p-group, metabelian, nilpotent (class 2), monomial
Aliases: C23.31C24, C22.71C25, C42.568C23, C4.1322- 1+4, Q8○3(C4×D4), D4○3(C4×Q8), C4○3(D4×Q8), (D4×Q8)⋊35C2, D4⋊12(C4○D4), C4○3(D4⋊6D4), D4⋊6D4⋊48C2, C4○3(Q8⋊5D4), C4○3(Q8⋊3Q8), Q8⋊11(C4○D4), Q8⋊5D4⋊38C2, Q8⋊3Q8⋊34C2, C4⋊C4.480C23, (C2×C4).167C24, C4⋊Q8.337C22, (C2×D4).504C23, (C4×D4).356C22, C22⋊C4.93C23, (C2×Q8).438C23, (C4×Q8).324C22, C4⋊D4.221C22, (C2×C42).938C22, C22⋊Q8.227C22, C2.17(C2×2- 1+4), C42⋊2C2.15C22, (C22×C4).1204C23, C4.4D4.172C22, C42.C2.150C22, (C22×Q8).498C22, C4○2(C22.46C24), C4○3(C22.50C24), C23.36C23⋊24C2, C22.50C24⋊46C2, C42⋊C2.222C22, C22.46C24⋊45C2, C23.37C23⋊33C2, C22.D4.28C22, (C2×C4×Q8)⋊58C2, (C4×C4○D4)⋊24C2, C4.273(C2×C4○D4), C22.40(C2×C4○D4), C2.42(C22×C4○D4), (C2×C4⋊C4).962C22, (C2×C4○D4).326C22, SmallGroup(128,2214)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C22.71C25
G = < a,b,c,d,e,f,g | a2=b2=d2=f2=1, c2=e2=a, g2=ba=ab, dcd=ac=ca, fdf=ad=da, ae=ea, af=fa, ag=ga, ece-1=bc=cb, bd=db, be=eb, bf=fb, bg=gb, cf=fc, cg=gc, de=ed, dg=gd, ef=fe, eg=ge, fg=gf >
Subgroups: 700 in 547 conjugacy classes, 400 normal (20 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C2×C4, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C2×Q8, C4○D4, C2×C42, C2×C4⋊C4, C42⋊C2, C4×D4, C4×D4, C4×Q8, C4×Q8, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C42.C2, C42⋊2C2, C4⋊Q8, C22×Q8, C2×C4○D4, C2×C4○D4, C2×C4×Q8, C4×C4○D4, C4×C4○D4, C23.36C23, C23.37C23, D4⋊6D4, Q8⋊5D4, D4×Q8, C22.46C24, C22.50C24, Q8⋊3Q8, C22.71C25
Quotients: C1, C2, C22, C23, C4○D4, C24, C2×C4○D4, 2- 1+4, C25, C22×C4○D4, C2×2- 1+4, C22.71C25
►On 64 points
(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 51)(2 52)(3 49)(4 50)(5 36)(6 33)(7 34)(8 35)(9 53)(10 54)(11 55)(12 56)(13 57)(14 58)(15 59)(16 60)(17 61)(18 62)(19 63)(20 64)(21 37)(22 38)(23 39)(24 40)(25 41)(26 42)(27 43)(28 44)(29 45)(30 46)(31 47)(32 48)
(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 19)(2 18)(3 17)(4 20)(5 37)(6 40)(7 39)(8 38)(9 15)(10 14)(11 13)(12 16)(21 36)(22 35)(23 34)(24 33)(25 31)(26 30)(27 29)(28 32)(41 47)(42 46)(43 45)(44 48)(49 61)(50 64)(51 63)(52 62)(53 59)(54 58)(55 57)(56 60)
(1 13 3 15)(2 58 4 60)(5 41 7 43)(6 26 8 28)(9 19 11 17)(10 64 12 62)(14 50 16 52)(18 54 20 56)(21 31 23 29)(22 48 24 46)(25 34 27 36)(30 38 32 40)(33 42 35 44)(37 47 39 45)(49 59 51 57)(53 63 55 61)
(1 11)(2 12)(3 9)(4 10)(5 45)(6 46)(7 47)(8 48)(13 17)(14 18)(15 19)(16 20)(21 25)(22 26)(23 27)(24 28)(29 36)(30 33)(31 34)(32 35)(37 41)(38 42)(39 43)(40 44)(49 53)(50 54)(51 55)(52 56)(57 61)(58 62)(59 63)(60 64)
(1 21 49 39)(2 22 50 40)(3 23 51 37)(4 24 52 38)(5 17 34 63)(6 18 35 64)(7 19 36 61)(8 20 33 62)(9 27 55 41)(10 28 56 42)(11 25 53 43)(12 26 54 44)(13 31 59 45)(14 32 60 46)(15 29 57 47)(16 30 58 48)
