p-group, metabelian, nilpotent (class 2), monomial
Aliases: C42.307C23, C4.1272- (1+4), C4.1802+ (1+4), (C8×D4)⋊51C2, C8⋊9D4⋊48C2, C8⋊4Q8⋊46C2, C4⋊D4.30C4, C22⋊Q8.30C4, C4⋊C8.372C22, C42.237(C2×C4), (C4×C8).346C22, (C2×C8).447C23, (C2×C4).689C24, C42⋊2C2.4C4, C4.4D4.23C4, C22.5(C8○D4), C42.C2.23C4, (C4×Q8).62C22, C42.6C4⋊56C2, (C4×D4).306C22, C23.46(C22×C4), C8⋊C4.106C22, C2.37(Q8○M4(2)), C22⋊C8.149C22, (C2×C42).796C22, C22.211(C23×C4), (C22×C8).454C22, C22.D4.11C4, (C22×C4).1289C23, C42.7C22⋊29C2, C42⋊C2.91C22, C42.6C22⋊32C2, (C2×M4(2)).252C22, C23.36C23.16C2, C2.47(C23.33C23), (C2×C4⋊C8)⋊52C2, C2.37(C2×C8○D4), C4⋊C4.124(C2×C4), (C2×D4).185(C2×C4), C22⋊C4.45(C2×C4), (C2×C4).87(C22×C4), (C2×Q8).129(C2×C4), (C22×C4).367(C2×C4), SmallGroup(128,1724)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 252 in 178 conjugacy classes, 126 normal (40 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×11], C22, C22 [×2], C22 [×8], C8 [×8], C2×C4 [×6], C2×C4 [×6], C2×C4 [×9], D4 [×5], Q8, C23, C23 [×2], C42 [×4], C42 [×2], C22⋊C4 [×10], C4⋊C4 [×2], C4⋊C4 [×8], C2×C8 [×8], C2×C8 [×4], M4(2) [×2], C22×C4 [×3], C22×C4 [×2], C2×D4, C2×D4 [×2], C2×Q8, C4×C8 [×2], C8⋊C4 [×4], C22⋊C8 [×6], C4⋊C8 [×2], C4⋊C8 [×8], C2×C42, C42⋊C2 [×2], C4×D4, C4×D4 [×2], C4×Q8, C4⋊D4, C22⋊Q8, C22.D4 [×2], C4.4D4, C42.C2, C42⋊2C2 [×2], C22×C8 [×2], C22×C8 [×2], C2×M4(2) [×2], C2×C4⋊C8, C42.6C22 [×2], C42.6C4, C42.7C22 [×2], C8×D4 [×2], C8⋊9D4 [×4], C8⋊4Q8 [×2], C23.36C23, C42.307C23
Quotients:
C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], C23 [×15], C22×C4 [×14], C24, C8○D4 [×2], C23×C4, 2+ (1+4), 2- (1+4), C23.33C23, C2×C8○D4, Q8○M4(2), C42.307C23
Generators and relations
G = < a,b,c,d,e | a4=b4=d2=e2=1, c2=b, ab=ba, ac=ca, dad=a-1b2, eae=ab2, bc=cb, bd=db, be=eb, cd=dc, ece=a2b2c, ede=a2b2d >
►On 64 points
(1 14 51 57)(2 15 52 58)(3 16 53 59)(4 9 54 60)(5 10 55 61)(6 11 56 62)(7 12 49 63)(8 13 50 64)(17 47 27 33)(18 48 28 34)(19 41 29 35)(20 42 30 36)(21 43 31 37)(22 44 32 38)(23 45 25 39)(24 46 26 40)
(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 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 16)(2 9)(3 10)(4 11)(5 12)(6 13)(7 14)(8 15)(17 45)(18 46)(19 47)(20 48)(21 41)(22 42)(23 43)(24 44)(25 37)(26 38)(27 39)(28 40)(29 33)(30 34)(31 35)(32 36)(49 57)(50 58)(51 59)(52 60)(53 61)(54 62)(55 63)(56 64)
(1 43)(2 34)(3 45)(4 36)(5 47)(6 38)(7 41)(8 40)(9 24)(10 31)(11 18)(12 25)(13 20)(14 27)(15 22)(16 29)(17 57)(19 59)(21 61)(23 63)(26 60)(28 62)(30 64)(32 58)(33 55)(35 49)(37 51)(39 53)(42 54)(44 56)(46 50)(48 52)
G:=sub<Sym(64)| (1,14,51,57)(2,15,52,58)(3,16,53,59)(4,9,54,60)(5,10,55,61)(6,11,56,62)(7,12,49,63)(8,13,50,64)(17,47,27,33)(18,48,28,34)(19,41,29,35)(20,42,30,36)(21,43,31,37)(22,44,32,38)(23,45,25,39)(24,46,26,40), (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), (1,2,3,4,5,6,7,8)(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), (1,16)(2,9)(3,10)(4,11)(5,12)(6,13)(7,14)(8,15)(17,45)(18,46)(19,47)(20,48)(21,41)(22,42)(23,43)(24,44)(25,37)(26,38)(27,39)(28,40)(29,33)(30,34)(31,35)(32,36)(49,57)(50,58)(51,59)(52,60)(53,61)(54,62)(55,63)(56,64), (1,43)(2,34)(3,45)(4,36)(5,47)(6,38)(7,41)(8,40)(9,24)(10,31)(11,18)(12,25)(13,20)(14,27)(15,22)(16,29)(17,57)(19,59)(21,61)(23,63)(26,60)(28,62)(30,64)(32,58)(33,55)(35,49)(37,51)(39,53)(42,54)(44,56)(46,50)(48,52)>;
