direct product, p-group, metabelian, nilpotent (class 3), monomial
Aliases: C2×C42.28C22, C42.236D4, C42.362C23, C4⋊Q8⋊62C22, C4⋊C4.85C23, C8⋊C4⋊62C22, (C2×C8).451C23, (C2×C4).330C24, (C2×D4).99C23, (C22×C4).456D4, C23.873(C2×D4), (C2×Q8).87C23, Q8⋊C4⋊97C22, C4.21(C4.4D4), (C22×C8).457C22, (C2×C42).843C22, C22.590(C22×D4), D4⋊C4.200C22, C22.123(C8⋊C22), (C22×C4).1552C23, C4.4D4.134C22, C22.83(C4.4D4), (C22×D4).366C22, (C22×Q8).299C22, C22.112(C8.C22), (C2×C4⋊Q8)⋊35C2, (C2×C8⋊C4)⋊37C2, C4.39(C2×C4○D4), (C2×C4).510(C2×D4), C2.37(C2×C8⋊C22), (C2×Q8⋊C4)⋊57C2, C2.41(C2×C4.4D4), C2.37(C2×C8.C22), (C2×D4⋊C4).38C2, (C2×C4).709(C4○D4), (C2×C4⋊C4).621C22, (C2×C4.4D4).39C2, SmallGroup(128,1864)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 452 in 218 conjugacy classes, 100 normal (18 characteristic)
C1, C2 [×3], C2 [×4], C2 [×2], C4 [×4], C4 [×10], C22, C22 [×6], C22 [×10], C8 [×4], C2×C4 [×2], C2×C4 [×8], C2×C4 [×16], D4 [×6], Q8 [×14], C23, C23 [×8], C42 [×4], C22⋊C4 [×8], C4⋊C4 [×4], C4⋊C4 [×6], C2×C8 [×4], C2×C8 [×4], C22×C4 [×3], C22×C4 [×3], C2×D4 [×2], C2×D4 [×5], C2×Q8 [×2], C2×Q8 [×13], C24, C8⋊C4 [×4], D4⋊C4 [×8], Q8⋊C4 [×8], C2×C42, C2×C22⋊C4 [×2], C2×C4⋊C4 [×2], C2×C4⋊C4, C4.4D4 [×4], C4.4D4 [×2], C4⋊Q8 [×4], C4⋊Q8 [×2], C22×C8 [×2], C22×D4, C22×Q8, C22×Q8, C2×C8⋊C4, C2×D4⋊C4 [×2], C2×Q8⋊C4 [×2], C42.28C22 [×8], C2×C4.4D4, C2×C4⋊Q8, C2×C42.28C22
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×4], C24, C4.4D4 [×4], C8⋊C22 [×2], C8.C22 [×2], C22×D4, C2×C4○D4 [×2], C42.28C22 [×4], C2×C4.4D4, C2×C8⋊C22, C2×C8.C22, C2×C42.28C22
Generators and relations
G = < a,b,c,d,e | a2=b4=c4=d2=1, e2=c, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd=b-1c2, ebe-1=bc2, dcd=c-1, ce=ec, ede-1=b2c-1d >
►On 64 points
(1 49)(2 50)(3 51)(4 52)(5 53)(6 54)(7 55)(8 56)(9 61)(10 62)(11 63)(12 64)(13 57)(14 58)(15 59)(16 60)(17 31)(18 32)(19 25)(20 26)(21 27)(22 28)(23 29)(24 30)(33 47)(34 48)(35 41)(36 42)(37 43)(38 44)(39 45)(40 46)
(1 64 17 38)(2 61 18 35)(3 58 19 40)(4 63 20 37)(5 60 21 34)(6 57 22 39)(7 62 23 36)(8 59 24 33)(9 32 41 50)(10 29 42 55)(11 26 43 52)(12 31 44 49)(13 28 45 54)(14 25 46 51)(15 30 47 56)(16 27 48 53)
(1 3 5 7)(2 4 6 8)(9 11 13 15)(10 12 14 16)(17 19 21 23)(18 20 22 24)(25 27 29 31)(26 28 30 32)(33 35 37 39)(34 36 38 40)(41 43 45 47)(42 44 46 48)(49 51 53 55)(50 52 54 56)(57 59 61 63)(58 60 62 64)
(1 31)(2 52)(3 29)(4 50)(5 27)(6 56)(7 25)(8 54)(9 33)(10 62)(11 39)(12 60)(13 37)(14 58)(15 35)(16 64)(17 49)(18 26)(19 55)(20 32)(21 53)(22 30)(23 51)(24 28)(34 44)(36 42)(38 48)(40 46)(41 59)(43 57)(45 63)(47 61)
