p-group, metabelian, nilpotent (class 2), monomial
Aliases: C24.50C23, C23.554C24, C22.2452- 1+4, C22.3292+ 1+4, (C2×C42).86C22, (C22×C4).164C23, C23.Q8.23C2, C23.4Q8.16C2, C23.11D4.31C2, C23.81C23⋊70C2, C23.83C23⋊70C2, C24.C22.45C2, C23.63C23⋊121C2, C23.65C23⋊109C2, C2.C42.271C22, C2.51(C22.33C24), C2.61(C22.36C24), C2.34(C22.35C24), C2.35(C22.34C24), C2.106(C23.36C23), (C4×C4⋊C4)⋊114C2, (C2×C4).179(C4○D4), (C2×C4⋊C4).895C22, C22.426(C2×C4○D4), (C2×C22⋊C4).235C22, SmallGroup(128,1386)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C23.554C24
G = < a,b,c,d,e,f,g | a2=b2=c2=1, d2=c, e2=f2=a, g2=b, ab=ba, ac=ca, ede-1=ad=da, ae=ea, gfg-1=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, dg=gd, eg=ge >
Subgroups: 356 in 189 conjugacy classes, 88 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, C23.63C23, C24.C22, C23.65C23, C23.Q8, C23.11D4, C23.81C23, C23.4Q8, C23.83C23, C23.554C24
Quotients: C1, C2, C22, C23, C4○D4, C24, C2×C4○D4, 2+ 1+4, 2- 1+4, C23.36C23, C22.33C24, C22.34C24, C22.35C24, C22.36C24, C23.554C24
►On 64 points
Generators in S64
(1 25)(2 26)(3 27)(4 28)(5 54)(6 55)(7 56)(8 53)(9 52)(10 49)(11 50)(12 51)(13 48)(14 45)(15 46)(16 47)(17 42)(18 43)(19 44)(20 41)(21 39)(22 40)(23 37)(24 38)(29 33)(30 34)(31 35)(32 36)(57 64)(58 61)(59 62)(60 63)
(1 61)(2 62)(3 63)(4 64)(5 35)(6 36)(7 33)(8 34)(9 39)(10 40)(11 37)(12 38)(13 19)(14 20)(15 17)(16 18)(21 52)(22 49)(23 50)(24 51)(25 58)(26 59)(27 60)(28 57)(29 56)(30 53)(31 54)(32 55)(41 45)(42 46)(43 47)(44 48)
(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 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 17 25 42)(2 43 26 18)(3 19 27 44)(4 41 28 20)(5 24 54 38)(6 39 55 21)(7 22 56 40)(8 37 53 23)(9 32 52 36)(10 33 49 29)(11 30 50 34)(12 35 51 31)(13 60 48 63)(14 64 45 57)(15 58 46 61)(16 62 47 59)
(1 51 25 12)(2 21 26 39)(3 49 27 10)(4 23 28 37)(5 13 54 48)(6 20 55 41)(7 15 56 46)(8 18 53 43)(9 62 52 59)(11 64 50 57)(14 32 45 36)(16 30 47 34)(17 29 42 33)(19 31 44 35)(22 60 40 63)(24 58 38 61)
(1 34 61 8)(2 35 62 5)(3 36 63 6)(4 33 64 7)(9 13 39 19)(10 14 40 20)(11 15 37 17)(12 16 38 18)(21 44 52 48)(22 41 49 45)(23 42 50 46)(24 43 51 47)(25 30 58 53)(26 31 59 54)(27 32 60 55)(28 29 57 56)
