p-group, metabelian, nilpotent (class 2), monomial
Aliases: C24.298C23, C23.385C24, C22.1872+ (1+4), C22.1392- (1+4), C23.182(C2×D4), (C22×C4).382D4, C23.39(C4○D4), (C2×C42).41C22, C23.7Q8⋊53C2, C23.11D4⋊26C2, (C22×C4).521C23, (C23×C4).371C22, C22.265(C22×D4), C24.C22⋊62C2, C23.10D4.12C2, (C22×D4).523C22, C23.63C23⋊60C2, C23.65C23⋊67C2, C2.56(C22.19C24), C4.124(C22.D4), C2.C42.138C22, C2.33(C22.46C24), C2.10(C22.31C24), C2.20(C22.50C24), C2.30(C22.47C24), C2.12(C22.49C24), (C2×C4×D4).55C2, (C2×C4).1191(C2×D4), (C2×C42⋊C2)⋊29C2, (C2×C4).121(C4○D4), (C2×C4⋊C4).256C22, C22.262(C2×C4○D4), C2.30(C2×C22.D4), (C2×C22⋊C4).151C22, SmallGroup(128,1217)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 500 in 270 conjugacy classes, 104 normal (42 characteristic)
C1, C2 [×7], C2 [×4], C4 [×4], C4 [×14], C22 [×7], C22 [×20], C2×C4 [×10], C2×C4 [×46], D4 [×8], C23, C23 [×4], C23 [×12], C42 [×6], C22⋊C4 [×16], C4⋊C4 [×16], C22×C4 [×5], C22×C4 [×12], C22×C4 [×10], C2×D4 [×6], C24 [×2], C2.C42 [×10], C2×C42, C2×C42 [×2], C2×C22⋊C4 [×2], C2×C22⋊C4 [×8], C2×C4⋊C4 [×5], C2×C4⋊C4 [×4], C42⋊C2 [×4], C4×D4 [×4], C23×C4 [×2], C22×D4, C23.7Q8 [×3], C23.63C23 [×2], C24.C22 [×2], C23.65C23 [×2], C23.10D4 [×2], C23.11D4 [×2], C2×C42⋊C2, C2×C4×D4, C24.298C23
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×8], C24, C22.D4 [×4], C22×D4, C2×C4○D4 [×4], 2+ (1+4), 2- (1+4), C2×C22.D4, C22.19C24, C22.31C24, C22.46C24, C22.47C24, C22.49C24, C22.50C24, C24.298C23
Generators and relations
G = < a,b,c,d,e,f,g | a2=b2=c2=d2=f2=1, e2=d, g2=b, ab=ba, eae-1=ac=ca, faf=ad=da, ag=ga, bc=cb, bd=db, fef=geg-1=be=eb, bf=fb, bg=gb, cd=dc, ce=ec, cf=fc, cg=gc, de=ed, df=fd, dg=gd, fg=gf >
(2 52)(4 50)(5 34)(6 8)(7 36)(10 22)(12 24)(14 26)(16 28)(17 19)(18 32)(20 30)(29 31)(33 35)(37 39)(38 64)(40 62)(42 54)(44 56)(45 47)(46 60)(48 58)(57 59)(61 63)
(1 11)(2 12)(3 9)(4 10)(5 38)(6 39)(7 40)(8 37)(13 41)(14 42)(15 43)(16 44)(17 45)(18 46)(19 47)(20 48)(21 49)(22 50)(23 51)(24 52)(25 53)(26 54)(27 55)(28 56)(29 57)(30 58)(31 59)(32 60)(33 63)(34 64)(35 61)(36 62)
(1 51)(2 52)(3 49)(4 50)(5 36)(6 33)(7 34)(8 35)(9 21)(10 22)(11 23)(12 24)(13 25)(14 26)(15 27)(16 28)(17 29)(18 30)(19 31)(20 32)(37 61)(38 62)(39 63)(40 64)(41 53)(42 54)(43 55)(44 56)(45 57)(46 58)(47 59)(48 60)
(1 3)(2 4)(5 7)(6 8)(9 11)(10 12)(13 15)(14 16)(17 19)(18 20)(21 23)(22 24)(25 27)(26 28)(29 31)(30 32)(33 35)(34 36)(37 39)(38 40)(41 43)(42 44)(45 47)(46 48)(49 51)(50 52)(53 55)(54 56)(57 59)(58 60)(61 63)(62 64)
(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 47)(2 20)(3 45)(4 18)(5 42)(6 15)(7 44)(8 13)(9 17)(10 46)(11 19)(12 48)(14 38)(16 40)(21 29)(22 58)(23 31)(24 60)(25 35)(26 62)(27 33)(28 64)(30 50)(32 52)(34 56)(36 54)(37 41)(39 43)(49 57)(51 59)(53 61)(55 63)
(1 15 11 43)(2 44 12 16)(3 13 9 41)(4 42 10 14)(5 46 38 18)(6 19 39 47)(7 48 40 20)(8 17 37 45)(21 53 49 25)(22 26 50 54)(23 55 51 27)(24 28 52 56)(29 61 57 35)(30 36 58 62)(31 63 59 33)(32 34 60 64)