G:=Group( (1,14,51,57)(2,15,52,58)(3,16,53,59)(4,9,54,60)(5,10,55,61)(6,11,56,62)(7,12,49,63)(8,13,50,64)(17,47,27,33)(18,48,28,34)(19,41,29,35)(20,42,30,36)(21,43,31,37)(22,44,32,38)(23,45,25,39)(24,46,26,40), (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), (1,2,3,4,5,6,7,8)(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), (1,16)(2,9)(3,10)(4,11)(5,12)(6,13)(7,14)(8,15)(17,45)(18,46)(19,47)(20,48)(21,41)(22,42)(23,43)(24,44)(25,37)(26,38)(27,39)(28,40)(29,33)(30,34)(31,35)(32,36)(49,57)(50,58)(51,59)(52,60)(53,61)(54,62)(55,63)(56,64), (1,43)(2,34)(3,45)(4,36)(5,47)(6,38)(7,41)(8,40)(9,24)(10,31)(11,18)(12,25)(13,20)(14,27)(15,22)(16,29)(17,57)(19,59)(21,61)(23,63)(26,60)(28,62)(30,64)(32,58)(33,55)(35,49)(37,51)(39,53)(42,54)(44,56)(46,50)(48,52) );
G=PermutationGroup([(1,14,51,57),(2,15,52,58),(3,16,53,59),(4,9,54,60),(5,10,55,61),(6,11,56,62),(7,12,49,63),(8,13,50,64),(17,47,27,33),(18,48,28,34),(19,41,29,35),(20,42,30,36),(21,43,31,37),(22,44,32,38),(23,45,25,39),(24,46,26,40)], [(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)], [(1,2,3,4,5,6,7,8),(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)], [(1,16),(2,9),(3,10),(4,11),(5,12),(6,13),(7,14),(8,15),(17,45),(18,46),(19,47),(20,48),(21,41),(22,42),(23,43),(24,44),(25,37),(26,38),(27,39),(28,40),(29,33),(30,34),(31,35),(32,36),(49,57),(50,58),(51,59),(52,60),(53,61),(54,62),(55,63),(56,64)], [(1,43),(2,34),(3,45),(4,36),(5,47),(6,38),(7,41),(8,40),(9,24),(10,31),(11,18),(12,25),(13,20),(14,27),(15,22),(16,29),(17,57),(19,59),(21,61),(23,63),(26,60),(28,62),(30,64),(32,58),(33,55),(35,49),(37,51),(39,53),(42,54),(44,56),(46,50),(48,52)])
Matrix representation ►G ⊆ GL6(𝔽17)
| 13 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 1 | 0 | 0 |
| 0 | 0 | 0 | 13 | 0 | 0 |
| 0 | 0 | 9 | 8 | 1 | 13 |
| 0 | 0 | 7 | 9 | 9 | 16 |
| 13 | 0 | 0 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 |
| 9 | 0 | 0 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 9 | 8 | 14 | 13 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 6 | 12 | 9 | 9 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 1 | 0 | 0 |
| 0 | 0 | 2 | 13 | 0 | 0 |
| 0 | 0 | 9 | 8 | 1 | 13 |
| 0 | 0 | 11 | 9 | 0 | 16 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 13 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 4 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 15 | 2 | 13 | 16 |
| 0 | 0 | 10 | 3 | 15 | 4 |
G:=sub<GL(6,GF(17))| [13,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,9,7,0,0,1,13,8,9,0,0,0,0,1,9,0,0,0,0,13,16],[13,0,0,0,0,0,0,13,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[9,0,0,0,0,0,0,9,0,0,0,0,0,0,0,9,16,6,0,0,0,8,0,12,0,0,1,14,0,9,0,0,0,13,0,9],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,4,2,9,11,0,0,1,13,8,9,0,0,0,0,1,0,0,0,0,0,13,16],[0,13,0,0,0,0,4,0,0,0,0,0,0,0,16,0,15,10,0,0,4,1,2,3,0,0,0,0,13,15,0,0,0,0,16,4] >;
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | ··· | 4P | 8A | ··· | 8H | 8I | ··· | 8T |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 8 | ··· | 8 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | - | ||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C4 | C4 | C4 | C8○D4 | 2+ (1+4) | 2- (1+4) | Q8○M4(2) |
| kernel | C42.307C23 | C2×C4⋊C8 | C42.6C22 | C42.6C4 | C42.7C22 | C8×D4 | C8⋊9D4 | C8⋊4Q8 | C23.36C23 | C4⋊D4 | C22⋊Q8 | C22.D4 | C4.4D4 | C42.C2 | C42⋊2C2 | C22 | C4 | C4 | C2 |
| # reps | 1 | 1 | 2 | 1 | 2 | 2 | 4 | 2 | 1 | 2 | 2 | 4 | 2 | 2 | 4 | 8 | 1 | 1 | 2 |
In GAP, Magma, Sage, TeX
C_4^2._{307}C_2^3 % in TeX
G:=Group("C4^2.307C2^3"); // GroupNames label
G:=SmallGroup(128,1724);
// by ID
G=gap.SmallGroup(128,1724);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,219,675,1018,80,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^4=d^2=e^2=1,c^2=b,a*b=b*a,a*c=c*a,d*a*d=a^-1*b^2,e*a*e=a*b^2,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e=a^2*b^2*c,e*d*e=a^2*b^2*d>;
// generators/relations