p-group, metabelian, nilpotent (class 2), monomial
Aliases: C24.19C23, C23.416C24, C22.2102+ (1+4), (C22×C4).84C23, C23.4Q8⋊17C2, C23.147(C4○D4), C23.34D4⋊33C2, C23.11D4⋊35C2, (C2×C42).531C22, (C23×C4).105C22, C24.C22⋊71C2, C23.23D4.29C2, C23.10D4.13C2, (C22×D4).155C22, C23.63C23⋊70C2, C23.83C23⋊30C2, C24.3C22.40C2, C2.33(C22.45C24), C2.C42.164C22, C2.44(C22.47C24), C2.20(C22.34C24), C2.14(C22.53C24), C2.59(C23.36C23), (C4×C4⋊C4)⋊78C2, (C4×C22⋊C4)⋊78C2, (C2×C4).137(C4○D4), (C2×C4⋊C4).863C22, C22.293(C2×C4○D4), (C2×C22⋊C4).164C22, SmallGroup(128,1248)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 452 in 227 conjugacy classes, 92 normal (82 characteristic)
C1, C2 [×7], C2 [×3], C4 [×17], C22 [×7], C22 [×17], C2×C4 [×8], C2×C4 [×39], D4 [×4], C23, C23 [×2], C23 [×13], C42 [×5], C22⋊C4 [×17], C4⋊C4 [×10], C22×C4 [×13], C22×C4 [×5], C2×D4 [×5], C24 [×2], C2.C42 [×12], C2×C42 [×4], C2×C22⋊C4 [×11], C2×C4⋊C4 [×6], C23×C4, C22×D4, C4×C22⋊C4, C4×C4⋊C4, C23.34D4, C23.23D4, C23.63C23 [×2], C24.C22 [×4], C24.3C22, C23.10D4, C23.11D4, C23.4Q8, C23.83C23, C23.416C24
Quotients:
C1, C2 [×15], C22 [×35], C23 [×15], C4○D4 [×10], C24, C2×C4○D4 [×5], 2+ (1+4) [×2], C23.36C23 [×2], C22.34C24, C22.45C24 [×2], C22.47C24, C22.53C24, C23.416C24
Generators and relations
G = < a,b,c,d,e,f,g | a2=b2=c2=f2=1, d2=ca=ac, e2=a, g2=b, ab=ba, ede-1=gdg-1=ad=da, ae=ea, af=fa, ag=ga, bc=cb, fdf=bd=db, be=eb, bf=fb, bg=gb, cd=dc, fef=ce=ec, cf=fc, cg=gc, eg=ge, fg=gf >
(1 27)(2 28)(3 25)(4 26)(5 60)(6 57)(7 58)(8 59)(9 47)(10 48)(11 45)(12 46)(13 51)(14 52)(15 49)(16 50)(17 43)(18 44)(19 41)(20 42)(21 40)(22 37)(23 38)(24 39)(29 36)(30 33)(31 34)(32 35)(53 63)(54 64)(55 61)(56 62)
(1 58)(2 59)(3 60)(4 57)(5 25)(6 26)(7 27)(8 28)(9 43)(10 44)(11 41)(12 42)(13 39)(14 40)(15 37)(16 38)(17 47)(18 48)(19 45)(20 46)(21 52)(22 49)(23 50)(24 51)(29 56)(30 53)(31 54)(32 55)(33 63)(34 64)(35 61)(36 62)
(1 25)(2 26)(3 27)(4 28)(5 58)(6 59)(7 60)(8 57)(9 45)(10 46)(11 47)(12 48)(13 49)(14 50)(15 51)(16 52)(17 41)(18 42)(19 43)(20 44)(21 38)(22 39)(23 40)(24 37)(29 34)(30 35)(31 36)(32 33)(53 61)(54 62)(55 63)(56 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 20 27 42)(2 43 28 17)(3 18 25 44)(4 41 26 19)(5 10 60 48)(6 45 57 11)(7 12 58 46)(8 47 59 9)(13 32 51 35)(14 36 52 29)(15 30 49 33)(16 34 50 31)(21 56 40 62)(22 63 37 53)(23 54 38 64)(24 61 39 55)
(1 50)(2 24)(3 52)(4 22)(5 40)(6 15)(7 38)(8 13)(9 30)(10 54)(11 32)(12 56)(14 25)(16 27)(17 63)(18 34)(19 61)(20 36)(21 60)(23 58)(26 37)(28 39)(29 42)(31 44)(33 47)(35 45)(41 55)(43 53)(46 62)(48 64)(49 57)(51 59)
(1 30 58 53)(2 34 59 64)(3 32 60 55)(4 36 57 62)(5 61 25 35)(6 56 26 29)(7 63 27 33)(8 54 28 31)(9 23 43 50)(10 39 44 13)(11 21 41 52)(12 37 42 15)(14 45 40 19)(16 47 38 17)(18 51 48 24)(20 49 46 22)