G:=sub<Sym(64)| (2,52)(4,50)(5,34)(6,8)(7,36)(10,22)(12,24)(14,26)(16,28)(17,19)(18,32)(20,30)(29,31)(33,35)(37,39)(38,64)(40,62)(42,54)(44,56)(45,47)(46,60)(48,58)(57,59)(61,63), (1,11)(2,12)(3,9)(4,10)(5,38)(6,39)(7,40)(8,37)(13,41)(14,42)(15,43)(16,44)(17,45)(18,46)(19,47)(20,48)(21,49)(22,50)(23,51)(24,52)(25,53)(26,54)(27,55)(28,56)(29,57)(30,58)(31,59)(32,60)(33,63)(34,64)(35,61)(36,62), (1,51)(2,52)(3,49)(4,50)(5,36)(6,33)(7,34)(8,35)(9,21)(10,22)(11,23)(12,24)(13,25)(14,26)(15,27)(16,28)(17,29)(18,30)(19,31)(20,32)(37,61)(38,62)(39,63)(40,64)(41,53)(42,54)(43,55)(44,56)(45,57)(46,58)(47,59)(48,60), (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,15)(14,16)(17,19)(18,20)(21,23)(22,24)(25,27)(26,28)(29,31)(30,32)(33,35)(34,36)(37,39)(38,40)(41,43)(42,44)(45,47)(46,48)(49,51)(50,52)(53,55)(54,56)(57,59)(58,60)(61,63)(62,64), (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), (1,47)(2,20)(3,45)(4,18)(5,42)(6,15)(7,44)(8,13)(9,17)(10,46)(11,19)(12,48)(14,38)(16,40)(21,29)(22,58)(23,31)(24,60)(25,35)(26,62)(27,33)(28,64)(30,50)(32,52)(34,56)(36,54)(37,41)(39,43)(49,57)(51,59)(53,61)(55,63), (1,15,11,43)(2,44,12,16)(3,13,9,41)(4,42,10,14)(5,46,38,18)(6,19,39,47)(7,48,40,20)(8,17,37,45)(21,53,49,25)(22,26,50,54)(23,55,51,27)(24,28,52,56)(29,61,57,35)(30,36,58,62)(31,63,59,33)(32,34,60,64)>;
G:=Group( (2,52)(4,50)(5,34)(6,8)(7,36)(10,22)(12,24)(14,26)(16,28)(17,19)(18,32)(20,30)(29,31)(33,35)(37,39)(38,64)(40,62)(42,54)(44,56)(45,47)(46,60)(48,58)(57,59)(61,63), (1,11)(2,12)(3,9)(4,10)(5,38)(6,39)(7,40)(8,37)(13,41)(14,42)(15,43)(16,44)(17,45)(18,46)(19,47)(20,48)(21,49)(22,50)(23,51)(24,52)(25,53)(26,54)(27,55)(28,56)(29,57)(30,58)(31,59)(32,60)(33,63)(34,64)(35,61)(36,62), (1,51)(2,52)(3,49)(4,50)(5,36)(6,33)(7,34)(8,35)(9,21)(10,22)(11,23)(12,24)(13,25)(14,26)(15,27)(16,28)(17,29)(18,30)(19,31)(20,32)(37,61)(38,62)(39,63)(40,64)(41,53)(42,54)(43,55)(44,56)(45,57)(46,58)(47,59)(48,60), (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,15)(14,16)(17,19)(18,20)(21,23)(22,24)(25,27)(26,28)(29,31)(30,32)(33,35)(34,36)(37,39)(38,40)(41,43)(42,44)(45,47)(46,48)(49,51)(50,52)(53,55)(54,56)(57,59)(58,60)(61,63)(62,64), (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), (1,47)(2,20)(3,45)(4,18)(5,42)(6,15)(7,44)(8,13)(9,17)(10,46)(11,19)(12,48)(14,38)(16,40)(21,29)(22,58)(23,31)(24,60)(25,35)(26,62)(27,33)(28,64)(30,50)(32,52)(34,56)(36,54)(37,41)(39,43)(49,57)(51,59)(53,61)(55,63), (1,15,11,43)(2,44,12,16)(3,13,9,41)(4,42,10,14)(5,46,38,18)(6,19,39,47)(7,48,40,20)(8,17,37,45)(21,53,49,25)(22,26,50,54)(23,55,51,27)(24,28,52,56)(29,61,57,35)(30,36,58,62)(31,63,59,33)(32,34,60,64) );
