p-group, metabelian, nilpotent (class 2), monomial
Aliases: C23.83C24, C42.125C23, C22.142C25, C22.52- 1+4, D4⋊3Q8⋊39C2, C4⋊C4.327C23, (C2×C4).132C24, C4⋊Q8.227C22, (C2×D4).333C23, (C4×D4).252C22, C22⋊C4.56C23, (C2×Q8).311C23, (C4×Q8).238C22, C4⋊D4.236C22, (C2×C42).970C22, (C22×C4).402C23, C22⋊Q8.126C22, C2.48(C2×2- 1+4), C2.54(C2.C25), C42⋊2C2.24C22, C22.58C24⋊2C2, C4.4D4.182C22, C42.C2.166C22, C42⋊C2.246C22, C23.36C23⋊53C2, C22.35C24⋊20C2, C22.47C24⋊36C2, C22.56C24⋊11C2, C22.46C24⋊36C2, C22.57C24⋊13C2, C22.33C24⋊18C2, C22.D4.18C22, (C2×C42.C2)⋊50C2, (C2×C4⋊C4).723C22, SmallGroup(128,2285)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C22.142C25
G = < a,b,c,d,e,f,g | a2=b2=c2=g2=1, d2=b, e2=ba=ab, f2=a, dcd-1=gcg=ac=ca, fdf-1=ad=da, ae=ea, af=fa, ag=ga, ece-1=fcf-1=bc=cb, ede-1=bd=db, be=eb, bf=fb, bg=gb, dg=gd, ef=fe, eg=ge, fg=gf >
Subgroups: 620 in 466 conjugacy classes, 380 normal (16 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C42, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×Q8, C2×C42, C2×C4⋊C4, C42⋊C2, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C42.C2, C42.C2, C42⋊2C2, C4⋊Q8, C2×C42.C2, C23.36C23, C22.33C24, C22.35C24, C22.46C24, C22.47C24, D4⋊3Q8, C22.56C24, C22.57C24, C22.58C24, C22.142C25
Quotients: C1, C2, C22, C23, C24, 2- 1+4, C25, C2×2- 1+4, C2.C25, C22.142C25
►On 64 points
Generators in S64
(1 51)(2 52)(3 49)(4 50)(5 36)(6 33)(7 34)(8 35)(9 21)(10 22)(11 23)(12 24)(13 25)(14 26)(15 27)(16 28)(17 29)(18 30)(19 31)(20 32)(37 61)(38 62)(39 63)(40 64)(41 53)(42 54)(43 55)(44 56)(45 57)(46 58)(47 59)(48 60)
(1 3)(2 4)(5 7)(6 8)(9 11)(10 12)(13 15)(14 16)(17 19)(18 20)(21 23)(22 24)(25 27)(26 28)(29 31)(30 32)(33 35)(34 36)(37 39)(38 40)(41 43)(42 44)(45 47)(46 48)(49 51)(50 52)(53 55)(54 56)(57 59)(58 60)(61 63)(62 64)
(1 19)(2 32)(3 17)(4 30)(5 56)(6 41)(7 54)(8 43)(9 57)(10 46)(11 59)(12 48)(13 63)(14 40)(15 61)(16 38)(18 50)(20 52)(21 45)(22 58)(23 47)(24 60)(25 39)(26 64)(27 37)(28 62)(29 49)(31 51)(33 53)(34 42)(35 55)(36 44)
(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 57 49 47)(2 60 50 46)(3 59 51 45)(4 58 52 48)(5 16 34 26)(6 15 35 25)(7 14 36 28)(8 13 33 27)(9 31 23 17)(10 30 24 20)(11 29 21 19)(12 32 22 18)(37 41 63 55)(38 44 64 54)(39 43 61 53)(40 42 62 56)
(1 55 51 43)(2 44 52 56)(3 53 49 41)(4 42 50 54)(5 30 36 18)(6 19 33 31)(7 32 34 20)(8 17 35 29)(9 25 21 13)(10 14 22 26)(11 27 23 15)(12 16 24 28)(37 45 61 57)(38 58 62 46)(39 47 63 59)(40 60 64 48)
(1 11)(2 12)(3 9)(4 10)(5 38)(6 39)(7 40)(8 37)(13 41)(14 42)(15 43)(16 44)(17 45)(18 46)(19 47)(20 48)(21 49)(22 50)(23 51)(24 52)(25 53)(26 54)(27 55)(28 56)(29 57)(30 58)(31 59)(32 60)(33 63)(34 64)(35 61)(36 62)
G:=sub<Sym(64)| (1,51)(2,52)(3,49)(4,50)(5,36)(6,33)(7,34)(8,35)(9,21)(10,22)(11,23)(12,24)(13,25)(14,26)(15,27)(16,28)(17,29)(18,30)(19,31)(20,32)(37,61)(38,62)(39,63)(40,64)(41,53)(42,54)(43,55)(44,56)(45,57)(46,58)(47,59)(48,60), (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,15)(14,16)(17,19)(18,20)(21,23)(22,24)(25,27)(26,28)(29,31)(30,32)(33,35)(34,36)(37,39)(38,40)(41,43)(42,44)(45,47)(46,48)(49,51)(50,52)(53,55)(54,56)(57,59)(58,60)(61,63)(62,64), (1,19)