p-group, metabelian, nilpotent (class 2), monomial
Aliases: C24.347C23, C23.495C24, C22.2772+ (1+4), C23.58(C4○D4), (C2×C42).78C22, C23.8Q8⋊74C2, C23.11D4⋊50C2, (C22×C4).117C23, (C23×C4).128C22, C24.C22⋊95C2, C23.23D4.41C2, C23.10D4.27C2, (C22×D4).181C22, C23.83C23⋊51C2, C2.34(C22.32C24), C24.3C22.53C2, C23.63C23⋊100C2, C2.63(C22.45C24), C2.C42.494C22, C2.97(C23.36C23), C2.72(C22.47C24), C2.27(C22.53C24), (C4×C4⋊C4)⋊108C2, (C4×C22⋊C4)⋊18C2, (C2×C4).404(C4○D4), (C2×C4⋊C4).336C22, C22.371(C2×C4○D4), (C2×C22⋊C4).198C22, SmallGroup(128,1327)
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 [×18], C4⋊C4 [×10], C22×C4 [×13], C22×C4 [×4], 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, C23.8Q8, C23.23D4, C23.63C23 [×2], C24.C22 [×4], C24.3C22, C23.10D4, C23.11D4 [×2], C23.83C23, C24.347C23
Quotients:
C1, C2 [×15], C22 [×35], C23 [×15], C4○D4 [×10], C24, C2×C4○D4 [×5], 2+ (1+4) [×2], C23.36C23 [×2], C22.32C24, C22.45C24, C22.47C24 [×2], C22.53C24, C24.347C23
Generators and relations
G = < a,b,c,d,e,f,g | a2=b2=c2=d2=1, e2=bcd, f2=d, g2=cb=bc, eae-1=gag-1=ab=ba, ac=ca, ad=da, faf-1=acd, bd=db, geg-1=be=eb, bf=fb, bg=gb, cd=dc, ce=ec, cf=fc, cg=gc, fef-1=de=ed, df=fd, dg=gd, gfg-1=cdf >
(1 34)(2 61)(3 36)(4 63)(5 44)(6 15)(7 42)(8 13)(9 62)(10 33)(11 64)(12 35)(14 40)(16 38)(17 25)(18 56)(19 27)(20 54)(21 29)(22 60)(23 31)(24 58)(26 48)(28 46)(30 52)(32 50)(37 41)(39 43)(45 53)(47 55)(49 57)(51 59)
(1 11)(2 12)(3 9)(4 10)(5 40)(6 37)(7 38)(8 39)(13 43)(14 44)(15 41)(16 42)(17 47)(18 48)(19 45)(20 46)(21 51)(22 52)(23 49)(24 50)(25 55)(26 56)(27 53)(28 54)(29 59)(30 60)(31 57)(32 58)(33 63)(34 64)(35 61)(36 62)
(1 53)(2 54)(3 55)(4 56)(5 21)(6 22)(7 23)(8 24)(9 25)(10 26)(11 27)(12 28)(13 58)(14 59)(15 60)(16 57)(17 62)(18 63)(19 64)(20 61)(29 44)(30 41)(31 42)(32 43)(33 48)(34 45)(35 46)(36 47)(37 52)(38 49)(39 50)(40 51)
(1 25)(2 26)(3 27)(4 28)(5 49)(6 50)(7 51)(8 52)(9 53)(10 54)(11 55)(12 56)(13 30)(14 31)(15 32)(16 29)(17 34)(18 35)(19 36)(20 33)(21 38)(22 39)(23 40)(24 37)(41 58)(42 59)(43 60)(44 57)(45 62)(46 63)(47 64)(48 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)
(1 33 25 20)(2 17 26 34)(3 35 27 18)(4 19 28 36)(5 58 49 41)(6 42 50 59)(7 60 51 43)(8 44 52 57)(9 61 53 48)(10 45 54 62)(11 63 55 46)(12 47 56 64)(13 38 30 21)(14 22 31 39)(15 40 32 23)(16 24 29 37)
(1 8 27 50)(2 40 28 21)(3 6 25 52)(4 38 26 23)(5 54 51 12)(7 56 49 10)(9 37 55 22)(11 39 53 24)(13 45 32 64)(14 20 29 35)(15 47 30 62)(16 18 31 33)(17 60 36 41)(19 58 34 43)(42 48 57 63)(44 46 59 61)
