direct product, p-group, metabelian, nilpotent (class 4), monomial
Aliases: C22×SD32, C16⋊3C23, C8.10C24, Q16⋊1C23, D8.1C23, C23.64D8, (C2×C4).95D8, C8.55(C2×D4), C4.22(C2×D8), (C2×C8).263D4, (C22×C16)⋊12C2, (C2×C16)⋊20C22, C22.76(C2×D8), C2.25(C22×D8), C4.16(C22×D4), (C2×C8).572C23, (C22×Q16)⋊14C2, (C2×Q16)⋊49C22, (C22×D8).10C2, (C22×C4).622D4, (C2×D8).150C22, (C22×C8).542C22, (C2×C4).873(C2×D4), SmallGroup(128,2141)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 500 in 200 conjugacy classes, 100 normal (11 characteristic)
C1, C2, C2 [×6], C2 [×4], C4, C4 [×3], C4 [×4], C22 [×7], C22 [×16], C8, C8 [×3], C2×C4 [×6], C2×C4 [×6], D4 [×10], Q8 [×10], C23, C23 [×10], C16 [×4], C2×C8 [×6], D8 [×4], D8 [×6], Q16 [×4], Q16 [×6], C22×C4, C22×C4, C2×D4 [×9], C2×Q8 [×9], C24, C2×C16 [×6], SD32 [×16], C22×C8, C2×D8 [×6], C2×D8 [×3], C2×Q16 [×6], C2×Q16 [×3], C22×D4, C22×Q8, C22×C16, C2×SD32 [×12], C22×D8, C22×Q16, C22×SD32
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D8 [×4], C2×D4 [×6], C24, SD32 [×4], C2×D8 [×6], C22×D4, C2×SD32 [×6], C22×D8, C22×SD32
Generators and relations
G = < a,b,c,d | a2=b2=c16=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c7 >
►On 64 points
(1 21)(2 22)(3 23)(4 24)(5 25)(6 26)(7 27)(8 28)(9 29)(10 30)(11 31)(12 32)(13 17)(14 18)(15 19)(16 20)(33 60)(34 61)(35 62)(36 63)(37 64)(38 49)(39 50)(40 51)(41 52)(42 53)(43 54)(44 55)(45 56)(46 57)(47 58)(48 59)
(1 34)(2 35)(3 36)(4 37)(5 38)(6 39)(7 40)(8 41)(9 42)(10 43)(11 44)(12 45)(13 46)(14 47)(15 48)(16 33)(17 57)(18 58)(19 59)(20 60)(21 61)(22 62)(23 63)(24 64)(25 49)(26 50)(27 51)(28 52)(29 53)(30 54)(31 55)(32 56)
(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)(2 41)(3 48)(4 39)(5 46)(6 37)(7 44)(8 35)(9 42)(10 33)(11 40)(12 47)(13 38)(14 45)(15 36)(16 43)(17 49)(18 56)(19 63)(20 54)(21 61)(22 52)(23 59)(24 50)(25 57)(26 64)(27 55)(28 62)(29 53)(30 60)(31 51)(32 58)
G:=sub<Sym(64)| (1,21)(2,22)(3,23)(4,24)(5,25)(6,26)(7,27)(8,28)(9,29)(10,30)(11,31)(12,32)(13,17)(14,18)(15,19)(16,20)(33,60)(34,61)(35,62)(36,63)(37,64)(38,49)(39,50)(40,51)(41,52)(42,53)(43,54)(44,55)(45,56)(46,57)(47,58)(48,59), (1,34)(2,35)(3,36)(4,37)(5,38)(6,39)(7,40)(8,41)(9,42)(10,43)(11,44)(12,45)(13,46)(14,47)(15,48)(16,33)(17,57)(18,58)(19,59)(20,60)(21,61)(22,62)(23,63)(24,64)(25,49)(26,50)(27,51)(28,52)(29,53)(30,54)(31,55)(32,56), (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)(2,41)(3,48)(4,39)(5,46)(6,37)(7,44)(8,35)(9,42)(10,33)(11,40)(12,47)(13,38)(14,45)(15,36)(16,43)(17,49)(18,56)(19,63)(20,54)(21,61)(22,52)(23,59)(24,50)(25,57)(26,64)(27,55)(28,62)(29,53)(30,60)(31,51)(32,58)>;
