p-group, metabelian, nilpotent (class 2), monomial
Aliases: C23.32C24, C22.72C25, C42.569C23, C4.1872+ 1+4, C4○2Q82, Q82⋊16C2, Q8○3(C4×Q8), Q8⋊12(C4○D4), C4○3(Q8⋊6D4), Q8⋊6D4⋊34C2, C4⋊C4.481C23, (C2×C4).168C24, C4⋊Q8.338C22, (C4×D4).357C22, (C2×D4).462C23, C22⋊C4.94C23, (C4×Q8).325C22, (C2×Q8).487C23, C4⋊D4.222C22, C4⋊1D4.184C22, (C2×C42).939C22, C2.25(C2×2+ 1+4), (C22×C4).1205C23, C22.26C24⋊32C2, C4.4D4.173C22, C4○2(C22.49C24), C22.49C24⋊29C2, C42⋊C2.343C22, (C4×C4○D4)⋊25C2, C4.274(C2×C4○D4), C2.43(C22×C4○D4), (C2×C4○D4).327C22, SmallGroup(128,2215)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C22.72C25
G = < a,b,c,d,e,f,g | a2=b2=c2=1, d2=e2=f2=a, g2=ba=ab, dcd-1=ac=ca, fdf-1=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: 892 in 601 conjugacy classes, 400 normal (6 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C2×C4, C2×C4, C2×C4, D4, Q8, Q8, C23, C42, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×Q8, C4○D4, C2×C42, C42⋊C2, C4×D4, C4×Q8, C4⋊D4, C4.4D4, C4⋊1D4, C4⋊Q8, C2×C4○D4, C4×C4○D4, C22.26C24, Q8⋊6D4, C22.49C24, Q82, C22.72C25
Quotients: C1, C2, C22, C23, C4○D4, C24, C2×C4○D4, 2+ 1+4, C25, C22×C4○D4, C2×2+ 1+4, C22.72C25
►On 64 points
Generators in S64
(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 46)(2 45)(3 48)(4 47)(5 10)(6 9)(7 12)(8 11)(13 22)(14 21)(15 24)(16 23)(17 28)(18 27)(19 26)(20 25)(29 52)(30 51)(31 50)(32 49)(33 53)(34 56)(35 55)(36 54)(37 58)(38 57)(39 60)(40 59)(41 64)(42 63)(43 62)(44 61)
(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 13 3 15)(2 14 4 16)(5 41 7 43)(6 42 8 44)(9 19 11 17)(10 20 12 18)(21 31 23 29)(22 32 24 30)(25 34 27 36)(26 35 28 33)(37 47 39 45)(38 48 40 46)(49 59 51 57)(50 60 52 58)(53 63 55 61)(54 64 56 62)
(1 9 3 11)(2 12 4 10)(5 45 7 47)(6 48 8 46)(13 19 15 17)(14 18 16 20)(21 27 23 25)(22 26 24 28)(29 34 31 36)(30 33 32 35)(37 43 39 41)(38 42 40 44)(49 55 51 53)(50 54 52 56)(57 63 59 61)(58 62 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), (1,46)(2,45)(3,48)(4,47)(5,10)(6,9)(7,12)(8,11)(13,22)(14,21)(15,24)(16,23)(17,28)(18,27)(19,26)(20,25)(29,52)(30,51)(31,50)(32,49)(33,53)(34,56)(35,55)(36,54)(37,58)(38,57)(39,60)(40,59)(41,64)(42,63)(43,62)(44,61), (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), (1,13,3,15)(2,14,4,16)(5,41,7,43)(6,42,8,44)(9,19,11,17)(10,20,12,18)(21,31,23,29)(22,32,24,30)(25,34,27,36)(26,35,28,33)(37,47,39,45)(38,48,40,46)(49,59,51,57)(50,60,52,58)(53,63,55,61)(54,64,56,62), (1,9,3,11)(2,12,4,10)(5,45,7,47)(6,48,8,46)(13,19,15,17)(14,18,16,20)(21,27,23,25)(22,26,24,28)(29,34,31,36)(30,33,32,35)(37,43,39,41)(38,42,40,44)(49,55,51,53)(50,54,52,56)(57,63,59,61)(58,62,60,64), (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), (1,46)(2,45)(3,48)(4,47)(5,10)(6,9)(7,12)(8,11)(13,22)(14,21)(15,24)(16,23)(17,28)(18,27)(19,26)(20,25)(29,52)(30,51)(31,50)(32,49)(33,53)(34,56)(35,55)(36,54)(37,58)(38,57)(39,60)(40,59)(41,64)(42,63)(43,62)(44,61), (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), (1,13,3,15)(2,14,4,16)(5,41,7,43