 39)(13 40)(14 33)(15 34)(16 35)(17 55)(18 56)(19 49)(20 50)(21 51)(22 52)(23 53)(24 54)(25 42)(26 43)(27 44)(28 45)(29 46)(30 47)(31 48)(32 41)
(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 34 5 38)(2 33 6 37)(3 40 7 36)(4 39 8 35)(9 61 13 57)(10 60 14 64)(11 59 15 63)(12 58 16 62)(17 31 21 27)(18 30 22 26)(19 29 23 25)(20 28 24 32)(41 50 45 54)(42 49 46 53)(43 56 47 52)(44 55 48 51)
G:=sub<Sym(64)| (1,52)(2,53)(3,54)(4,55)(5,56)(6,49)(7,50)(8,51)(9,28)(10,29)(11,30)(12,31)(13,32)(14,25)(15,26)(16,27)(17,62)(18,63)(19,64)(20,57)(21,58)(22,59)(23,60)(24,61)(33,42)(34,43)(35,44)(36,45)(37,46)(38,47)(39,48)(40,41), (1,5)(2,64)(3,7)(4,58)(6,60)(8,62)(9,13)(10,33)(11,15)(12,35)(14,37)(16,39)(17,51)(18,22)(19,53)(20,24)(21,55)(23,49)(25,46)(26,30)(27,48)(28,32)(29,42)(31,44)(34,38)(36,40)(41,45)(43,47)(50,54)(52,56)(57,61)(59,63), (1,59)(2,60)(3,61)(4,62)(5,63)(6,64)(7,57)(8,58)(9,36)(10,37)(11,38)(12,39)(13,40)(14,33)(15,34)(16,35)(17,55)(18,56)(19,49)(20,50)(21,51)(22,52)(23,53)(24,54)(25,42)(26,43)(27,44)(28,45)(29,46)(30,47)(31,48)(32,41), (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), (1,34,5,38)(2,33,6,37)(3,40,7,36)(4,39,8,35)(9,61,13,57)(10,60,14,64)(11,59,15,63)(12,58,16,62)(17,31,21,27)(18,30,22,26)(19,29,23,25)(20,28,24,32)(41,50,45,54)(42,49,46,53)(43,56,47,52)(44,55,48,51)>;
G:=Group( (1,52)(2,53)(3,54)(4,55)(5,56)(6,49)(7,50)(8,51)(9,28)(10,29)(11,30)(12,31)(13,32)(14,25)(15,26)(16,27)(17,62)(18,63)(19,64)(20,57)(21,58)(22,59)(23,60)(24,61)(33,42)(34,43)(35,44)(36,45)(37,46)(38,47)(39,48)(40,41), (1,5)(2,64)(3,7)(4,58)(6,60)(8,62)(9,13)(10,33)(11,15)(12,35)(14,37)(16,39)(17,51)(18,22)(19,53)(20,24)(21,55)(23,49)(25,46)(26,30)(27,48)(28,32)(29,42)(31,44)(34,38)(36,40)(41,45)(43,47)(50,54)(52,56)(57,61)(59,63), (1,59)(2,60)(3,61)(4,62)(5,63)(6,64)(7,57)(8,58)(9,36)(10,37)(11,38)(12,39)(13,40)(14,33)(15,34)(16,35)(17,55)(18,56)(19,49)(20,50)(21,51)(22,52)(23,53)(24,54)(25,42)(26,43)(27,44)(28,45)(29,46)(30,47)(31,48)(32,41), (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), (1,34,5,38)(2,33,6,37)(3,40,7,36)(4,39,8,35)(9,61,13,57)(10,60,14,64)(11,59,15,63)(12,58,16,62)(17,31,21,27)(18,30,22,26)(19,29,23,25)(20,28,24,32)(41,50,45,54)(42,49,46,53)(43,56,47,52)(44,55,48,51) );
G=PermutationGroup([(1,52),(2,53),(3,54),(4,55),(5,56),(6,49),(7,50),(8,51),(9,28),(10,29),(11,30),(12,31),(13,32),(14,25),(15,26),(16,27),(17,62),(18,63),(19,64),(20,57),(21,58),(22,59),(23,60),(24,61),(33,42),(34,43),(35,44),(36,45),(37,46),(38,47),(39,48),(40,41)], [(1,5),(2,64),(3,7),(4,58),(6,60),(8,62),(9,13),(10,33),(11,15),(12,35),(14,37),(16,39),(17,51),(18,22),(19,53),(20,24),(21,55),(23,49),(25,46),(26,30),(27,48),(28,32),(29,42),(31,44),(34,38),(36,40),(41,45),(43,47),(50,54),(52,56),(57,61),(59,63)], [(1,59),(2,60),(3,61),(4,62),(5,63),(6,64),(7,57),(8,58),(9,36),(10,37),(11,38),(12,39),(13,40),(14,33),(15,34),(16,35),(17,55),(18,56),(19,49),(20,50),(21,51),(22,52),(23,53),(24,54),(25,42),(26,43),(27,44),(28,45),(29,46),(30,47),(31,48),(32,41)], [(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)], [(1,34,5,38),(2,33,6,37),(3,40,7,36),(4,39,8,35),(9,61,13,57),(10,60,14,64),(11,59,15,63),(12,58,16,62),(17,31,21,27),(18,30,22,26),(19,29,23,25),(20,28,24,32),(41,50,45,54),(42,49,46,53),(43,56,47,52),(44,55,48,51)])
Matrix representation ►G ⊆ GL5(𝔽17)
| 16 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 16 |
| 16 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 16 | 16 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 16 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 2 | 0 | 0 | 0 |
| 0 | 0 | 9 | 0 | 0 |
| 0 | 0 | 0 | 16 | 15 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(5,GF(17))| [16,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,16],[16,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,1,16,0,0,0,0,16],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,16,0,0,0,0,0,16],[1,0,0,0,0,0,2,0,0,0,0,0,9,0,0,0,0,0,16,0,0,0,0,15,1],[1,0,0,0,0,0,0,16,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1] >;
38 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 4A | 4B | 4C | 4D | 4E | ··· | 4N | 4O | 4P | 4Q | 4R | 8A | ··· | 8H |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 8 | ··· | 8 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 8 | 4 | ··· | 4 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | - | - |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | Q16 | C8.C22 |
| kernel | C2×C22⋊Q16 | C2×C22⋊C8 | C2×Q8⋊C4 | C22⋊Q16 | C2×C22⋊Q8 | C22×Q16 | Q8×C23 | C22×C4 | C2×Q8 | C24 | C23 | C22 |
| # reps | 1 | 1 | 2 | 8 | 1 | 2 | 1 | 3 | 8 | 1 | 8 | 2 |
In GAP, Magma, Sage, TeX
C_2\times C_2^2\rtimes Q_{16} % in TeX
G:=Group("C2xC2^2:Q16"); // GroupNames label
G:=SmallGroup(128,1731);
// by ID
G=gap.SmallGroup(128,1731);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,456,758,2804,1411,172]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^8=1,e^2=d^4,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,d*b*d^-1=b*c=c*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^-1>;
// generators/relations