p-group, metabelian, nilpotent (class 2), monomial
Aliases: C24.49C23, C23.553C24, C22.3282+ 1+4, C22.2442- 1+4, C23⋊Q8⋊34C2, C23.73(C4○D4), (C2×C42).85C22, C23.8Q8⋊91C2, C23.11D4⋊70C2, (C23×C4).146C22, (C22×C4).163C23, C23.84C23⋊8C2, C23.10D4.36C2, C23.23D4.48C2, (C22×D4).205C22, (C22×Q8).163C22, C24.C22⋊110C2, C23.81C23⋊69C2, C23.67C23⋊75C2, C2.51(C22.32C24), C23.63C23⋊120C2, C2.C42.270C22, C2.50(C22.33C24), C2.60(C22.36C24), C2.105(C23.36C23), (C4×C22⋊C4)⋊97C2, (C2×C4).178(C4○D4), (C2×C4⋊C4).378C22, C22.425(C2×C4○D4), (C2×C22⋊C4).474C22, SmallGroup(128,1385)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C23.553C24
G = < a,b,c,d,e,f,g | a2=b2=c2=e2=1, d2=ca=ac, f2=a, g2=b, ab=ba, ede=ad=da, ae=ea, gfg-1=af=fa, ag=ga, bc=cb, fdf-1=bd=db, be=eb, bf=fb, bg=gb, cd=dc, fef-1=ce=ec, cf=fc, cg=gc, dg=gd, eg=ge >
Subgroups: 468 in 221 conjugacy classes, 88 normal (82 characteristic)
C1, C2, C2, C4, C22, C22, C2×C4, C2×C4, D4, Q8, C23, C23, C23, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×Q8, C24, C2.C42, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C23×C4, C22×D4, C22×Q8, C4×C22⋊C4, C23.8Q8, C23.23D4, C23.63C23, C24.C22, C23.67C23, C23⋊Q8, C23.10D4, C23.11D4, C23.81C23, C23.84C23, C23.553C24
Quotients: C1, C2, C22, C23, C4○D4, C24, C2×C4○D4, 2+ 1+4, 2- 1+4, C23.36C23, C22.32C24, C22.33C24, C22.36C24, C23.553C24
►On 64 points
Generators in S64
(1 30)(2 31)(3 32)(4 29)(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 40)(18 37)(19 38)(20 39)(21 44)(22 41)(23 42)(24 43)(25 36)(26 33)(27 34)(28 35)(53 63)(54 64)(55 61)(56 62)
(1 58)(2 59)(3 60)(4 57)(5 32)(6 29)(7 30)(8 31)(9 23)(10 24)(11 21)(12 22)(13 20)(14 17)(15 18)(16 19)(25 56)(26 53)(27 54)(28 55)(33 63)(34 64)(35 61)(36 62)(37 49)(38 50)(39 51)(40 52)(41 46)(42 47)(43 48)(44 45)
(1 32)(2 29)(3 30)(4 31)(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 38)(18 39)(19 40)(20 37)(21 42)(22 43)(23 44)(24 41)(25 34)(26 35)(27 36)(28 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 22)(2 42)(3 24)(4 44)(5 48)(6 11)(7 46)(8 9)(10 60)(12 58)(13 28)(14 36)(15 26)(16 34)(17 62)(18 53)(19 64)(20 55)(21 29)(23 31)(25 52)(27 50)(30 41)(32 43)(33 49)(35 51)(37 63)(38 54)(39 61)(40 56)(45 57)(47 59)
(1 50 30 16)(2 39 31 20)(3 52 32 14)(4 37 29 18)(5 17 60 40)(6 15 57 49)(7 19 58 38)(8 13 59 51)(9 33 47 26)(10 64 48 54)(11 35 45 28)(12 62 46 56)(21 61 44 55)(22 36 41 25)(23 63 42 53)(24 34 43 27)
(1 26 58 53)(2 27 59 54)(3 28 60 55)(4 25 57 56)(5 61 32 35)(6 62 29 36)(7 63 30 33)(8 64 31 34)(9 19 23 16)(10 20 24 13)(11 17 21 14)(12 18 22 15)(37 41 49 46)(38 42 50 47)(39 43 51 48)(40 44 52 45)