)(6,42,8,44)(9,19,11,17)(10,20,12,18)(21,31,23,29)(22,32,24,30)(25,34,27,36)(26,35,28,33)(37,47,39,45)(38,48,40,46)(49,59,51,57)(50,60,52,58)(53,63,55,61)(54,64,56,62), (1,9,3,11)(2,12,4,10)(5,45,7,47)(6,48,8,46)(13,19,15,17)(14,18,16,20)(21,27,23,25)(22,26,24,28)(29,34,31,36)(30,33,32,35)(37,43,39,41)(38,42,40,44)(49,55,51,53)(50,54,52,56)(57,63,59,61)(58,62,60,64), (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)], [(1,46),(2,45),(3,48),(4,47),(5,10),(6,9),(7,12),(8,11),(13,22),(14,21),(15,24),(16,23),(17,28),(18,27),(19,26),(20,25),(29,52),(30,51),(31,50),(32,49),(33,53),(34,56),(35,55),(36,54),(37,58),(38,57),(39,60),(40,59),(41,64),(42,63),(43,62),(44,61)], [(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)], [(1,13,3,15),(2,14,4,16),(5,41,7,43),(6,42,8,44),(9,19,11,17),(10,20,12,18),(21,31,23,29),(22,32,24,30),(25,34,27,36),(26,35,28,33),(37,47,39,45),(38,48,40,46),(49,59,51,57),(50,60,52,58),(53,63,55,61),(54,64,56,62)], [(1,9,3,11),(2,12,4,10),(5,45,7,47),(6,48,8,46),(13,19,15,17),(14,18,16,20),(21,27,23,25),(22,26,24,28),(29,34,31,36),(30,33,32,35),(37,43,39,41),(38,42,40,44),(49,55,51,53),(50,54,52,56),(57,63,59,61),(58,62,60,64)], [(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 | ··· | 2L | 4A | 4B | 4C | 4D | 4E | ··· | 4AB | 4AC | ··· | 4AK |
| order | 1 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 |
| size | 1 | 1 | 1 | 1 | 4 | ··· | 4 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 4 | ··· | 4 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 4 |
| type | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4○D4 | 2+ 1+4 |
| kernel | C22.72C25 | C4×C4○D4 | C22.26C24 | Q8⋊6D4 | C22.49C24 | Q82 | Q8 | C4 |
| # reps | 1 | 6 | 9 | 6 | 9 | 1 | 16 | 2 |
Matrix representation of C22.72C25 ►in GL4(𝔽5) generated by
| 4 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 4 |
| 3 | 1 | 0 | 0 |
| 2 | 2 | 0 | 0 |
| 0 | 0 | 0 | 2 |
| 0 | 0 | 3 | 0 |
| 3 | 1 | 0 | 0 |
| 0 | 2 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 4 |
| 2 | 0 | 0 | 0 |
| 0 | 2 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
| 4 | 3 | 0 | 0 |
| 1 | 1 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 4 |
| 3 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 |
| 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 3 |
G:=sub<GL(4,GF(5))| [4,0,0,0,0,4,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,4,0,0,0,0,4],[3,2,0,0,1,2,0,0,0,0,0,3,0,0,2,0],[3,0,0,0,1,2,0,0,0,0,4,0,0,0,0,4],[2,0,0,0,0,2,0,0,0,0,0,1,0,0,1,0],[4,1,0,0,3,1,0,0,0,0,4,0,0,0,0,4],[3,0,0,0,0,3,0,0,0,0,3,0,0,0,0,3] >;
C22.72C25 in GAP, Magma, Sage, TeX
C_2^2._{72}C_2^5 % in TeX
G:=Group("C2^2.72C2^5"); // GroupNames label
G:=SmallGroup(128,2215);
// by ID
G=gap.SmallGroup(128,2215);
# by ID
G:=PCGroup([7,-2,2,2,2,2,-2,2,477,232,1430,184,570,136,242]);
// Polycyclic
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=1,d^2=e^2=f^2=a,g^2=b*a=a*b,d*c*d^-1=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=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