G:=sub<Sym(64)| (1,34)(2,61)(3,36)(4,63)(5,44)(6,15)(7,42)(8,13)(9,62)(10,33)(11,64)(12,35)(14,40)(16,38)(17,25)(18,56)(19,27)(20,54)(21,29)(22,60)(23,31)(24,58)(26,48)(28,46)(30,52)(32,50)(37,41)(39,43)(45,53)(47,55)(49,57)(51,59), (1,11)(2,12)(3,9)(4,10)(5,40)(6,37)(7,38)(8,39)(13,43)(14,44)(15,41)(16,42)(17,47)(18,48)(19,45)(20,46)(21,51)(22,52)(23,49)(24,50)(25,55)(26,56)(27,53)(28,54)(29,59)(30,60)(31,57)(32,58)(33,63)(34,64)(35,61)(36,62), (1,53)(2,54)(3,55)(4,56)(5,21)(6,22)(7,23)(8,24)(9,25)(10,26)(11,27)(12,28)(13,58)(14,59)(15,60)(16,57)(17,62)(18,63)(19,64)(20,61)(29,44)(30,41)(31,42)(32,43)(33,48)(34,45)(35,46)(36,47)(37,52)(38,49)(39,50)(40,51), (1,25)(2,26)(3,27)(4,28)(5,49)(6,50)(7,51)(8,52)(9,53)(10,54)(11,55)(12,56)(13,30)(14,31)(15,32)(16,29)(17,34)(18,35)(19,36)(20,33)(21,38)(22,39)(23,40)(24,37)(41,58)(42,59)(43,60)(44,57)(45,62)(46,63)(47,64)(48,61), (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), (1,33,25,20)(2,17,26,34)(3,35,27,18)(4,19,28,36)(5,58,49,41)(6,42,50,59)(7,60,51,43)(8,44,52,57)(9,61,53,48)(10,45,54,62)(11,63,55,46)(12,47,56,64)(13,38,30,21)(14,22,31,39)(15,40,32,23)(16,24,29,37), (1,8,27,50)(2,40,28,21)(3,6,25,52)(4,38,26,23)(5,54,51,12)(7,56,49,10)(9,37,55,22)(11,39,53,24)(13,45,32,64)(14,20,29,35)(15,47,30,62)(16,18,31,33)(17,60,36,41)(19,58,34,43)(42,48,57,63)(44,46,59,61)>;
G:=Group( (1,34)(2,61)(3,36)(4,63)(5,44)(6,15)(7,42)(8,13)(9,62)(10,33)(11,64)(12,35)(14,40)(16,38)(17,25)(18,56)(19,27)(20,54)(21,29)(22,60)(23,31)(24,58)(26,48)(28,46)(30,52)(32,50)(37,41)(39,43)(45,53)(47,55)(49,57)(51,59), (1,11)(2,12)(3,9)(4,10)(5,40)(6,37)(7,38)(8,39)(13,43)(14,44)(15,41)(16,42)(17,47)(18,48)(19,45)(20,46)(21,51)(22,52)(23,49)(24,50)(25,55)(26,56)(27,53)(28,54)(29,59)(30,60)(31,57)(32,58)(33,63)(34,64)(35,61)(36,62), (1,53)(2,54)(3,55)(4,56)(5,21)(6,22)(7,23)(8,24)(9,25)(10,26)(11,27)(12,28)(13,58)(14,59)(15,60)(16,57)(17,62)(18,63)(19,64)(20,61)(29,44)(30,41)(31,42)(32,43)(33,48)(34,45)(35,46)(36,47)(37,52)(38,49)(39,50)(40,51), (1,25)(2,26)(3,27)(4,28)(5,49)(6,50)(7,51)(8,52)(9,53)(10,54)(11,55)(12,56)(13,30)(14,31)(15,32)(16,29)(17,34)(18,35)(19,36)(20,33)(21,38)(22,39)(23,40)(24,37)(41,58)(42,59)(43,60)(44,57)(45,62)(46,63)(47,64)(48,61), (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), (1,33,25,20)(2,17,26,34)(3,35,27,18)(4,19,28,36)(5,58,49,41)(6,42,50,59)(7,60,51,43)(8,44,52,57)(9,61,53,48)(10,45,54,62)(11,63,55,46)(12,47,56,64)(13,38,30,21)(14,22,31,39)(15,40,32,23)(16,24,29,37), (1,8,27,50)(2,40,28,21)(3,6,25,52)(4,38,26,23)(5,54,51,12)(7,56,49,10)(9,37,55,22)(11,39,53,24)(13,45,32,64)(14,20,29,35)(15,47,30,62)(16,18,31,33)(17,60,36,41)(19,58,34,43)(42,48,57,63)(44,46,59,61) );
G=PermutationGroup([(1,34),(2,61),(3,36),(4,63),(5,44),(6,15),(7,42),(8,13),(9,62),(10,33),(11,64),(12,35),(14,40),(16,38),(17,25),(18,56),(19,27),(20,54),(21,29),(22,60),(23,31),(24,58),(26,48),(28,46),(30,52),(32,50),(37,41),(39,43),(45,53),(47,55),(49,57),(51,59)], [(1,11),(2,12),(3,9),(4,10),(5,40),(6,37),(7,38),(8,39),(13,43),(14,44),(15,41),(16,42),(17,47),(18,48),(19,45),(20,46),(21,51),(22,52),(23,49),(24,50),(25,55),(26,56),(27,53),(28,54),(29,59),(30,60),(31,57),(32,58),(33,63),(34,64),(35,61),(36,62)], [(1,53),(2,54),(3,55),(4,56),(5,21),(6,22),(7,23),(8,24),(9,25),(10,26),(11,27),(12,28),(13,58),(14,59),(15,60),(16,57),(17,62),(18,63),(19,64),(20,61),(29,44),(30,41),(31,42),(32,43),(33