G:=sub<Sym(64)| (1,30)(2,31)(3,32)(4,29)(5,60)(6,57)(7,58)(8,59)(9,47)(10,48)(11,45)(12,46)(13,51)(14,52)(15,49)(16,50)(17,40)(18,37)(19,38)(20,39)(21,44)(22,41)(23,42)(24,43)(25,36)(26,33)(27,34)(28,35)(53,63)(54,64)(55,61)(56,62), (1,58)(2,59)(3,60)(4,57)(5,32)(6,29)(7,30)(8,31)(9,23)(10,24)(11,21)(12,22)(13,20)(14,17)(15,18)(16,19)(25,56)(26,53)(27,54)(28,55)(33,63)(34,64)(35,61)(36,62)(37,49)(38,50)(39,51)(40,52)(41,46)(42,47)(43,48)(44,45), (1,32)(2,29)(3,30)(4,31)(5,58)(6,59)(7,60)(8,57)(9,45)(10,46)(11,47)(12,48)(13,49)(14,50)(15,51)(16,52)(17,38)(18,39)(19,40)(20,37)(21,42)(22,43)(23,44)(24,41)(25,34)(26,35)(27,36)(28,33)(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), (1,22)(2,42)(3,24)(4,44)(5,48)(6,11)(7,46)(8,9)(10,60)(12,58)(13,28)(14,36)(15,26)(16,34)(17,62)(18,53)(19,64)(20,55)(21,29)(23,31)(25,52)(27,50)(30,41)(32,43)(33,49)(35,51)(37,63)(38,54)(39,61)(40,56)(45,57)(47,59), (1,50,30,16)(2,39,31,20)(3,52,32,14)(4,37,29,18)(5,17,60,40)(6,15,57,49)(7,19,58,38)(8,13,59,51)(9,33,47,26)(10,64,48,54)(11,35,45,28)(12,62,46,56)(21,61,44,55)(22,36,41,25)(23,63,42,53)(24,34,43,27), (1,26,58,53)(2,27,59,54)(3,28,60,55)(4,25,57,56)(5,61,32,35)(6,62,29,36)(7,63,30,33)(8,64,31,34)(9,19,23,16)(10,20,24,13)(11,17,21,14)(12,18,22,15)(37,41,49,46)(38,42,50,47)(39,43,51,48)(40,44,52,45)>;
G:=Group( (1,30)(2,31)(3,32)(4,29)(5,60)(6,57)(7,58)(8,59)(9,47)(10,48)(11,45)(12,46)(13,51)(14,52)(15,49)(16,50)(17,40)(18,37)(19,38)(20,39)(21,44)(22,41)(23,42)(24,43)(25,36)(26,33)(27,34)(28,35)(53,63)(54,64)(55,61)(56,62), (1,58)(2,59)(3,60)(4,57)(5,32)(6,29)(7,30)(8,31)(9,23)(10,24)(11,21)(12,22)(13,20)(14,17)(15,18)(16,19)(25,56)(26,53)(27,54)(28,55)(33,63)(34,64)(35,61)(36,62)(37,49)(38,50)(39,51)(40,52)(41,46)(42,47)(43,48)(44,45), (1,32)(2,29)(3,30)(4,31)(5,58)(6,59)(7,60)(8,57)(9,45)(10,46)(11,47)(12,48)(13,49)(14,50)(15,51)(16,52)(17,38)(18,39)(19,40)(20,37)(21,42)(22,43)(23,44)(24,41)(25,34)(26,35)(27,36)(28,33)(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), (1,22)(2,42)(3,24)(4,44)(5,48)(6,11)(7,46)(8,9)(10,60)(12,58)(13,28)(14,36)(15,26)(16,34)(17,62)(18,53)(19,64)(20,55)(21,29)(23,31)(25,52)(27,50)(30,41)(32,43)(33,49)(35,51)(37,63)(38,54)(39,61)(40,56)(45,57)(47,59), (1,50,30,16)(2,39,31,20)(3,52,32,14)(4,37,29,18)(5,17,60,40)(6,15,57,49)(7,19,58,38)(8,13,59,51)(9,33,47,26)(10,64,48,54)(11,35,45,28)(12,62,46,56)(21,61,44,55)(22,36,41,25)(23,63,42,53)(24,34,43,27), (1,26,58,53)(2,27,59,54)(3,28,60,55)(4,25,57,56)(5,61,32,35)(6,62,29,36)(7,63,30,33)(8,64,31,34)(9,19,23,16)(10,20,24,13)(11,17,21,14)(12,18,22,15)(37,41,49,46)(38,42,50,47)(39,43,51,48)(40,44,52,45) );
G=PermutationGroup([[(1,30),(2,31),(3,32),(4,29),(5,60),(6,57),(7,58),(8,59),(9,47),(10,48),(11,45),(12,46),(13,51),(14,52),(15,49),(16,50),(17,40),(18,37),(19,38),(20,39),(21,44),(22,41),(23,42),(24,43),(25,36),(26,33),(27,34),(28,35),(53,63),(54,64),(55,61),(56,62)], [(1,58),(2,59),(3,60),(4,57),(5,32),(6,29),(7,30),(8,31),(9,23),(10,24),(11,21),(12,22),(13,20),(14,17),(15,18),(16,19),(25,56),(26,53),(27,54),(28,55),(33,63),(34,64),(35,61),(36,62),(37,49),(38,50),(39,51),(40,52),(41,46),(42,47),(43,48),(44,45)], [(1,32),(2,29),(3,30),(4,31),(5,58),(6,59),(7,60),(8,57),(9,45),(10,46),(11,47),(12,48),(13,49),(14,50),(15,51),(16,52),(17,38),(18,39),(19,40),(20,37),(21,42),(22,43),(23,44),(24,41),(25,34),(26,35),(27,36),(28,33),(