G:=sub<Sym(64)| (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), (1,51)(2,52)(3,49)(4,50)(5,36)(6,33)(7,34)(8,35)(9,53)(10,54)(11,55)(12,56)(13,57)(14,58)(15,59)(16,60)(17,61)(18,62)(19,63)(20,64)(21,37)(22,38)(23,39)(24,40)(25,41)(26,42)(27,43)(28,44)(29,45)(30,46)(31,47)(32,48), (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), (1,19)(2,18)(3,17)(4,20)(5,37)(6,40)(7,39)(8,38)(9,15)(10,14)(11,13)(12,16)(21,36)(22,35)(23,34)(24,33)(25,31)(26,30)(27,29)(28,32)(41,47)(42,46)(43,45)(44,48)(49,61)(50,64)(51,63)(52,62)(53,59)(54,58)(55,57)(56,60), (1,13,3,15)(2,58,4,60)(5,41,7,43)(6,26,8,28)(9,19,11,17)(10,64,12,62)(14,50,16,52)(18,54,20,56)(21,31,23,29)(22,48,24,46)(25,34,27,36)(30,38,32,40)(33,42,35,44)(37,47,39,45)(49,59,51,57)(53,63,55,61), (1,11)(2,12)(3,9)(4,10)(5,45)(6,46)(7,47)(8,48)(13,17)(14,18)(15,19)(16,20)(21,25)(22,26)(23,27)(24,28)(29,36)(30,33)(31,34)(32,35)(37,41)(38,42)(39,43)(40,44)(49,53)(50,54)(51,55)(52,56)(57,61)(58,62)(59,63)(60,64), (1,21,49,39)(2,22,50,40)(3,23,51,37)(4,24,52,38)(5,17,34,63)(6,18,35,64)(7,19,36,61)(8,20,33,62)(9,27,55,41)(10,28,56,42)(11,25,53,43)(12,26,54,44)(13,31,59,45)(14,32,60,46)(15,29,57,47)(16,30,58,48)>;
G:=Group( (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), (1,51)(2,52)(3,49)(4,50)(5,36)(6,33)(7,34)(8,35)(9,53)(10,54)(11,55)(12,56)(13,57)(14,58)(15,59)(16,60)(17,61)(18,62)(19,63)(20,64)(21,37)(22,38)(23,39)(24,40)(25,41)(26,42)(27,43)(28,44)(29,45)(30,46)(31,47)(32,48), (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), (1,19)(2,18)(3,17)(4,20)(5,37)(6,40)(7,39)(8,38)(9,15)(10,14)(11,13)(12,16)(21,36)(22,35)(23,34)(24,33)(25,31)(26,30)(27,29)(28,32)(41,47)(42,46)(43,45)(44,48)(49,61)(50,64)(51,63)(52,62)(53,59)(54,58)(55,57)(56,60), (1,13,3,15)(2,58,4,60)(5,41,7,43)(6,26,8,28)(9,19,11,17)(10,64,12,62)(14,50,16,52)(18,54,20,56)(21,31,23,29)(22,48,24,46)(25,34,27,36)(30,38,32,40)(33,42,35,44)(37,47,39,45)(49,59,51,57)(53,63,55,61), (1,11)(2,12)(3,9)(4,10)(5,45)(6,46)(7,47)(8,48)(13,17)(14,18)(15,19)(16,20)(21,25)(22,26)(23,27)(24,28)(29,36)(30,33)(31,34)(32,35)(37,41)(38,42)(39,43)(40,44)(49,53)(50,54)(51,55)(52,56)(57,61)(58,62)(59,63)(60,64), (1,21,49,39)(2,22,50,40)(3,23,51,37)(4,24,52,38)(5,17,34,63)(6,18,35,64)(7,19,36,61)(8,20,33,62)(9,27,55,41)(10,28,56,42)(11,25,53,43)(12,26,54,44)(13,31,59,45)(14,32,60,46)(15,29,57,47)(16,30,58,48) );
