p-group, metabelian, nilpotent (class 2), monomial
Aliases: C42.309C23, C4.1292- 1+4, C4.1822+ 1+4, (C8×D4)⋊53C2, (C8×Q8)⋊38C2, C8⋊6D4⋊48C2, C8⋊9D4⋊50C2, C8⋊4Q8⋊48C2, C4.42(C8○D4), C4⋊D4.32C4, C22⋊Q8.32C4, C4⋊C8.373C22, (C2×C4).691C24, C42.239(C2×C4), C42⋊2C2.6C4, (C2×C8).449C23, (C4×C8).348C22, C4.4D4.25C4, C22.8(C8○D4), C42.C2.25C4, (C4×D4).307C22, C23.48(C22×C4), (C22×C8).96C22, (C4×Q8).288C22, C8⋊C4.108C22, C42.12C4⋊60C2, C22⋊C8.242C22, (C22×C4).951C23, C22.213(C23×C4), (C2×C42).798C22, C22.D4.13C4, C42⋊C2.93C22, C42.7C22⋊31C2, C42.6C22⋊34C2, (C2×M4(2)).254C22, C23.36C23.18C2, C2.49(C23.33C23), (C2×C4⋊C8)⋊53C2, C2.39(C2×C8○D4), C4⋊C4.172(C2×C4), (C2×D4).186(C2×C4), C22⋊C4.46(C2×C4), (C2×C4).89(C22×C4), (C2×Q8).169(C2×C4), (C22×C4).369(C2×C4), SmallGroup(128,1726)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.309C23
G = < a,b,c,d,e | a4=b4=d2=e2=1, c2=b, ab=ba, ac=ca, dad=a-1, eae=ab2, bc=cb, bd=db, be=eb, cd=dc, ece=a2b2c, ede=a2d >
Subgroups: 252 in 182 conjugacy classes, 128 normal (52 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C2×C8, M4(2), C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4×C8, C4×C8, C8⋊C4, C22⋊C8, C22⋊C8, C4⋊C8, C4⋊C8, C2×C42, C42⋊C2, C4×D4, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C42.C2, C42⋊2C2, C22×C8, C2×M4(2), C2×C4⋊C8, C42.6C22, C42.12C4, C42.7C22, C8×D4, C8×D4, C8⋊9D4, C8⋊6D4, C8×Q8, C8⋊4Q8, C23.36C23, C42.309C23
Quotients: C1, C2, C4, C22, C2×C4, C23, C22×C4, C24, C8○D4, C23×C4, 2+ 1+4, 2- 1+4, C23.33C23, C2×C8○D4, C42.309C23
►On 64 points
(1 47 51 59)(2 48 52 60)(3 41 53 61)(4 42 54 62)(5 43 55 63)(6 44 56 64)(7 45 49 57)(8 46 50 58)(9 26 34 22)(10 27 35 23)(11 28 36 24)(12 29 37 17)(13 30 38 18)(14 31 39 19)(15 32 40 20)(16 25 33 21)
(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 47)(2 48)(3 41)(4 42)(5 43)(6 44)(7 45)(8 46)(9 18)(10 19)(11 20)(12 21)(13 22)(14 23)(15 24)(16 17)(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 14)(2 36)(3 16)(4 38)(5 10)(6 40)(7 12)(8 34)(9 50)(11 52)(13 54)(15 56)(17 61)(18 46)(19 63)(20 48)(21 57)(22 42)(23 59)(24 44)(25 45)(26 62)(27 47)(28 64)(29 41)(30 58)(31 43)(32 60)(33 53)(35 55)(37 49)(39 51)
G:=sub<Sym(64)| (1,47,51,59)(2,48,52,60)(3,41,53,61)(4,42,54,62)(5,43,55,63)(6,44,56,64)(7,45,49,57)(8,46,50,58)(9,26,34,22)(10,27,35,23)(11,28,36,24)(12,29,37,17)(13,30,38,18)(14,31,39,19)(15,32,40,20)(16,25,33,21), (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), (1,47)(2,48)(3,41)(4,42)(5,43)(6,44)(7,45)(8,46)(9,18)(10,19)(11,20)(12,21)(13,22)(14,23)(15,24)(16,17)(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), (1,14)(2,36)(3,16)(4,38)(5,10)(6,40)(7,12)(8,34)(9,50)(11,52)(13,54)(15,56)(17,61)(18,46)(19,63)(20,48)(21,57)(22,42)(23,59)(24,44)(25,45)(26,62)(27,47)(28,64)(29,41)(30,58)(31,43)(32,60)(33,53)(35,55)(37,49)(39,51)>;