,48),(34,45),(35,46),(36,47),(37,52),(38,49),(39,50),(40,51)], [(1,25),(2,26),(3,27),(4,28),(5,49),(6,50),(7,51),(8,52),(9,53),(10,54),(11,55),(12,56),(13,30),(14,31),(15,32),(16,29),(17,34),(18,35),(19,36),(20,33),(21,38),(22,39),(23,40),(24,37),(41,58),(42,59),(43,60),(44,57),(45,62),(46,63),(47,64),(48,61)], [(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)], [(1,33,25,20),(2,17,26,34),(3,35,27,18),(4,19,28,36),(5,58,49,41),(6,42,50,59),(7,60,51,43),(8,44,52,57),(9,61,53,48),(10,45,54,62),(11,63,55,46),(12,47,56,64),(13,38,30,21),(14,22,31,39),(15,40,32,23),(16,24,29,37)], [(1,8,27,50),(2,40,28,21),(3,6,25,52),(4,38,26,23),(5,54,51,12),(7,56,49,10),(9,37,55,22),(11,39,53,24),(13,45,32,64),(14,20,29,35),(15,47,30,62),(16,18,31,33),(17,60,36,41),(19,58,34,43),(42,48,57,63),(44,46,59,61)])
Matrix representation ►G ⊆ GL6(𝔽5)
| 3 | 4 | 0 | 0 | 0 | 0 |
| 3 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 3 | 0 | 0 |
| 0 | 0 | 2 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 4 |
| 0 | 0 | 0 | 0 | 3 | 3 |
| 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 |
| 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 |
| 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 | 3 | 0 | 0 | 0 | 0 |
| 1 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 0 | 0 | 0 |
| 0 | 0 | 0 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 4 |
| 0 | 0 | 0 | 0 | 0 | 3 |
| 3 | 0 | 0 | 0 | 0 | 0 |
| 3 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 |
| 3 | 4 | 0 | 0 | 0 | 0 |
| 3 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 0 | 3 | 3 |
G:=sub<GL(6,GF(5))| [3,3,0,0,0,0,4,2,0,0,0,0,0,0,0,2,0,0,0,0,3,0,0,0,0,0,0,0,2,3,0,0,0,0,4,3],[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],[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],[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],[1,1,0,0,0,0,3,4,0,0,0,0,0,0,2,0,0,0,0,0,0,2,0,0,0,0,0,0,2,0,0,0,0,0,4,3],[3,3,0,0,0,0,0,2,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[3,3,0,0,0,0,4,2,0,0,0,0,0,0,0,1,0,0,0,0,4,0,0,0,0,0,0,0,2,3,0,0,0,0,0,3] >;
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 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4○D4 | C4○D4 | 2+ (1+4) |
| kernel | C24.347C23 | C4×C22⋊C4 | C4×C4⋊C4 | C23.8Q8 | C23.23D4 | C23.63C23 | C24.C22 | C24.3C22 | C23.10D4 | C23.11D4 | C23.83C23 | C2×C4 | C23 | C22 |
| # reps | 1 | 1 | 1 | 1 | 1 | 2 | 4 | 1 | 1 | 2 | 1 | 16 | 4 | 2 |
In GAP, Magma, Sage, TeX
C_2^4._{347}C_2^3 % in TeX
G:=Group("C2^4.347C2^3"); // GroupNames label
G:=SmallGroup(128,1327);
// by ID
G=gap.SmallGroup(128,1327);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,2,560,253,792,758,723,352,675,136]);
// Polycyclic
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=1,e^2=b*c*d,f^2=d,g^2=c*b=b*c,e*a*e^-1=g*a*g^-1=a*b=b*a,a*c=c*a,a*d=d*a,f*a*f^-1=a*c*d,b*d=d*b,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,f*e*f^-1=d*e=e*d,d*f=f*d,d*g=g*d,g*f*g^-1=c*d*f>;
// generators/relations