G=PermutationGroup([(2,52),(4,50),(5,34),(6,8),(7,36),(10,22),(12,24),(14,26),(16,28),(17,19),(18,32),(20,30),(29,31),(33,35),(37,39),(38,64),(40,62),(42,54),(44,56),(45,47),(46,60),(48,58),(57,59),(61,63)], [(1,11),(2,12),(3,9),(4,10),(5,38),(6,39),(7,40),(8,37),(13,41),(14,42),(15,43),(16,44),(17,45),(18,46),(19,47),(20,48),(21,49),(22,50),(23,51),(24,52),(25,53),(26,54),(27,55),(28,56),(29,57),(30,58),(31,59),(32,60),(33,63),(34,64),(35,61),(36,62)], [(1,51),(2,52),(3,49),(4,50),(5,36),(6,33),(7,34),(8,35),(9,21),(10,22),(11,23),(12,24),(13,25),(14,26),(15,27),(16,28),(17,29),(18,30),(19,31),(20,32),(37,61),(38,62),(39,63),(40,64),(41,53),(42,54),(43,55),(44,56),(45,57),(46,58),(47,59),(48,60)], [(1,3),(2,4),(5,7),(6,8),(9,11),(10,12),(13,15),(14,16),(17,19),(18,20),(21,23),(22,24),(25,27),(26,28),(29,31),(30,32),(33,35),(34,36),(37,39),(38,40),(41,43),(42,44),(45,47),(46,48),(49,51),(50,52),(53,55),(54,56),(57,59),(58,60),(61,63),(62,64)], [(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)], [(1,47),(2,20),(3,45),(4,18),(5,42),(6,15),(7,44),(8,13),(9,17),(10,46),(11,19),(12,48),(14,38),(16,40),(21,29),(22,58),(23,31),(24,60),(25,35),(26,62),(27,33),(28,64),(30,50),(32,52),(34,56),(36,54),(37,41),(39,43),(49,57),(51,59),(53,61),(55,63)], [(1,15,11,43),(2,44,12,16),(3,13,9,41),(4,42,10,14),(5,46,38,18),(6,19,39,47),(7,48,40,20),(8,17,37,45),(21,53,49,25),(22,26,50,54),(23,55,51,27),(24,28,52,56),(29,61,57,35),(30,36,58,62),(31,63,59,33),(32,34,60,64)])
Matrix representation ►G ⊆ GL6(𝔽5)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 0 | 0 | 0 |
| 0 | 0 | 0 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 |
| 0 | 0 | 0 | 0 | 4 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 2 |
| 0 | 0 | 0 | 0 | 3 | 0 |
| 3 | 0 | 0 | 0 | 0 | 0 |
| 0 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 4 | 0 |
G:=sub<GL(6,GF(5))| [1,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,2,0,0,0,0,0,0,2,0,0,0,0,0,0,0,4,0,0,0,0,4,0],[1,0,0,0,0,0,0,4,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,3,0,0,0,0,2,0],[3,0,0,0,0,0,0,2,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,4,0,0,0,0,1,0] >;
38 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 4A | ··· | 4H | 4I | ··· | 4V | 4W | 4X | 4Y | 4Z |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 |
| size | 1 | 1 | ··· | 1 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | C4○D4 | C4○D4 | 2+ (1+4) | 2- (1+4) |
| kernel | C24.298C23 | C23.7Q8 | C23.63C23 | C24.C22 | C23.65C23 | C23.10D4 | C23.11D4 | C2×C42⋊C2 | C2×C4×D4 | C22×C4 | C2×C4 | C23 | C22 | C22 |
| # reps | 1 | 3 | 2 | 2 | 2 | 2 | 2 | 1 | 1 | 4 | 12 | 4 | 1 | 1 |
In GAP, Magma, Sage, TeX
C_2^4._{298}C_2^3 % in TeX
G:=Group("C2^4.298C2^3"); // GroupNames label
G:=SmallGroup(128,1217);
// by ID
G=gap.SmallGroup(128,1217);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,758,723,675,80]);
// Polycyclic
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=f^2=1,e^2=d,g^2=b,a*b=b*a,e*a*e^-1=a*c=c*a,f*a*f=a*d=d*a,a*g=g*a,b*c=c*b,b*d=d*b,f*e*f=g*e*g^-1=b*e=e*b,b*f=f*b,b*g=g*b,c*d=d*c,c*e=e*c,c*f=f*c,c*g=g*c,d*e=e*d,d*f=f*d,d*g=g*d,f*g=g*f>;
// generators/relations