p-group, metabelian, nilpotent (class 2), monomial
Aliases: C24.22C23, C23.425C24, C22.1662- 1+4, C22.2172+ 1+4, C42⋊5C4⋊16C2, (C22×C4).87C23, (C2×C42).540C22, C23.Q8.11C2, C23.11D4.14C2, C23.65C23⋊80C2, C23.81C23⋊31C2, C23.83C23⋊31C2, C23.63C23⋊77C2, C24.C22.26C2, C2.C42.173C22, C2.53(C22.46C24), C2.39(C22.36C24), C2.47(C22.47C24), C2.31(C22.50C24), C2.68(C23.36C23), (C4×C4⋊C4)⋊82C2, (C2×C4).144(C4○D4), (C2×C4⋊C4).287C22, C22.302(C2×C4○D4), (C2×C22⋊C4).50C22, SmallGroup(128,1257)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C23.425C24
G = < a,b,c,d,e,f,g | a2=b2=c2=1, d2=ca=ac, e2=f2=a, g2=ba=ab, 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: 356 in 199 conjugacy classes, 92 normal (82 characteristic)
C1, C2, C2, C4, C22, C22, C2×C4, C2×C4, C23, C23, C42, C22⋊C4, C4⋊C4, C22×C4, C24, C2.C42, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C4×C4⋊C4, C42⋊5C4, C23.63C23, C24.C22, C23.65C23, C23.Q8, C23.11D4, C23.81C23, C23.83C23, C23.425C24
Quotients: C1, C2, C22, C23, C4○D4, C24, C2×C4○D4, 2+ 1+4, 2- 1+4, C23.36C23, C22.36C24, C22.46C24, C22.47C24, C22.50C24, C23.425C24
►On 64 points
Generators in S64
(1 36)(2 33)(3 34)(4 35)(5 55)(6 56)(7 53)(8 54)(9 14)(10 15)(11 16)(12 13)(17 21)(18 22)(19 23)(20 24)(25 30)(26 31)(27 32)(28 29)(37 44)(38 41)(39 42)(40 43)(45 52)(46 49)(47 50)(48 51)(57 62)(58 63)(59 64)(60 61)
(1 58)(2 59)(3 60)(4 57)(5 32)(6 29)(7 30)(8 31)(9 42)(10 43)(11 44)(12 41)(13 38)(14 39)(15 40)(16 37)(17 45)(18 46)(19 47)(20 48)(21 52)(22 49)(23 50)(24 51)(25 53)(26 54)(27 55)(28 56)(33 64)(34 61)(35 62)(36 63)
(1 34)(2 35)(3 36)(4 33)(5 53)(6 54)(7 55)(8 56)(9 16)(10 13)(11 14)(12 15)(17 23)(18 24)(19 21)(20 22)(25 32)(26 29)(27 30)(28 31)(37 42)(38 43)(39 44)(40 41)(45 50)(46 51)(47 52)(48 49)(57 64)(58 61)(59 62)(60 63)
(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 18 36 22)(2 23 33 19)(3 20 34 24)(4 21 35 17)(5 39 55 42)(6 43 56 40)(7 37 53 44)(8 41 54 38)(9 32 14 27)(10 28 15 29)(11 30 16 25)(12 26 13 31)(45 57 52 62)(46 63 49 58)(47 59 50 64)(48 61 51 60)
(1 13 36 12)(2 39 33 42)(3 15 34 10)(4 37 35 44)(5 17 55 21)(6 46 56 49)(7 19 53 23)(8 48 54 51)(9 59 14 64)(11 57 16 62)(18 28 22 29)(20 26 24 31)(25 50 30 47)(27 52 32 45)(38 63 41 58)(40 61 43 60)
(1 25 63 7)(2 31 64 54)(3 27 61 5)(4 29 62 56)(6 35 28 57)(8 33 26 59)(9 51 39 20)(10 45 40 21)(11 49 37 18)(12 47 38 23)(13 50 41 19)(14 48 42 24)(15 52 43 17)(16 46 44 22)(30 58 53 36)(32 60 55 34)
G:=sub<Sym(64)| (1,36)(2,33)(3,34)(4,35)(5,55)(6,56)(7,53)(8,54)(9,14)(10,15)(11,16)(12,13)(17,21)(18,22)(19,23)(20,24)(25,30)(26,31)(27,32)(28,29)(37,44)(38,41)(39,42)(40,43)(45,52)(46,49)(47,50)(48,51)(57,62)(58,63)(59,64)(60,61), (1,58)(2,59)(3,60)(4,57)(5,32)(6,29)(7,30)(8,31)(9,42)(10,43)(11,44)(12,41)(13,38)(14,39)(15,40)(16,37)(17,45)(18,46)(19,47)(20,48)(21,52)(22,49)(23,50)(24,51)(25,53)(26,54)(27,55)(28,56)(33,64)(34,61)(35,62)(36,63), (1,34