G:=Group( (1,21)(2,22)(3,23)(4,24)(5,25)(6,26)(7,27)(8,28)(9,29)(10,30)(11,31)(12,32)(13,17)(14,18)(15,19)(16,20)(33,60)(34,61)(35,62)(36,63)(37,64)(38,49)(39,50)(40,51)(41,52)(42,53)(43,54)(44,55)(45,56)(46,57)(47,58)(48,59), (1,34)(2,35)(3,36)(4,37)(5,38)(6,39)(7,40)(8,41)(9,42)(10,43)(11,44)(12,45)(13,46)(14,47)(15,48)(16,33)(17,57)(18,58)(19,59)(20,60)(21,61)(22,62)(23,63)(24,64)(25,49)(26,50)(27,51)(28,52)(29,53)(30,54)(31,55)(32,56), (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)(2,41)(3,48)(4,39)(5,46)(6,37)(7,44)(8,35)(9,42)(10,33)(11,40)(12,47)(13,38)(14,45)(15,36)(16,43)(17,49)(18,56)(19,63)(20,54)(21,61)(22,52)(23,59)(24,50)(25,57)(26,64)(27,55)(28,62)(29,53)(30,60)(31,51)(32,58) );
G=PermutationGroup([(1,21),(2,22),(3,23),(4,24),(5,25),(6,26),(7,27),(8,28),(9,29),(10,30),(11,31),(12,32),(13,17),(14,18),(15,19),(16,20),(33,60),(34,61),(35,62),(36,63),(37,64),(38,49),(39,50),(40,51),(41,52),(42,53),(43,54),(44,55),(45,56),(46,57),(47,58),(48,59)], [(1,34),(2,35),(3,36),(4,37),(5,38),(6,39),(7,40),(8,41),(9,42),(10,43),(11,44),(12,45),(13,46),(14,47),(15,48),(16,33),(17,57),(18,58),(19,59),(20,60),(21,61),(22,62),(23,63),(24,64),(25,49),(26,50),(27,51),(28,52),(29,53),(30,54),(31,55),(32,56)], [(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),(2,41),(3,48),(4,39),(5,46),(6,37),(7,44),(8,35),(9,42),(10,33),(11,40),(12,47),(13,38),(14,45),(15,36),(16,43),(17,49),(18,56),(19,63),(20,54),(21,61),(22,52),(23,59),(24,50),(25,57),(26,64),(27,55),(28,62),(29,53),(30,60),(31,51),(32,58)])
Matrix representation ►G ⊆ GL4(𝔽17) generated by
| 16 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 |
| 16 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 7 | 1 |
| 0 | 0 | 16 | 7 |
| 1 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(17))| [16,0,0,0,0,16,0,0,0,0,16,0,0,0,0,16],[16,0,0,0,0,1,0,0,0,0,16,0,0,0,0,16],[1,0,0,0,0,1,0,0,0,0,7,16,0,0,1,7],[1,0,0,0,0,16,0,0,0,0,16,0,0,0,0,1] >;
44 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 8A | ··· | 8H | 16A | ··· | 16P |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 16 | ··· | 16 |
| size | 1 | 1 | ··· | 1 | 8 | 8 | 8 | 8 | 2 | 2 | 2 | 2 | 8 | 8 | 8 | 8 | 2 | ··· | 2 | 2 | ··· | 2 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | D4 | D4 | D8 | D8 | SD32 |
| kernel | C22×SD32 | C22×C16 | C2×SD32 | C22×D8 | C22×Q16 | C2×C8 | C22×C4 | C2×C4 | C23 | C22 |
| # reps | 1 | 1 | 12 | 1 | 1 | 3 | 1 | 6 | 2 | 16 |
In GAP, Magma, Sage, TeX
C_2^2\times SD_{32} % in TeX
G:=Group("C2^2xSD32"); // GroupNames label
G:=SmallGroup(128,2141);
// by ID
G=gap.SmallGroup(128,2141);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,-2,-2,448,253,1684,851,242,4037,2028,124]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^2=c^16=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^7>;
// generators/relations