direct product, p-group, metabelian, nilpotent (class 3), monomial, rational
Aliases: C22×C8.C22, C4.6C25, C8.1C24, Q16⋊3C23, D4.3C24, Q8.3C24, SD16⋊2C23, C24.186D4, M4(2)⋊5C23, (C2×Q8)⋊21C23, (Q8×C23)⋊14C2, C2.41(D4×C23), C4.32(C22×D4), (C2×C8).297C23, (C2×C4).612C24, (C2×Q16)⋊58C22, (C22×Q16)⋊22C2, (C22×SD16)⋊9C2, C4○D4.33C23, C23.711(C2×D4), (C22×C4).537D4, (C2×SD16)⋊60C22, (C2×D4).491C23, (C22×M4(2))⋊7C2, (C22×Q8)⋊69C22, C22.53(C22×D4), (C2×M4(2))⋊57C22, (C23×C4).623C22, (C22×C8).297C22, (C22×C4).1223C23, (C22×D4).604C22, (C2×C4).668(C2×D4), (C22×C4○D4).29C2, (C2×C4○D4).336C22, SmallGroup(128,2311)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 1068 in 732 conjugacy classes, 436 normal (14 characteristic)
C1, C2, C2 [×6], C2 [×8], C4, C4 [×7], C4 [×12], C22 [×11], C22 [×28], C8 [×8], C2×C4 [×28], C2×C4 [×50], D4 [×4], D4 [×22], Q8 [×12], Q8 [×34], C23, C23 [×6], C23 [×14], C2×C8 [×12], M4(2) [×16], SD16 [×32], Q16 [×32], C22×C4 [×2], C22×C4 [×12], C22×C4 [×27], C2×D4 [×6], C2×D4 [×15], C2×Q8 [×34], C2×Q8 [×45], C4○D4 [×16], C4○D4 [×24], C24, C24, C22×C8 [×2], C2×M4(2) [×12], C2×SD16 [×24], C2×Q16 [×24], C8.C22 [×64], C23×C4, C23×C4 [×2], C22×D4, C22×D4, C22×Q8, C22×Q8 [×14], C22×Q8 [×7], C2×C4○D4 [×12], C2×C4○D4 [×6], C22×M4(2), C22×SD16 [×2], C22×Q16 [×2], C2×C8.C22 [×24], Q8×C23, C22×C4○D4, C22×C8.C22
Quotients:
C1, C2 [×31], C22 [×155], D4 [×8], C23 [×155], C2×D4 [×28], C24 [×31], C8.C22 [×4], C22×D4 [×14], C25, C2×C8.C22 [×6], D4×C23, C22×C8.C22
Generators and relations
G = < a,b,c,d,e | a2=b2=c8=d2=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd=c3, ece=c5, ede=c4d >
►On 64 points
(1 39)(2 40)(3 33)(4 34)(5 35)(6 36)(7 37)(8 38)(9 55)(10 56)(11 49)(12 50)(13 51)(14 52)(15 53)(16 54)(17 64)(18 57)(19 58)(20 59)(21 60)(22 61)(23 62)(24 63)(25 45)(26 46)(27 47)(28 48)(29 41)(30 42)(31 43)(32 44)
(1 11)(2 12)(3 13)(4 14)(5 15)(6 16)(7 9)(8 10)(17 42)(18 43)(19 44)(20 45)(21 46)(22 47)(23 48)(24 41)(25 59)(26 60)(27 61)(28 62)(29 63)(30 64)(31 57)(32 58)(33 51)(34 52)(35 53)(36 54)(37 55)(38 56)(39 49)(40 50)
(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 11)(2 14)(3 9)(4 12)(5 15)(6 10)(7 13)(8 16)(17 44)(18 47)(19 42)(20 45)(21 48)(22 43)(23 46)(24 41)(25 59)(26 62)(27 57)(28 60)(29 63)(30 58)(31 61)(32 64)(33 55)(34 50)(35 53)(36 56)(37 51)(38 54)(39 49)(40 52)
(1 57)(2 62)(3 59)(4 64)(5 61)(6 58)(7 63)(8 60)(9 29)(10 26)(11 31)(12 28)(13 25)(14 30)(15 27)(16 32)(17 34)(18 39)(19 36)(20 33)(21 38)(22 35)(23 40)(24 37)(41 55)(42 52)(43 49)(44 54)(45 51)(46 56)(47 53)(48 50)
G:=sub<Sym(64)| (1,39)(2,40)(3,33)(4,34)(5,35)(6,36)(7,37)(8,38)(9,55)(10,56)(11,49)(12,50)(13,51)(14,52)(15,53)(16,54)(17,64)(18,57)(19,58)(20,59)(21,60)(22,61)(23,62)(24,63)(25,45)(26,46)(27,47)(28,48)(29,41)(30,42)(31,43)(32,44), (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,9)(8,10)(17,42)(18,43)(19,44)(20,45)(21,46)(22,47)(23,48)(24,41)(25,59)(26,60)(27,61)(28,62)(29,63)(30,64)(31,57)(32,58)(33,51)(34,52)(35,53)(36,54)(37,55)(38,56)(39,49)(40,50), (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), (1,11)(2,14)(3,9)(4,12)(5,15)(6,10)(7,13)(8,16)(17,44)(18,47)(19,42)(20,45)(21,48)(22,43)(23,46)(24,41)(25,59)(26,62)(27,57)(28,60)(29,63)(30,58)(31,61)(32,64)(33,55)(34,50)(35,53)(36,56)(37,51)(38,54)(39,49)(40,52), (1,57)(2,62)(3,59)(4,64)(5,61)(6,58)(7,63)(8,60)(9,29)(10,26)(11,31)(12,28)(13,25)(14,30)(15,27)(16,32)(17,34)(18,39)(19,36)(20,33)(21,38)(22,35)(23,40)(24,37)(41,55)(42,52)(43,49)(44,54)(45,51)(46,56)(47,53)(48,50)>;