)(2,35)(3,36)(4,33)(5,53)(6,54)(7,55)(8,56)(9,16)(10,13)(11,14)(12,15)(17,23)(18,24)(19,21)(20,22)(25,32)(26,29)(27,30)(28,31)(37,42)(38,43)(39,44)(40,41)(45,50)(46,51)(47,52)(48,49)(57,64)(58,61)(59,62)(60,63), (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), (1,18,36,22)(2,23,33,19)(3,20,34,24)(4,21,35,17)(5,39,55,42)(6,43,56,40)(7,37,53,44)(8,41,54,38)(9,32,14,27)(10,28,15,29)(11,30,16,25)(12,26,13,31)(45,57,52,62)(46,63,49,58)(47,59,50,64)(48,61,51,60), (1,13,36,12)(2,39,33,42)(3,15,34,10)(4,37,35,44)(5,17,55,21)(6,46,56,49)(7,19,53,23)(8,48,54,51)(9,59,14,64)(11,57,16,62)(18,28,22,29)(20,26,24,31)(25,50,30,47)(27,52,32,45)(38,63,41,58)(40,61,43,60), (1,25,63,7)(2,31,64,54)(3,27,61,5)(4,29,62,56)(6,35,28,57)(8,33,26,59)(9,51,39,20)(10,45,40,21)(11,49,37,18)(12,47,38,23)(13,50,41,19)(14,48,42,24)(15,52,43,17)(16,46,44,22)(30,58,53,36)(32,60,55,34)>;
G:=Group( (1,36)(2,33)(3,34)(4,35)(5,55)(6,56)(7,53)(8,54)(9,14)(10,15)(11,16)(12,13)(17,21)(18,22)(19,23)(20,24)(25,30)(26,31)(27,32)(28,29)(37,44)(38,41)(39,42)(40,43)(45,52)(46,49)(47,50)(48,51)(57,62)(58,63)(59,64)(60,61), (1,58)(2,59)(3,60)(4,57)(5,32)(6,29)(7,30)(8,31)(9,42)(10,43)(11,44)(12,41)(13,38)(14,39)(15,40)(16,37)(17,45)(18,46)(19,47)(20,48)(21,52)(22,49)(23,50)(24,51)(25,53)(26,54)(27,55)(28,56)(33,64)(34,61)(35,62)(36,63), (1,34)(2,35)(3,36)(4,33)(5,53)(6,54)(7,55)(8,56)(9,16)(10,13)(11,14)(12,15)(17,23)(18,24)(19,21)(20,22)(25,32)(26,29)(27,30)(28,31)(37,42)(38,43)(39,44)(40,41)(45,50)(46,51)(47,52)(48,49)(57,64)(58,61)(59,62)(60,63), (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), (1,18,36,22)(2,23,33,19)(3,20,34,24)(4,21,35,17)(5,39,55,42)(6,43,56,40)(7,37,53,44)(8,41,54,38)(9,32,14,27)(10,28,15,29)(11,30,16,25)(12,26,13,31)(45,57,52,62)(46,63,49,58)(47,59,50,64)(48,61,51,60), (1,13,36,12)(2,39,33,42)(3,15,34,10)(4,37,35,44)(5,17,55,21)(6,46,56,49)(7,19,53,23)(8,48,54,51)(9,59,14,64)(11,57,16,62)(18,28,22,29)(20,26,24,31)(25,50,30,47)(27,52,32,45)(38,63,41,58)(40,61,43,60), (1,25,63,7)(2,31,64,54)(3,27,61,5)(4,29,62,56)(6,35,28,57)(8,33,26,59)(9,51,39,20)(10,45,40,21)(11,49,37,18)(12,47,38,23)(13,50,41,19)(14,48,42,24)(15,52,43,17)(16,46,44,22)(30,58,53,36)(32,60,55,34) );
G=PermutationGroup([[(1,36),(2,33),(3,34),(4,35),(5,55),(6,56),(7,53),(8,54),(9,14),(10,15),(11,16),(12,13),(17,21),(18,22),(19,23),(20,24),(25,30),(26,31),(27,32),(28,29),(37,44),(38,41),(39,42),(40,43),(45,52),(46,49),(47,50),(48,51),(57,62),(58,63),(59,64),(60,61)], [(1,58),(2,59),(3,60),(4,57),(5,32),(6,29),(7,30),(8,31),(9,42),(10,43),(11,44),(12,41),(13,38),(14,39),(15,40),(16,37),(17,45),(18,46),(19,47),(20,48),(21,52),(22,49),(23,50),(24,51),(25,53),(26,54),(27,55),(28,56),(33,64),(34,61),(35,62),(36,63)], [(1,34),(2,35),(3,36),(4,33),(5,53),(6,54),(7,55),(8,56),(9,16),(10,13),(11,14),(12,15),(17,23),(18,24),(19,21),(20,22),(25,32),(26,29),(27,30),(28,31),(37,42),(38,43),(39,44),