p-group, metabelian, nilpotent (class 2), monomial
Aliases: C23.375C24, C24.292C23, C22.1332- 1+4, C4⋊C4.334D4, C2.36(D4⋊6D4), C2.30(Q8⋊5D4), (C2×C42).34C22, C23.Q8.6C2, C23.142(C4○D4), (C23×C4).363C22, (C22×C4).519C23, C23.11D4.4C2, C22.255(C22×D4), C23.8Q8.17C2, C23.65C23⋊64C2, C23.63C23⋊53C2, C23.83C23⋊13C2, C23.81C23⋊19C2, C2.47(C22.19C24), C24.C22.16C2, C2.C42.131C22, C2.41(C23.36C23), C2.27(C22.46C24), C2.14(C22.35C24), (C4×C4⋊C4)⋊64C2, (C2×C4).344(C2×D4), (C2×C42.C2)⋊7C2, (C2×C4).370(C4○D4), (C2×C4⋊C4).852C22, C22.252(C2×C4○D4), (C2×C42⋊C2).39C2, (C2×C22⋊C4).145C22, SmallGroup(128,1207)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C23.375C24
G = < a,b,c,d,e,f,g | a2=b2=c2=1, d2=f2=a, e2=ba=ab, g2=b, ac=ca, ede-1=gdg-1=ad=da, ae=ea, 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, eg=ge, fg=gf >
Subgroups: 404 in 238 conjugacy classes, 100 normal (82 characteristic)
C1, C2, C2, C4, C22, C22, C2×C4, C2×C4, C23, C23, C23, C42, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C24, C2.C42, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C42⋊C2, C42.C2, C23×C4, C4×C4⋊C4, C23.8Q8, C23.63C23, C24.C22, C23.65C23, C23.Q8, C23.11D4, C23.81C23, C23.83C23, C2×C42⋊C2, C2×C42.C2, C23.375C24
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C22×D4, C2×C4○D4, 2- 1+4, C22.19C24, C23.36C23, C22.35C24, D4⋊6D4, Q8⋊5D4, C22.46C24, C23.375C24
►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 49)(2 50)(3 51)(4 52)(5 28)(6 25)(7 26)(8 27)(9 56)(10 53)(11 54)(12 55)(13 60)(14 57)(15 58)(16 59)(17 34)(18 35)(19 36)(20 33)(21 47)(22 48)(23 45)(24 46)(29 61)(30 62)(31 63)(32 64)(37 41)(38 42)(39 43)(40 44)
(1 35)(2 36)(3 33)(4 34)(5 30)(6 31)(7 32)(8 29)(9 58)(10 59)(11 60)(12 57)(13 54)(14 55)(15 56)(16 53)(17 52)(18 49)(19 50)(20 51)(21 41)(22 42)(23 43)(24 44)(25 63)(26 64)(27 61)(28 62)(37 47)(38 48)(39 45)(40 46)
(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 11 51 56)(2 10 52 55)(3 9 49 54)(4 12 50 53)(5 46 26 22)(6 45 27 21)(7 48 28 24)(8 47 25 23)(13 33 58 18)(14 36 59 17)(15 35 60 20)(16 34 57 19)(29 37 63 43)(30 40 64 42)(31 39 61 41)(32 38 62 44)
(1 44 3 42)(2 37 4 39)(5 11 7 9)(6 55 8 53)(10 25 12 27)(13 64 15 62)(14 29 16 31)(17 23 19 21)(18 46 20 48)(22 35 24 33)(26 56 28 54)(30 60 32 58)(34 45 36 47)(38 49 40 51)(41 52 43 50)(57 61 59 63)
(1 53 49 10)(2 56 50 9)(3 55 51 12)(4 54 52 11)(5 37 28 41)(6 40 25 44)(7 39 26 43)(8 38 27 42)(13 17 60 34)(14 20 57 33)(15 19 58 36)(16 18 59 35)(21 30 47 62)(22 29 48 61)(23 32 45 64)(24 31 46 63)