G:=sub<Sym(64)| (1,27)(2,28)(3,25)(4,26)(5,60)(6,57)(7,58)(8,59)(9,47)(10,48)(11,45)(12,46)(13,51)(14,52)(15,49)(16,50)(17,43)(18,44)(19,41)(20,42)(21,40)(22,37)(23,38)(24,39)(29,36)(30,33)(31,34)(32,35)(53,63)(54,64)(55,61)(56,62), (1,58)(2,59)(3,60)(4,57)(5,25)(6,26)(7,27)(8,28)(9,43)(10,44)(11,41)(12,42)(13,39)(14,40)(15,37)(16,38)(17,47)(18,48)(19,45)(20,46)(21,52)(22,49)(23,50)(24,51)(29,56)(30,53)(31,54)(32,55)(33,63)(34,64)(35,61)(36,62), (1,25)(2,26)(3,27)(4,28)(5,58)(6,59)(7,60)(8,57)(9,45)(10,46)(11,47)(12,48)(13,49)(14,50)(15,51)(16,52)(17,41)(18,42)(19,43)(20,44)(21,38)(22,39)(23,40)(24,37)(29,34)(30,35)(31,36)(32,33)(53,61)(54,62)(55,63)(56,64), (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), (1,20,27,42)(2,43,28,17)(3,18,25,44)(4,41,26,19)(5,10,60,48)(6,45,57,11)(7,12,58,46)(8,47,59,9)(13,32,51,35)(14,36,52,29)(15,30,49,33)(16,34,50,31)(21,56,40,62)(22,63,37,53)(23,54,38,64)(24,61,39,55), (1,50)(2,24)(3,52)(4,22)(5,40)(6,15)(7,38)(8,13)(9,30)(10,54)(11,32)(12,56)(14,25)(16,27)(17,63)(18,34)(19,61)(20,36)(21,60)(23,58)(26,37)(28,39)(29,42)(31,44)(33,47)(35,45)(41,55)(43,53)(46,62)(48,64)(49,57)(51,59), (1,30,58,53)(2,34,59,64)(3,32,60,55)(4,36,57,62)(5,61,25,35)(6,56,26,29)(7,63,27,33)(8,54,28,31)(9,23,43,50)(10,39,44,13)(11,21,41,52)(12,37,42,15)(14,45,40,19)(16,47,38,17)(18,51,48,24)(20,49,46,22)>;
G:=Group( (1,27)(2,28)(3,25)(4,26)(5,60)(6,57)(7,58)(8,59)(9,47)(10,48)(11,45)(12,46)(13,51)(14,52)(15,49)(16,50)(17,43)(18,44)(19,41)(20,42)(21,40)(22,37)(23,38)(24,39)(29,36)(30,33)(31,34)(32,35)(53,63)(54,64)(55,61)(56,62), (1,58)(2,59)(3,60)(4,57)(5,25)(6,26)(7,27)(8,28)(9,43)(10,44)(11,41)(12,42)(13,39)(14,40)(15,37)(16,38)(17,47)(18,48)(19,45)(20,46)(21,52)(22,49)(23,50)(24,51)(29,56)(30,53)(31,54)(32,55)(33,63)(34,64)(35,61)(36,62), (1,25)(2,26)(3,27)(4,28)(5,58)(6,59)(7,60)(8,57)(9,45)(10,46)(11,47)(12,48)(13,49)(14,50)(15,51)(16,52)(17,41)(18,42)(19,43)(20,44)(21,38)(22,39)(23,40)(24,37)(29,34)(30,35)(31,36)(32,33)(53,61)(54,62)(55,63)(56,64), (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), (1,20,27,42)(2,43,28,17)(3,18,25,44)(4,41,26,19)(5,10,60,48)(6,45,57,11)(7,12,58,46)(8,47,59,9)(13,32,51,35)(14,36,52,29)(15,30,49,33)(16,34,50,31)(21,56,40,62)(22,63,37,53)(23,54,38,64)(24,61,39,55), (1,50)(2,24)(3,52)(4,22)(5,40)(6,15)(7,38)(8,13)(9,30)(10,54)(11,32)(12,56)(14,25)(16,27)(17,63)(18,34)(19,61)(20,36)(21,60)(23,58)(26,37)(28,39)(29,42)(31,44)(33,47)(35,45)(41,55)(43,53)(46,62)(48,64)(49,57)(51,59), (1,30,58,53)(2,34,59,64)(3,32,60,55)(4,36,57,62)(5,61,25,35)(6,56,26,29)(7,63,27,33)(8,54,28,31)(9,23,43,50)(10,39,44,13)(11,21,41,52)(12,37,42,15)(14,45,40,19)(16,47,38,17)(18,51,48,24)(20,49,46,22) );
G=PermutationGroup([(1,27),(2,28),(3,25),(4,26),(5,60),(6,57),(7,58),(8,59),(9,47),(10,48),(11,45),(12,46),(13,51),(14,52),(15,49),(16,50),(17,43),(18,44),(19,41),(20,42),(21,40),(22,37),(23,38),(24,39),(29,36),(30,33),(31,34),(32,35),(53,63),(54,64),(55,61),(56,62)], [(1,58),(2,59),(3,60),(4,57),(5,25),(6,26),(7,27),(8,28),(9,43),(10,44),(11,41),(12,42),(13,39),(14,40),(15,37),(16,38),(17,47),(18,48),(19,45),(20,46),(21,52),(22,49),(23,50),(24,51),(29,56),(30,53),(31,54),(32,55),(33,63),(34,64),(35,61),(36,62)], [(1,25),(2,26),(3,27),(4,28),(5,58),(6,59),(7,60),(8,57),(9,45),(10,46),(11,47),(12,48),(13,49),(14,50),(15,51),(16,52),(17,41),(18,42),(19,43),(20,44),(21,38),(22,39),(23,40),(24,37),(29,34),(30,35),(31,36),(32,33),(53,61),(54,62),(55,63),(56,64