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)], [(1,22),(2,42),(3,24),(4,44),(5,48),(6,11),(7,46),(8,9),(10,60),(12,58),(13,28),(14,36),(15,26),(16,34),(17,62),(18,53),(19,64),(20,55),(21,29),(23,31),(25,52),(27,50),(30,41),(32,43),(33,49),(35,51),(37,63),(38,54),(39,61),(40,56),(45,57),(47,59)], [(1,50,30,16),(2,39,31,20),(3,52,32,14),(4,37,29,18),(5,17,60,40),(6,15,57,49),(7,19,58,38),(8,13,59,51),(9,33,47,26),(10,64,48,54),(11,35,45,28),(12,62,46,56),(21,61,44,55),(22,36,41,25),(23,63,42,53),(24,34,43,27)], [(1,26,58,53),(2,27,59,54),(3,28,60,55),(4,25,57,56),(5,61,32,35),(6,62,29,36),(7,63,30,33),(8,64,31,34),(9,19,23,16),(10,20,24,13),(11,17,21,14),(12,18,22,15),(37,41,49,46),(38,42,50,47),(39,43,51,48),(40,44,52,45)]])
32 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 4A | 4B | 4C | 4D | 4E | ··· | 4N | 4O | ··· | 4U |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 |
| size | 1 | 1 | ··· | 1 | 4 | 4 | 8 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 8 | ··· | 8 |
32 irreducible representations
| dim | 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 | C2 | C4○D4 | C4○D4 | 2+ 1+4 | 2- 1+4 |
| kernel | C23.553C24 | C4×C22⋊C4 | C23.8Q8 | C23.23D4 | C23.63C23 | C24.C22 | C23.67C23 | C23⋊Q8 | C23.10D4 | C23.11D4 | C23.81C23 | C23.84C23 | C2×C4 | C23 | C22 | C22 |
| # reps | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 2 | 2 | 2 | 1 | 1 | 8 | 4 | 3 | 1 |
Matrix representation of C23.553C24 ►in GL8(𝔽5)
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 4 |
| 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 4 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 3 | 0 | 0 | 0 | 0 | 0 | 0 |
| 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 |
| 0 | 0 | 0 | 0 | 3 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 3 |
| 0 | 0 | 0 | 0 | 0 | 0 | 2 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 | 0 | 0 |
| 3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 4 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(8,GF(5))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,2,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,2,0,0,0,0,3,0,0,0,0,0,0,0,0,2,0,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,3,3,0,0,0,0,0,0,4,2,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,3,0],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,2,1,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[3,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0] >;
C23.553C24 in GAP, Magma, Sage, TeX
C_2^3._{553}C_2^4 % in TeX
G:=Group("C2^3.553C2^4"); // GroupNames label
G:=SmallGroup(128,1385);
// by ID
G=gap.SmallGroup(128,1385);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,2,560,253,758,723,100,185,136]);
// Polycyclic
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=e^2=1,d^2=c*a=a*c,f^2=a,g^2=b,a*b=b*a,e*d*e=a*d=d*a,a*e=e*a,g*f*g^-1=a*f=f*a,a*g=g*a,b*c=c*b,f*d*f^-1=b*d=d*b,b*e=e*b,b*f=f*b,b*g=g*b,c*d=d*c,f*e*f^-1=c*e=e*c,c*f=f*c,c*g=g*c,d*g=g*d,e*g=g*e>;
// generators/relations