(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)
G:=sub<Sym(64)| (1,49)(2,50)(3,51)(4,52)(5,53)(6,54)(7,55)(8,56)(9,61)(10,62)(11,63)(12,64)(13,57)(14,58)(15,59)(16,60)(17,31)(18,32)(19,25)(20,26)(21,27)(22,28)(23,29)(24,30)(33,47)(34,48)(35,41)(36,42)(37,43)(38,44)(39,45)(40,46), (1,64,17,38)(2,61,18,35)(3,58,19,40)(4,63,20,37)(5,60,21,34)(6,57,22,39)(7,62,23,36)(8,59,24,33)(9,32,41,50)(10,29,42,55)(11,26,43,52)(12,31,44,49)(13,28,45,54)(14,25,46,51)(15,30,47,56)(16,27,48,53), (1,3,5,7)(2,4,6,8)(9,11,13,15)(10,12,14,16)(17,19,21,23)(18,20,22,24)(25,27,29,31)(26,28,30,32)(33,35,37,39)(34,36,38,40)(41,43,45,47)(42,44,46,48)(49,51,53,55)(50,52,54,56)(57,59,61,63)(58,60,62,64), (1,31)(2,52)(3,29)(4,50)(5,27)(6,56)(7,25)(8,54)(9,33)(10,62)(11,39)(12,60)(13,37)(14,58)(15,35)(16,64)(17,49)(18,26)(19,55)(20,32)(21,53)(22,30)(23,51)(24,28)(34,44)(36,42)(38,48)(40,46)(41,59)(43,57)(45,63)(47,61), (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)>;
G:=Group( (1,49)(2,50)(3,51)(4,52)(5,53)(6,54)(7,55)(8,56)(9,61)(10,62)(11,63)(12,64)(13,57)(14,58)(15,59)(16,60)(17,31)(18,32)(19,25)(20,26)(21,27)(22,28)(23,29)(24,30)(33,47)(34,48)(35,41)(36,42)(37,43)(38,44)(39,45)(40,46), (1,64,17,38)(2,61,18,35)(3,58,19,40)(4,63,20,37)(5,60,21,34)(6,57,22,39)(7,62,23,36)(8,59,24,33)(9,32,41,50)(10,29,42,55)(11,26,43,52)(12,31,44,49)(13,28,45,54)(14,25,46,51)(15,30,47,56)(16,27,48,53), (1,3,5,7)(2,4,6,8)(9,11,13,15)(10,12,14,16)(17,19,21,23)(18,20,22,24)(25,27,29,31)(26,28,30,32)(33,35,37,39)(34,36,38,40)(41,43,45,47)(42,44,46,48)(49,51,53,55)(50,52,54,56)(57,59,61,63)(58,60,62,64), (1,31)(2,52)(3,29)(4,50)(5,27)(6,56)(7,25)(8,54)(9,33)(10,62)(11,39)(12,60)(13,37)(14,58)(15,35)(16,64)(17,49)(18,26)(19,55)(20,32)(21,53)(22,30)(23,51)(24,28)(34,44)(36,42)(38,48)(40,46)(41,59)(43,57)(45,63)(47,61), (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) );
G=PermutationGroup([(1,49),(2,50),(3,51),(4,52),(5,53),(6,54),(7,55),(8,56),(9,61),(10,62),(11,63),(12,64),(13,57),(14,58),(15,59),(16,60),(17,31),(18,32),(19,25),(20,26),(21,27),(22,28),(23,29),(24,30),(33,47),(34,48),(35,41),(36,42),(37,43),(38,44),(39,45),(40,46)], [(1,64,17,38),(2,61,18,35),(3,58,19,40),(4,63,20,37),(5,60,21,34),(6,57,22,39),(7,62,23,36),(8,59,24,33),(9,32,41,50),(10,29,42,55),(11,26,43,52),(12,31,44,49),(13,28,45,54),(14,25,46,51),(15,30,47,56),(16,27,48,53)], [(1,3,5,7),(2,4,6,8),(9,11,13,15),(10,12,14,16),(17,19,21,23),(18,20,22,24),(25,27,29,31),(26,28,30,32),(33,35,37,39),(34,36,38,40),(41,43,45,47),(42,44,46,48),(49,51,53,55),(50,52,54,56),(57,59,61,63),(58,60,62,64)], [(1,31),(2,52),(3,29),(4,50),(5,27),(6,56),(7,25),(8,54),(9,33),(10,62),(11,39),(12,60),(13,37),(14,58),(15,35),(16,64),(17,49),(18,26),(19,55),(20,32),(21,53),(22,30),(23,51),(24,28),(34,44),(36,42),(38,48),(40,46),(41,59),(43,57),(45,63),(47,61)], [(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)])