(2,32)(3,17)(4,30)(5,56)(6,41)(7,54)(8,43)(9,57)(10,46)(11,59)(12,48)(13,63)(14,40)(15,61)(16,38)(18,50)(20,52)(21,45)(22,58)(23,47)(24,60)(25,39)(26,64)(27,37)(28,62)(29,49)(31,51)(33,53)(34,42)(35,55)(36,44), (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), (1,57,49,47)(2,60,50,46)(3,59,51,45)(4,58,52,48)(5,16,34,26)(6,15,35,25)(7,14,36,28)(8,13,33,27)(9,31,23,17)(10,30,24,20)(11,29,21,19)(12,32,22,18)(37,41,63,55)(38,44,64,54)(39,43,61,53)(40,42,62,56), (1,55,51,43)(2,44,52,56)(3,53,49,41)(4,42,50,54)(5,30,36,18)(6,19,33,31)(7,32,34,20)(8,17,35,29)(9,25,21,13)(10,14,22,26)(11,27,23,15)(12,16,24,28)(37,45,61,57)(38,58,62,46)(39,47,63,59)(40,60,64,48), (1,11)(2,12)(3,9)(4,10)(5,38)(6,39)(7,40)(8,37)(13,41)(14,42)(15,43)(16,44)(17,45)(18,46)(19,47)(20,48)(21,49)(22,50)(23,51)(24,52)(25,53)(26,54)(27,55)(28,56)(29,57)(30,58)(31,59)(32,60)(33,63)(34,64)(35,61)(36,62)>;
G:=Group( (1,51)(2,52)(3,49)(4,50)(5,36)(6,33)(7,34)(8,35)(9,21)(10,22)(11,23)(12,24)(13,25)(14,26)(15,27)(16,28)(17,29)(18,30)(19,31)(20,32)(37,61)(38,62)(39,63)(40,64)(41,53)(42,54)(43,55)(44,56)(45,57)(46,58)(47,59)(48,60), (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,15)(14,16)(17,19)(18,20)(21,23)(22,24)(25,27)(26,28)(29,31)(30,32)(33,35)(34,36)(37,39)(38,40)(41,43)(42,44)(45,47)(46,48)(49,51)(50,52)(53,55)(54,56)(57,59)(58,60)(61,63)(62,64), (1,19)(2,32)(3,17)(4,30)(5,56)(6,41)(7,54)(8,43)(9,57)(10,46)(11,59)(12,48)(13,63)(14,40)(15,61)(16,38)(18,50)(20,52)(21,45)(22,58)(23,47)(24,60)(25,39)(26,64)(27,37)(28,62)(29,49)(31,51)(33,53)(34,42)(35,55)(36,44), (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), (1,57,49,47)(2,60,50,46)(3,59,51,45)(4,58,52,48)(5,16,34,26)(6,15,35,25)(7,14,36,28)(8,13,33,27)(9,31,23,17)(10,30,24,20)(11,29,21,19)(12,32,22,18)(37,41,63,55)(38,44,64,54)(39,43,61,53)(40,42,62,56), (1,55,51,43)(2,44,52,56)(3,53,49,41)(4,42,50,54)(5,30,36,18)(6,19,33,31)(7,32,34,20)(8,17,35,29)(9,25,21,13)(10,14,22,26)(11,27,23,15)(12,16,24,28)(37,45,61,57)(38,58,62,46)(39,47,63,59)(40,60,64,48), (1,11)(2,12)(3,9)(4,10)(5,38)(6,39)(7,40)(8,37)(13,41)(14,42)(15,43)(16,44)(17,45)(18,46)(19,47)(20,48)(21,49)(22,50)(23,51)(24,52)(25,53)(26,54)(27,55)(28,56)(29,57)(30,58)(31,59)(32,60)(33,63)(34,64)(35,61)(36,62) );
G=PermutationGroup([[(1,51),(2,52),(3,49),(4,50),(5,36),(6,33),(7,34),(8,35),(9,21),(10,22),(11,23),(12,24),(13,25),(14,26),(15,27),(16,28),(17,29),(18,30),(19,31),(20,32),(37,61),(38,62),(39,63),(40,64),(41,53),(42,54),(43,55),(44,56),(45,57),(46,58),(47,59),(48,60)], [(1,3),(2,4),(5,7),(6,8),(9,11),(10,12),(13,15),(14,16),(17,19),(18,20),(21,23),(22,24),(25,27),(26,28),(29,31),(30,32),(33,35),(34,36),(37,39),(38,40),(41,43),(42,44),(45,47),(46,48),(49,51),(50,52),(53,55),(54,56),(57,59),(58,60),(61,63),(62,64)], [(1,19),(2,32),(3,17),(4,30),(5,56),(6,41),(7,54),(8,43),(9,57),(10,46),(11,59),(12,48),(13,63),(14,40),(15,61),(16,38),(18,50),(20,52),(21,45),(22,58),(23,47),(24,60),(25,39),(26,64),(27,37),(28,62),(29,49),(31,51),(33,53),(34,42),(35,55),(36,44)], [(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)], [(1,57,49,47),(2,60,50,46),(3,59,51,45),(4,58,52,48),(5,16,34,26),(6,15,35,25),