G:=Group( (1,47,51,59)(2,48,52,60)(3,41,53,61)(4,42,54,62)(5,43,55,63)(6,44,56,64)(7,45,49,57)(8,46,50,58)(9,26,34,22)(10,27,35,23)(11,28,36,24)(12,29,37,17)(13,30,38,18)(14,31,39,19)(15,32,40,20)(16,25,33,21), (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), (1,47)(2,48)(3,41)(4,42)(5,43)(6,44)(7,45)(8,46)(9,18)(10,19)(11,20)(12,21)(13,22)(14,23)(15,24)(16,17)(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), (1,14)(2,36)(3,16)(4,38)(5,10)(6,40)(7,12)(8,34)(9,50)(11,52)(13,54)(15,56)(17,61)(18,46)(19,63)(20,48)(21,57)(22,42)(23,59)(24,44)(25,45)(26,62)(27,47)(28,64)(29,41)(30,58)(31,43)(32,60)(33,53)(35,55)(37,49)(39,51) );
G=PermutationGroup([[(1,47,51,59),(2,48,52,60),(3,41,53,61),(4,42,54,62),(5,43,55,63),(6,44,56,64),(7,45,49,57),(8,46,50,58),(9,26,34,22),(10,27,35,23),(11,28,36,24),(12,29,37,17),(13,30,38,18),(14,31,39,19),(15,32,40,20),(16,25,33,21)], [(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)], [(1,47),(2,48),(3,41),(4,42),(5,43),(6,44),(7,45),(8,46),(9,18),(10,19),(11,20),(12,21),(13,22),(14,23),(15,24),(16,17),(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)], [(1,14),(2,36),(3,16),(4,38),(5,10),(6,40),(7,12),(8,34),(9,50),(11,52),(13,54),(15,56),(17,61),(18,46),(19,63),(20,48),(21,57),(22,42),(23,59),(24,44),(25,45),(26,62),(27,47),(28,64),(29,41),(30,58),(31,43),(32,60),(33,53),(35,55),(37,49),(39,51)]])
50 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | ··· | 4J | 4K | ··· | 4R | 8A | ··· | 8P | 8Q | ··· | 8X |
| 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 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | - | ||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C4 | C4 | C4 | C8○D4 | C8○D4 | 2+ 1+4 | 2- 1+4 |
| kernel | C42.309C23 | C2×C4⋊C8 | C42.6C22 | C42.12C4 | C42.7C22 | C8×D4 | C8⋊9D4 | C8⋊6D4 | C8×Q8 | C8⋊4Q8 | C23.36C23 | C4⋊D4 | C22⋊Q8 | C22.D4 | C4.4D4 | C42.C2 | C42⋊2C2 | C4 | C22 | C4 | C4 |
| # reps | 1 | 1 | 2 | 1 | 2 | 3 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 2 | 2 | 4 | 8 | 8 | 1 | 1 |
Matrix representation of C42.309C23 ►in GL4(𝔽17) generated by
| 1 | 15 | 0 | 0 |
| 1 | 16 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 1 |
| 4 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 13 | 0 |
| 0 | 0 | 0 | 13 |
| 15 | 0 | 0 | 0 |
| 0 | 15 | 0 | 0 |
| 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 9 |
| 1 | 15 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 |
| 16 | 0 | 0 | 0 |
| 16 | 1 | 0 | 0 |
| 0 | 0 | 0 | 16 |
| 0 | 0 | 16 | 0 |
G:=sub<GL(4,GF(17))| [1,1,0,0,15,16,0,0,0,0,16,0,0,0,0,1],[4,0,0,0,0,4,0,0,0,0,13,0,0,0,0,13],[15,0,0,0,0,15,0,0,0,0,8,0,0,0,0,9],[1,0,0,0,15,16,0,0,0,0,16,0,0,0,0,16],[16,16,0,0,0,1,0,0,0,0,0,16,0,0,16,0] >;
C42.309C23 in GAP, Magma, Sage, TeX
C_4^2._{309}C_2^3 % in TeX
G:=Group("C4^2.309C2^3"); // GroupNames label
G:=SmallGroup(128,1726);
// by ID
G=gap.SmallGroup(128,1726);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,891,675,1018,80,124]);
// 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,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*d>;
// generators/relations