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), (1,49)(2,50)(3,51)(4,52)(5,28)(6,25)(7,26)(8,27)(9,56)(10,53)(11,54)(12,55)(13,60)(14,57)(15,58)(16,59)(17,34)(18,35)(19,36)(20,33)(21,47)(22,48)(23,45)(24,46)(29,61)(30,62)(31,63)(32,64)(37,41)(38,42)(39,43)(40,44), (1,35)(2,36)(3,33)(4,34)(5,30)(6,31)(7,32)(8,29)(9,58)(10,59)(11,60)(12,57)(13,54)(14,55)(15,56)(16,53)(17,52)(18,49)(19,50)(20,51)(21,41)(22,42)(23,43)(24,44)(25,63)(26,64)(27,61)(28,62)(37,47)(38,48)(39,45)(40,46), (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), (1,11,51,56)(2,10,52,55)(3,9,49,54)(4,12,50,53)(5,46,26,22)(6,45,27,21)(7,48,28,24)(8,47,25,23)(13,33,58,18)(14,36,59,17)(15,35,60,20)(16,34,57,19)(29,37,63,43)(30,40,64,42)(31,39,61,41)(32,38,62,44), (1,44,3,42)(2,37,4,39)(5,11,7,9)(6,55,8,53)(10,25,12,27)(13,64,15,62)(14,29,16,31)(17,23,19,21)(18,46,20,48)(22,35,24,33)(26,56,28,54)(30,60,32,58)(34,45,36,47)(38,49,40,51)(41,52,43,50)(57,61,59,63), (1,53,49,10)(2,56,50,9)(3,55,51,12)(4,54,52,11)(5,37,28,41)(6,40,25,44)(7,39,26,43)(8,38,27,42)(13,17,60,34)(14,20,57,33)(15,19,58,36)(16,18,59,35)(21,30,47,62)(22,29,48,61)(23,32,45,64)(24,31,46,63)>;
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), (1,49)(2,50)(3,51)(4,52)(5,28)(6,25)(7,26)(8,27)(9,56)(10,53)(11,54)(12,55)(13,60)(14,57)(15,58)(16,59)(17,34)(18,35)(19,36)(20,33)(21,47)(22,48)(23,45)(24,46)(29,61)(30,62)(31,63)(32,64)(37,41)(38,42)(39,43)(40,44), (1,35)(2,36)(3,33)(4,34)(5,30)(6,31)(7,32)(8,29)(9,58)(10,59)(11,60)(12,57)(13,54)(14,55)(15,56)(16,53)(17,52)(18,49)(19,50)(20,51)(21,41)(22,42)(23,43)(24,44)(25,63)(26,64)(27,61)(28,62)(37,47)(38,48)(39,45)(40,46), (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), (1,11,51,56)(2,10,52,55)(3,9,49,54)(4,12,50,53)(5,46,26,22)(6,45,27,21)(7,48,28,24)(8,47,25,23)(13,33,58,18)(14,36,59,17)(15,35,60,20)(16,34,57,19)(29,37,63,43)(30,40,64,42)(31,39,61,41)(32,38,62,44), (1,44,3,42)(2,37,4,39)(5,11,7,9)(6,55,8,53)(10,25,12,27)(13,64,15,62)(14,29,16,31)(17,23,19,21)(18,46,20,48)(22,35,24,33)(26,56,28,54)(30,60,32,58)(34,45,36,47)(38,49,40,51)(41,52,43,50)(57,61,59,63), (1,53,49,10)(2,56,50,9)(3,55,51,12)(4,54,52,11)(5,37,28,41)(6,40,25,44)(7,39,26,43)(8,38,27,42)(13,17,60,34)(14,20,57,33)(15,19,58,36)(16,18,59,35)(21,30,47,62)(22,29,48,61)(23,32,45,64)(24,31,46,63) );
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)], [(1,49),(2,50),(3,51),(4,52),(5,28),(6,25),(7,26),(8,27),(9,56),(10,53),(11,54),(12,55),(13,60),(14,57),(15,58),(16,59),(17,34),(18,35),(19,36),(20,33),(21,47),(22,48),(23,45),(24,46),(29,61),(30,62),(31,63),(32,64),(37,41),(38,42),(39,43),(40,44)], [(1,35),(2,36),(3,33),(4,34),(5,30),(6,31),(7,32),(8,29),(9,58),(10,59),(11,60),(12,57),(13,54),(14,55),(15,56),(16,53),(17,52),(18,49),(19,50),(20,51),(21,41),(22,42),(23,43),(24,44),(25,63),(26,64),(27,61),(28,62),(37