p-group, metabelian, nilpotent (class 2), monomial
Aliases: C23.27C42, C42.744C23, C4⋊C8.19C4, (C4×C8).17C4, C8⋊C8⋊20C2, C4.4(C8⋊C4), C4.54(C8○D4), C22⋊C8.18C4, (C22×C8).27C4, (C2×C4).61C42, C42.293(C2×C4), (C4×C8).305C22, C4.58(C2×M4(2)), (C2×C4).71M4(2), C22.4(C8⋊C4), C22.41(C2×C42), C2.7(C8○2M4(2)), C42.12C4.42C2, (C2×C42).1031C22, (C2×C4×C8).8C2, C2.6(C2×C8⋊C4), (C2×C8).118(C2×C4), (C2×C4).581(C22×C4), (C22×C4).469(C2×C4), SmallGroup(128,184)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C23.27C42
G = < a,b,c,d,e | a2=b2=c2=1, d4=b, e4=c, eae-1=ab=ba, ac=ca, ad=da, bc=cb, bd=db, be=eb, ede-1=cd=dc, ce=ec >
Subgroups: 124 in 98 conjugacy classes, 72 normal (12 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, C23, C42, C42, C2×C8, C2×C8, C22×C4, C22×C4, C4×C8, C22⋊C8, C4⋊C8, C2×C42, C22×C8, C8⋊C8, C2×C4×C8, C42.12C4, C23.27C42
Quotients: C1, C2, C4, C22, C2×C4, C23, C42, M4(2), C22×C4, C8⋊C4, C2×C42, C2×M4(2), C8○D4, C2×C8⋊C4, C8○2M4(2), C23.27C42
►On 64 points
Generators in S64
(9 13)(10 14)(11 15)(12 16)(25 29)(26 30)(27 31)(28 32)(41 45)(42 46)(43 47)(44 48)(49 53)(50 54)(51 55)(52 56)
(1 5)(2 6)(3 7)(4 8)(9 13)(10 14)(11 15)(12 16)(17 21)(18 22)(19 23)(20 24)(25 29)(26 30)(27 31)(28 32)(33 37)(34 38)(35 39)(36 40)(41 45)(42 46)(43 47)(44 48)(49 53)(50 54)(51 55)(52 56)(57 61)(58 62)(59 63)(60 64)
(1 20)(2 21)(3 22)(4 23)(5 24)(6 17)(7 18)(8 19)(9 41)(10 42)(11 43)(12 44)(13 45)(14 46)(15 47)(16 48)(25 55)(26 56)(27 49)(28 50)(29 51)(30 52)(31 53)(32 54)(33 58)(34 59)(35 60)(36 61)(37 62)(38 63)(39 64)(40 57)
(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 42 40 55 20 10 57 25)(2 11 33 26 21 43 58 56)(3 44 34 49 22 12 59 27)(4 13 35 28 23 45 60 50)(5 46 36 51 24 14 61 29)(6 15 37 30 17 47 62 52)(7 48 38 53 18 16 63 31)(8 9 39 32 19 41 64 54)
G:=sub<Sym(64)| (9,13)(10,14)(11,15)(12,16)(25,29)(26,30)(27,31)(28,32)(41,45)(42,46)(43,47)(44,48)(49,53)(50,54)(51,55)(52,56), (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16)(17,21)(18,22)(19,23)(20,24)(25,29)(26,30)(27,31)(28,32)(33,37)(34,38)(35,39)(36,40)(41,45)(42,46)(43,47)(44,48)(49,53)(50,54)(51,55)(52,56)(57,61)(58,62)(59,63)(60,64), (1,20)(2,21)(3,22)(4,23)(5,24)(6,17)(7,18)(8,19)(9,41)(10,42)(11,43)(12,44)(13,45)(14,46)(15,47)(16,48)(25,55)(26,56)(27,49)(28,50)(29,51)(30,52)(31,53)(32,54)(33,58)(34,59)(35,60)(36,61)(37,62)(38,63)(39,64)(40,57), (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), (1,42,40,55,20,10,57,25)(2,11,33,26,21,43,58,56)(3,44,34,49,22,12,59,27)(4,13,35,28,23,45,60,50)(5,46,36,51,24,14,61,29)(6,15,37,30,17,47,62,52)(7,48,38,53,18,16,63,31)(8,9,39,32,19,41,64,54)>;