G:=Group( (1,39)(2,40)(3,33)(4,34)(5,35)(6,36)(7,37)(8,38)(9,55)(10,56)(11,49)(12,50)(13,51)(14,52)(15,53)(16,54)(17,64)(18,57)(19,58)(20,59)(21,60)(22,61)(23,62)(24,63)(25,45)(26,46)(27,47)(28,48)(29,41)(30,42)(31,43)(32,44), (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,9)(8,10)(17,42)(18,43)(19,44)(20,45)(21,46)(22,47)(23,48)(24,41)(25,59)(26,60)(27,61)(28,62)(29,63)(30,64)(31,57)(32,58)(33,51)(34,52)(35,53)(36,54)(37,55)(38,56)(39,49)(40,50), (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), (1,11)(2,14)(3,9)(4,12)(5,15)(6,10)(7,13)(8,16)(17,44)(18,47)(19,42)(20,45)(21,48)(22,43)(23,46)(24,41)(25,59)(26,62)(27,57)(28,60)(29,63)(30,58)(31,61)(32,64)(33,55)(34,50)(35,53)(36,56)(37,51)(38,54)(39,49)(40,52), (1,57)(2,62)(3,59)(4,64)(5,61)(6,58)(7,63)(8,60)(9,29)(10,26)(11,31)(12,28)(13,25)(14,30)(15,27)(16,32)(17,34)(18,39)(19,36)(20,33)(21,38)(22,35)(23,40)(24,37)(41,55)(42,52)(43,49)(44,54)(45,51)(46,56)(47,53)(48,50) );
G=PermutationGroup([(1,39),(2,40),(3,33),(4,34),(5,35),(6,36),(7,37),(8,38),(9,55),(10,56),(11,49),(12,50),(13,51),(14,52),(15,53),(16,54),(17,64),(18,57),(19,58),(20,59),(21,60),(22,61),(23,62),(24,63),(25,45),(26,46),(27,47),(28,48),(29,41),(30,42),(31,43),(32,44)], [(1,11),(2,12),(3,13),(4,14),(5,15),(6,16),(7,9),(8,10),(17,42),(18,43),(19,44),(20,45),(21,46),(22,47),(23,48),(24,41),(25,59),(26,60),(27,61),(28,62),(29,63),(30,64),(31,57),(32,58),(33,51),(34,52),(35,53),(36,54),(37,55),(38,56),(39,49),(40,50)], [(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)], [(1,11),(2,14),(3,9),(4,12),(5,15),(6,10),(7,13),(8,16),(17,44),(18,47),(19,42),(20,45),(21,48),(22,43),(23,46),(24,41),(25,59),(26,62),(27,57),(28,60),(29,63),(30,58),(31,61),(32,64),(33,55),(34,50),(35,53),(36,56),(37,51),(38,54),(39,49),(40,52)], [(1,57),(2,62),(3,59),(4,64),(5,61),(6,58),(7,63),(8,60),(9,29),(10,26),(11,31),(12,28),(13,25),(14,30),(15,27),(16,32),(17,34),(18,39),(19,36),(20,33),(21,38),(22,35),(23,40),(24,37),(41,55),(42,52),(43,49),(44,54),(45,51),(46,56),(47,53),(48,50)])
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 | 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 |
| 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 | 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 | 2 | 0 | 0 | 0 | 0 | 0 | 0 |
| 16 | 16 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 15 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 4 | 13 |
| 0 | 0 | 0 | 0 | 0 | 13 | 0 | 13 |
| 0 | 0 | 0 | 0 | 13 | 0 | 0 | 4 |
| 0 | 0 | 0 | 0 | 13 | 4 | 0 | 4 |
| 16 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 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 | 1 | 16 | 1 |
| 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 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 | 16 | 1 | 15 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(8,GF(17))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16],[16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,16,0,0,0,0,0,0,2,16,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,15,16,0,0,0,0,0,0,0,0,0,0,13,13,0,0,0,0,0,13,0,4,0,0,0,0,4,0,0,0,0,0,0,0,13,13,4,4],[16,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,16,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,1,0,0,0,0,0,0,16,16,0,0,0,0,0,0,0,1],[16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,15,0,1] >;
44 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 2L | 2M | 2N | 2O | 4A | ··· | 4H | 4I | ··· | 4T | 8A | ··· | 8H |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | - |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | C8.C22 |
| kernel | C22×C8.C22 | C22×M4(2) | C22×SD16 | C22×Q16 | C2×C8.C22 | Q8×C23 | C22×C4○D4 | C22×C4 | C24 | C22 |
| # reps | 1 | 1 | 2 | 2 | 24 | 1 | 1 | 7 | 1 | 4 |
In GAP, Magma, Sage, TeX
C_2^2\times C_8.C_2^2
% in TeX
G:=Group("C2^2xC8.C2^2"); // GroupNames label
G:=SmallGroup(128,2311);
// by ID
G=gap.SmallGroup(128,2311);
# by ID
G:=PCGroup([7,-2,2,2,2,2,-2,-2,477,456,1430,4037,2028,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^8=d^2=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,d*c*d=c^3,e*c*e=c^5,e*d*e=c^4*d>;
// generators/relations