Matrix representation ►G ⊆ GL8(𝔽17)
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 8 | 0 | 0 | 0 | 0 | 0 | 0 |
| 4 | 16 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 12 | 0 | 10 |
| 0 | 0 | 0 | 0 | 12 | 2 | 10 | 10 |
| 0 | 0 | 0 | 0 | 5 | 12 | 10 | 0 |
| 0 | 0 | 0 | 0 | 14 | 2 | 5 | 3 |
| 16 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 1 | 16 | 15 |
| 0 | 0 | 0 | 0 | 16 | 0 | 1 | 1 |
| 16 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 13 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 16 | 1 | 2 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 16 |
| 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 2 | 10 | 3 |
| 0 | 0 | 0 | 0 | 5 | 12 | 10 | 0 |
| 0 | 0 | 0 | 0 | 15 | 5 | 0 | 7 |
| 0 | 0 | 0 | 0 | 2 | 14 | 15 | 3 |
G:=sub<GL(8,GF(17))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,4,0,0,0,0,0,0,8,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,2,12,5,14,0,0,0,0,12,2,12,2,0,0,0,0,0,10,10,5,0,0,0,0,10,10,0,3],[16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,16,1,16,0,0,0,0,1,0,1,0,0,0,0,0,0,0,16,1,0,0,0,0,0,0,15,1],[16,13,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,16,0,0,0,0,0,0,1,16,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,2,16],[4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,2,5,15,2,0,0,0,0,2,12,5,14,0,0,0,0,10,10,0,15,0,0,0,0,3,0,7,3] >;
32 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | ··· | 4N | 8A | ··· | 8H |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
| size | 1 | 1 | ··· | 1 | 8 | 8 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | C4○D4 | C8⋊C22 | C8.C22 |
| kernel | C2×C42.28C22 | C2×C8⋊C4 | C2×D4⋊C4 | C2×Q8⋊C4 | C42.28C22 | C2×C4.4D4 | C2×C4⋊Q8 | C42 | C22×C4 | C2×C4 | C22 | C22 |
| # reps | 1 | 1 | 2 | 2 | 8 | 1 | 1 | 2 | 2 | 8 | 2 | 2 |
In GAP, Magma, Sage, TeX
C_2\times C_4^2._{28}C_2^2 % in TeX
G:=Group("C2xC4^2.28C2^2"); // GroupNames label
G:=SmallGroup(128,1864);
// by ID
G=gap.SmallGroup(128,1864);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,232,758,723,100,2804,172,4037,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^4=c^4=d^2=1,e^2=c,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,d*b*d=b^-1*c^2,e*b*e^-1=b*c^2,d*c*d=c^-1,c*e=e*c,e*d*e^-1=b^2*c^-1*d>;
// generators/relations