)], [(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)], [(1,20,27,42),(2,43,28,17),(3,18,25,44),(4,41,26,19),(5,10,60,48),(6,45,57,11),(7,12,58,46),(8,47,59,9),(13,32,51,35),(14,36,52,29),(15,30,49,33),(16,34,50,31),(21,56,40,62),(22,63,37,53),(23,54,38,64),(24,61,39,55)], [(1,50),(2,24),(3,52),(4,22),(5,40),(6,15),(7,38),(8,13),(9,30),(10,54),(11,32),(12,56),(14,25),(16,27),(17,63),(18,34),(19,61),(20,36),(21,60),(23,58),(26,37),(28,39),(29,42),(31,44),(33,47),(35,45),(41,55),(43,53),(46,62),(48,64),(49,57),(51,59)], [(1,30,58,53),(2,34,59,64),(3,32,60,55),(4,36,57,62),(5,61,25,35),(6,56,26,29),(7,63,27,33),(8,54,28,31),(9,23,43,50),(10,39,44,13),(11,21,41,52),(12,37,42,15),(14,45,40,19),(16,47,38,17),(18,51,48,24),(20,49,46,22)])
Matrix representation ►G ⊆ GL6(𝔽5)
| 1 | 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 | 4 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 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 |
| 0 | 3 | 0 | 0 | 0 | 0 |
| 3 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 1 | 0 | 0 |
| 0 | 0 | 2 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 3 |
| 0 | 0 | 0 | 0 | 1 | 4 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 4 |
| 0 | 0 | 0 | 0 | 0 | 2 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 2 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 3 |
| 0 | 0 | 0 | 0 | 0 | 4 |
| 2 | 0 | 0 | 0 | 0 | 0 |
| 0 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 0 | 0 | 0 |
| 0 | 0 | 0 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 4 |
| 0 | 0 | 0 | 0 | 0 | 2 |
G:=sub<GL(6,GF(5))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[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],[0,3,0,0,0,0,3,0,0,0,0,0,0,0,3,2,0,0,0,0,1,2,0,0,0,0,0,0,1,1,0,0,0,0,3,4],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,3,0,0,0,0,0,4,2],[1,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,2,4,0,0,0,0,0,0,1,0,0,0,0,0,3,4],[2,0,0,0,0,0,0,2,0,0,0,0,0,0,3,0,0,0,0,0,0,3,0,0,0,0,0,0,3,0,0,0,0,0,4,2] >;
38 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 4A | ··· | 4H | 4I | ··· | 4X | 4Y | 4Z | 4AA |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | 4 | 4 |
| size | 1 | 1 | ··· | 1 | 4 | 4 | 8 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | 8 | 8 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4○D4 | C4○D4 | 2+ (1+4) |
| kernel | C23.416C24 | C4×C22⋊C4 | C4×C4⋊C4 | C23.34D4 | C23.23D4 | C23.63C23 | C24.C22 | C24.3C22 | C23.10D4 | C23.11D4 | C23.4Q8 | C23.83C23 | C2×C4 | C23 | C22 |
| # reps | 1 | 1 | 1 | 1 | 1 | 2 | 4 | 1 | 1 | 1 | 1 | 1 | 16 | 4 | 2 |
In GAP, Magma, Sage, TeX
C_2^3._{416}C_2^4 % in TeX
G:=Group("C2^3.416C2^4"); // GroupNames label
G:=SmallGroup(128,1248);
// by ID
G=gap.SmallGroup(128,1248);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,2,560,253,120,758,723,675,136]);
// Polycyclic
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=f^2=1,d^2=c*a=a*c,e^2=a,g^2=b,a*b=b*a,e*d*e^-1=g*d*g^-1=a*d=d*a,a*e=e*a,a*f=f*a,a*g=g*a,b*c=c*b,f*d*f=b*d=d*b,b*e=e*b,b*f=f*b,b*g=g*b,c*d=d*c,f*e*f=c*e=e*c,c*f=f*c,c*g=g*c,e*g=g*e,f*g=g*f>;
// generators/relations