G=PermutationGroup([[(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)], [(1,51),(2,52),(3,49),(4,50),(5,36),(6,33),(7,34),(8,35),(9,53),(10,54),(11,55),(12,56),(13,57),(14,58),(15,59),(16,60),(17,61),(18,62),(19,63),(20,64),(21,37),(22,38),(23,39),(24,40),(25,41),(26,42),(27,43),(28,44),(29,45),(30,46),(31,47),(32,48)], [(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)], [(1,19),(2,18),(3,17),(4,20),(5,37),(6,40),(7,39),(8,38),(9,15),(10,14),(11,13),(12,16),(21,36),(22,35),(23,34),(24,33),(25,31),(26,30),(27,29),(28,32),(41,47),(42,46),(43,45),(44,48),(49,61),(50,64),(51,63),(52,62),(53,59),(54,58),(55,57),(56,60)], [(1,13,3,15),(2,58,4,60),(5,41,7,43),(6,26,8,28),(9,19,11,17),(10,64,12,62),(14,50,16,52),(18,54,20,56),(21,31,23,29),(22,48,24,46),(25,34,27,36),(30,38,32,40),(33,42,35,44),(37,47,39,45),(49,59,51,57),(53,63,55,61)], [(1,11),(2,12),(3,9),(4,10),(5,45),(6,46),(7,47),(8,48),(13,17),(14,18),(15,19),(16,20),(21,25),(22,26),(23,27),(24,28),(29,36),(30,33),(31,34),(32,35),(37,41),(38,42),(39,43),(40,44),(49,53),(50,54),(51,55),(52,56),(57,61),(58,62),(59,63),(60,64)], [(1,21,49,39),(2,22,50,40),(3,23,51,37),(4,24,52,38),(5,17,34,63),(6,18,35,64),(7,19,36,61),(8,20,33,62),(9,27,55,41),(10,28,56,42),(11,25,53,43),(12,26,54,44),(13,31,59,45),(14,32,60,46),(15,29,57,47),(16,30,58,48)]])
50 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 4A | 4B | 4C | 4D | 4E | ··· | 4X | 4Y | ··· | 4AM |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 4 | ··· | 4 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4○D4 | C4○D4 | 2- 1+4 |
| kernel | C22.71C25 | C2×C4×Q8 | C4×C4○D4 | C23.36C23 | C23.37C23 | D4⋊6D4 | Q8⋊5D4 | D4×Q8 | C22.46C24 | C22.50C24 | Q8⋊3Q8 | D4 | Q8 | C4 |
| # reps | 1 | 2 | 4 | 6 | 3 | 3 | 2 | 1 | 6 | 3 | 1 | 8 | 8 | 2 |
Matrix representation of C22.71C25 ►in GL4(𝔽5) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 4 |
| 4 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 2 | 4 |
| 0 | 0 | 0 | 3 |
| 4 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 2 | 4 |
| 0 | 0 | 3 | 3 |
| 4 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 2 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 2 |
| 0 | 0 | 0 | 4 |
| 2 | 0 | 0 | 0 |
| 0 | 2 | 0 | 0 |
| 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 2 |
G:=sub<GL(4,GF(5))| [1,0,0,0,0,1,0,0,0,0,4,0,0,0,0,4],[4,0,0,0,0,4,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,2,0,0,0,4,3],[4,0,0,0,0,4,0,0,0,0,2,3,0,0,4,3],[4,0,0,0,0,1,0,0,0,0,2,0,0,0,0,2],[1,0,0,0,0,1,0,0,0,0,1,0,0,0,2,4],[2,0,0,0,0,2,0,0,0,0,2,0,0,0,0,2] >;
C22.71C25 in GAP, Magma, Sage, TeX
C_2^2._{71}C_2^5 % in TeX
G:=Group("C2^2.71C2^5"); // GroupNames label
G:=SmallGroup(128,2214);
// by ID
G=gap.SmallGroup(128,2214);
# by ID
G:=PCGroup([7,-2,2,2,2,2,-2,2,224,477,1430,184,570,242]);
// Polycyclic
G:=Group<a,b,c,d,e,f,g|a^2=b^2=d^2=f^2=1,c^2=e^2=a,g^2=b*a=a*b,d*c*d=a*c=c*a,f*d*f=a*d=d*a,a*e=e*a,a*f=f*a,a*g=g*a,e*c*e^-1=b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,b*g=g*b,c*f=f*c,c*g=g*c,d*e=e*d,d*g=g*d,e*f=f*e,e*g=g*e,f*g=g*f>;
// generators/relations