,47),(38,48),(39,45),(40,46)], [(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)], [(1,11,51,56),(2,10,52,55),(3,9,49,54),(4,12,50,53),(5,46,26,22),(6,45,27,21),(7,48,28,24),(8,47,25,23),(13,33,58,18),(14,36,59,17),(15,35,60,20),(16,34,57,19),(29,37,63,43),(30,40,64,42),(31,39,61,41),(32,38,62,44)], [(1,44,3,42),(2,37,4,39),(5,11,7,9),(6,55,8,53),(10,25,12,27),(13,64,15,62),(14,29,16,31),(17,23,19,21),(18,46,20,48),(22,35,24,33),(26,56,28,54),(30,60,32,58),(34,45,36,47),(38,49,40,51),(41,52,43,50),(57,61,59,63)], [(1,53,49,10),(2,56,50,9),(3,55,51,12),(4,54,52,11),(5,37,28,41),(6,40,25,44),(7,39,26,43),(8,38,27,42),(13,17,60,34),(14,20,57,33),(15,19,58,36),(16,18,59,35),(21,30,47,62),(22,29,48,61),(23,32,45,64),(24,31,46,63)]])
38 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 4A | ··· | 4H | 4I | ··· | 4X | 4Y | 4Z | 4AA | 4AB |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 |
| size | 1 | 1 | ··· | 1 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | C4○D4 | C4○D4 | 2- 1+4 |
| kernel | C23.375C24 | C4×C4⋊C4 | C23.8Q8 | C23.63C23 | C24.C22 | C23.65C23 | C23.Q8 | C23.11D4 | C23.81C23 | C23.83C23 | C2×C42⋊C2 | C2×C42.C2 | C4⋊C4 | C2×C4 | C23 | C22 |
| # reps | 1 | 1 | 2 | 3 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 12 | 4 | 2 |
Matrix representation of C23.375C24 ►in GL6(𝔽5)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 0 | 2 | 2 |
| 2 | 0 | 0 | 0 | 0 | 0 |
| 0 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 3 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 1 |
| 0 | 0 | 0 | 0 | 2 | 2 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 3 | 0 | 0 |
| 0 | 0 | 4 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 3 |
| 0 | 0 | 0 | 0 | 1 | 1 |
| 3 | 0 | 0 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 2 |
| 0 | 0 | 0 | 0 | 4 | 4 |
G:=sub<GL(6,GF(5))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,3,2,0,0,0,0,0,2],[2,0,0,0,0,0,0,2,0,0,0,0,0,0,1,3,0,0,0,0,0,4,0,0,0,0,0,0,3,2,0,0,0,0,1,2],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,3,4,0,0,0,0,3,2,0,0,0,0,0,0,4,1,0,0,0,0,3,1],[3,0,0,0,0,0,0,3,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,4,0,0,0,0,2,4] >;
C23.375C24 in GAP, Magma, Sage, TeX
C_2^3._{375}C_2^4 % in TeX
G:=Group("C2^3.375C2^4"); // GroupNames label
G:=SmallGroup(128,1207);
// by ID
G=gap.SmallGroup(128,1207);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,2,112,253,344,758,723,100,675,136]);
// Polycyclic
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=1,d^2=f^2=a,e^2=b*a=a*b,g^2=b,a*c=c*a,e*d*e^-1=g*d*g^-1=a*d=d*a,a*e=e*a,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,e*g=g*e,f*g=g*f>;
// generators/relations