(40,41),(45,50),(46,51),(47,52),(48,49),(57,64),(58,61),(59,62),(60,63)], [(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)], [(1,18,36,22),(2,23,33,19),(3,20,34,24),(4,21,35,17),(5,39,55,42),(6,43,56,40),(7,37,53,44),(8,41,54,38),(9,32,14,27),(10,28,15,29),(11,30,16,25),(12,26,13,31),(45,57,52,62),(46,63,49,58),(47,59,50,64),(48,61,51,60)], [(1,13,36,12),(2,39,33,42),(3,15,34,10),(4,37,35,44),(5,17,55,21),(6,46,56,49),(7,19,53,23),(8,48,54,51),(9,59,14,64),(11,57,16,62),(18,28,22,29),(20,26,24,31),(25,50,30,47),(27,52,32,45),(38,63,41,58),(40,61,43,60)], [(1,25,63,7),(2,31,64,54),(3,27,61,5),(4,29,62,56),(6,35,28,57),(8,33,26,59),(9,51,39,20),(10,45,40,21),(11,49,37,18),(12,47,38,23),(13,50,41,19),(14,48,42,24),(15,52,43,17),(16,46,44,22),(30,58,53,36),(32,60,55,34)]])
38 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 4A | ··· | 4H | 4I | ··· | 4Z | 4AA | 4AB | 4AC |
| order | 1 | 2 | ··· | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | 4 | 4 |
| size | 1 | 1 | ··· | 1 | 8 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | 8 | 8 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4○D4 | 2+ 1+4 | 2- 1+4 |
| kernel | C23.425C24 | C4×C4⋊C4 | C42⋊5C4 | C23.63C23 | C24.C22 | C23.65C23 | C23.Q8 | C23.11D4 | C23.81C23 | C23.83C23 | C2×C4 | C22 | C22 |
| # reps | 1 | 2 | 1 | 2 | 5 | 1 | 1 | 1 | 1 | 1 | 20 | 1 | 1 |
Matrix representation of C23.425C24 ►in GL6(𝔽5)
| 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 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 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 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 1 | 0 | 0 |
| 0 | 0 | 0 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 2 |
| 0 | 0 | 0 | 0 | 3 | 0 |
| 0 | 2 | 0 | 0 | 0 | 0 |
| 2 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 3 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 2 | 0 | 0 | 0 | 0 |
| 2 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 0 | 0 | 0 |
| 0 | 0 | 0 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 0 | 0 | 2 |
G:=sub<GL(6,GF(5))| [4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[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],[0,4,0,0,0,0,1,0,0,0,0,0,0,0,3,0,0,0,0,0,1,2,0,0,0,0,0,0,0,3,0,0,0,0,2,0],[0,2,0,0,0,0,2,0,0,0,0,0,0,0,4,0,0,0,0,0,3,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,2,0,0,0,0,2,0,0,0,0,0,0,0,4,1,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,2,0,0,0,0,0,0,2,0,0,0,0,0,0,2,0,0,0,0,0,0,2] >;
C23.425C24 in GAP, Magma, Sage, TeX
C_2^3._{425}C_2^4 % in TeX
G:=Group("C2^3.425C2^4"); // GroupNames label
G:=SmallGroup(128,1257);
// by ID
G=gap.SmallGroup(128,1257);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,2,560,253,120,758,723,100,675,192]);
// Polycyclic
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=1,d^2=c*a=a*c,e^2=f^2=a,g^2=b*a=a*b,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