G:=Group( (9,13)(10,14)(11,15)(12,16)(25,29)(26,30)(27,31)(28,32)(41,45)(42,46)(43,47)(44,48)(49,53)(50,54)(51,55)(52,56), (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16)(17,21)(18,22)(19,23)(20,24)(25,29)(26,30)(27,31)(28,32)(33,37)(34,38)(35,39)(36,40)(41,45)(42,46)(43,47)(44,48)(49,53)(50,54)(51,55)(52,56)(57,61)(58,62)(59,63)(60,64), (1,20)(2,21)(3,22)(4,23)(5,24)(6,17)(7,18)(8,19)(9,41)(10,42)(11,43)(12,44)(13,45)(14,46)(15,47)(16,48)(25,55)(26,56)(27,49)(28,50)(29,51)(30,52)(31,53)(32,54)(33,58)(34,59)(35,60)(36,61)(37,62)(38,63)(39,64)(40,57), (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), (1,42,40,55,20,10,57,25)(2,11,33,26,21,43,58,56)(3,44,34,49,22,12,59,27)(4,13,35,28,23,45,60,50)(5,46,36,51,24,14,61,29)(6,15,37,30,17,47,62,52)(7,48,38,53,18,16,63,31)(8,9,39,32,19,41,64,54) );
G=PermutationGroup([[(9,13),(10,14),(11,15),(12,16),(25,29),(26,30),(27,31),(28,32),(41,45),(42,46),(43,47),(44,48),(49,53),(50,54),(51,55),(52,56)], [(1,5),(2,6),(3,7),(4,8),(9,13),(10,14),(11,15),(12,16),(17,21),(18,22),(19,23),(20,24),(25,29),(26,30),(27,31),(28,32),(33,37),(34,38),(35,39),(36,40),(41,45),(42,46),(43,47),(44,48),(49,53),(50,54),(51,55),(52,56),(57,61),(58,62),(59,63),(60,64)], [(1,20),(2,21),(3,22),(4,23),(5,24),(6,17),(7,18),(8,19),(9,41),(10,42),(11,43),(12,44),(13,45),(14,46),(15,47),(16,48),(25,55),(26,56),(27,49),(28,50),(29,51),(30,52),(31,53),(32,54),(33,58),(34,59),(35,60),(36,61),(37,62),(38,63),(39,64),(40,57)], [(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)], [(1,42,40,55,20,10,57,25),(2,11,33,26,21,43,58,56),(3,44,34,49,22,12,59,27),(4,13,35,28,23,45,60,50),(5,46,36,51,24,14,61,29),(6,15,37,30,17,47,62,52),(7,48,38,53,18,16,63,31),(8,9,39,32,19,41,64,54)]])
56 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | ··· | 4L | 4M | ··· | 4R | 8A | ··· | 8P | 8Q | ··· | 8AF |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | ··· | 8 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | + | ||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C4 | C4 | M4(2) | C8○D4 |
| kernel | C23.27C42 | C8⋊C8 | C2×C4×C8 | C42.12C4 | C4×C8 | C22⋊C8 | C4⋊C8 | C22×C8 | C2×C4 | C4 |
| # reps | 1 | 4 | 1 | 2 | 4 | 8 | 8 | 4 | 8 | 16 |
Matrix representation of C23.27C42 ►in GL4(𝔽17) generated by
| 1 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 16 |
| 16 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 |
| 16 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 15 | 0 | 0 | 0 |
| 0 | 2 | 0 | 0 |
| 0 | 0 | 15 | 0 |
| 0 | 0 | 0 | 15 |
| 0 | 1 | 0 | 0 |
| 13 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
G:=sub<GL(4,GF(17))| [1,0,0,0,0,16,0,0,0,0,1,0,0,0,0,16],[16,0,0,0,0,16,0,0,0,0,16,0,0,0,0,16],[16,0,0,0,0,16,0,0,0,0,1,0,0,0,0,1],[15,0,0,0,0,2,0,0,0,0,15,0,0,0,0,15],[0,13,0,0,1,0,0,0,0,0,0,1,0,0,1,0] >;
C23.27C42 in GAP, Magma, Sage, TeX
C_2^3._{27}C_4^2 % in TeX
G:=Group("C2^3.27C4^2"); // GroupNames label
G:=SmallGroup(128,184);
// by ID
G=gap.SmallGroup(128,184);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,2,56,925,120,387,136,172]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=1,d^4=b,e^4=c,e*a*e^-1=a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,b*e=e*b,e*d*e^-1=c*d=d*c,c*e=e*c>;
// generators/relations