(7,14,36,28),(8,13,33,27),(9,31,23,17),(10,30,24,20),(11,29,21,19),(12,32,22,18),(37,41,63,55),(38,44,64,54),(39,43,61,53),(40,42,62,56)], [(1,55,51,43),(2,44,52,56),(3,53,49,41),(4,42,50,54),(5,30,36,18),(6,19,33,31),(7,32,34,20),(8,17,35,29),(9,25,21,13),(10,14,22,26),(11,27,23,15),(12,16,24,28),(37,45,61,57),(38,58,62,46),(39,47,63,59),(40,60,64,48)], [(1,11),(2,12),(3,9),(4,10),(5,38),(6,39),(7,40),(8,37),(13,41),(14,42),(15,43),(16,44),(17,45),(18,46),(19,47),(20,48),(21,49),(22,50),(23,51),(24,52),(25,53),(26,54),(27,55),(28,56),(29,57),(30,58),(31,59),(32,60),(33,63),(34,64),(35,61),(36,62)]])
38 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | ··· | 4AB |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | 2- 1+4 | C2.C25 |
| kernel | C22.142C25 | C2×C42.C2 | C23.36C23 | C22.33C24 | C22.35C24 | C22.46C24 | C22.47C24 | D4⋊3Q8 | C22.56C24 | C22.57C24 | C22.58C24 | C22 | C2 |
| # reps | 1 | 1 | 2 | 8 | 4 | 4 | 4 | 4 | 1 | 2 | 1 | 2 | 4 |
Matrix representation of C22.142C25 ►in GL8(𝔽5)
| 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 | 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 |
| 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 |
| 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 |
| 3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 2 | 2 | 4 | 0 | 0 | 0 | 0 |
| 1 | 4 | 3 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 4 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 4 | 3 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 2 | 0 | 3 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 |
| 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 |
| 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 2 | 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 | 2 | 0 | 0 | 0 | 0 | 0 |
| 0 | 2 | 0 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 4 | 3 | 0 | 0 | 0 | 0 |
| 2 | 3 | 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 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(8,GF(5))| [4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[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],[0,3,0,1,0,0,0,0,2,0,2,4,0,0,0,0,0,0,2,3,0,0,0,0,0,0,4,3,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0],[0,0,1,2,0,0,0,0,0,4,0,0,0,0,0,0,1,4,0,3,0,0,0,0,0,3,0,1,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,2,0,0,0,0,2,0,0,0,0,0,0,0,0,2,0,0],[2,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[3,0,0,0,0,0,0,0,0,3,0,2,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0],[0,1,0,2,0,0,0,0,1,0,4,3,0,0,0,0,0,0,4,0,0,0,0,0,0,0,3,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1] >;
C22.142C25 in GAP, Magma, Sage, TeX
C_2^2._{142}C_2^5 % in TeX
G:=Group("C2^2.142C2^5"); // GroupNames label
G:=SmallGroup(128,2285);
// by ID
G=gap.SmallGroup(128,2285);
# by ID
G:=PCGroup([7,-2,2,2,2,2,-2,2,477,456,1430,723,520,2019,570,136,1684]);
// Polycyclic
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=g^2=1,d^2=b,e^2=b*a=a*b,f^2=a,d*c*d^-1=g*c*g=a*c=c*a,f*d*f^-1=a*d=d*a,a*e=e*a,a*f=f*a,a*g=g*a,e*c*e^-1=f*c*f^-1=b*c=c*b,e*d*e^-1=b*d=d*b,b*e=e*b,b*f=f*b,b*g=g*b,d*g=g*d,e*f=f*e,e*g=g*e,f*g=g*f>;
// generators/relations