G:=sub<Sym(64)| (1,25)(2,26)(3,27)(4,28)(5,54)(6,55)(7,56)(8,53)(9,52)(10,49)(11,50)(12,51)(13,48)(14,45)(15,46)(16,47)(17,42)(18,43)(19,44)(20,41)(21,39)(22,40)(23,37)(24,38)(29,33)(30,34)(31,35)(32,36)(57,64)(58,61)(59,62)(60,63), (1,61)(2,62)(3,63)(4,64)(5,35)(6,36)(7,33)(8,34)(9,39)(10,40)(11,37)(12,38)(13,19)(14,20)(15,17)(16,18)(21,52)(22,49)(23,50)(24,51)(25,58)(26,59)(27,60)(28,57)(29,56)(30,53)(31,54)(32,55)(41,45)(42,46)(43,47)(44,48), (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), (1,2,3,4)(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), (1,17,25,42)(2,43,26,18)(3,19,27,44)(4,41,28,20)(5,24,54,38)(6,39,55,21)(7,22,56,40)(8,37,53,23)(9,32,52,36)(10,33,49,29)(11,30,50,34)(12,35,51,31)(13,60,48,63)(14,64,45,57)(15,58,46,61)(16,62,47,59), (1,51,25,12)(2,21,26,39)(3,49,27,10)(4,23,28,37)(5,13,54,48)(6,20,55,41)(7,15,56,46)(8,18,53,43)(9,62,52,59)(11,64,50,57)(14,32,45,36)(16,30,47,34)(17,29,42,33)(19,31,44,35)(22,60,40,63)(24,58,38,61), (1,34,61,8)(2,35,62,5)(3,36,63,6)(4,33,64,7)(9,13,39,19)(10,14,40,20)(11,15,37,17)(12,16,38,18)(21,44,52,48)(22,41,49,45)(23,42,50,46)(24,43,51,47)(25,30,58,53)(26,31,59,54)(27,32,60,55)(28,29,57,56)>;
G:=Group( (1,25)(2,26)(3,27)(4,28)(5,54)(6,55)(7,56)(8,53)(9,52)(10,49)(11,50)(12,51)(13,48)(14,45)(15,46)(16,47)(17,42)(18,43)(19,44)(20,41)(21,39)(22,40)(23,37)(24,38)(29,33)(30,34)(31,35)(32,36)(57,64)(58,61)(59,62)(60,63), (1,61)(2,62)(3,63)(4,64)(5,35)(6,36)(7,33)(8,34)(9,39)(10,40)(11,37)(12,38)(13,19)(14,20)(15,17)(16,18)(21,52)(22,49)(23,50)(24,51)(25,58)(26,59)(27,60)(28,57)(29,56)(30,53)(31,54)(32,55)(41,45)(42,46)(43,47)(44,48), (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), (1,2,3,4)(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), (1,17,25,42)(2,43,26,18)(3,19,27,44)(4,41,28,20)(5,24,54,38)(6,39,55,21)(7,22,56,40)(8,37,53,23)(9,32,52,36)(10,33,49,29)(11,30,50,34)(12,35,51,31)(13,60,48,63)(14,64,45,57)(15,58,46,61)(16,62,47,59), (1,51,25,12)(2,21,26,39)(3,49,27,10)(4,23,28,37)(5,13,54,48)(6,20,55,41)(7,15,56,46)(8,18,53,43)(9,62,52,59)(11,64,50,57)(14,32,45,36)(16,30,47,34)(17,29,42,33)(19,31,44,35)(22,60,40,63)(24,58,38,61), (1,34,61,8)(2,35,62,5)(3,36,63,6)(4,33,64,7)(9,13,39,19)(10,14,40,20)(11,15,37,17)(12,16,38,18)(21,44,52,48)(22,41,49,45)(23,42,50,46)(24,43,51,47)(25,30,58,53)(26,31,59,54)(27,32,60,55)(28,29,57,56) );
G=PermutationGroup([[(1,25),(2,26),(3,27),(4,28),(5,54),(6,55),(7,56),(8,53),(9,52),(10,49),(11,50),(12,51),(13,48),(14,45),(15,46),(16,47),(17,42),(18,43),(19,44),(20,41),(21,39),(22,40),(23,37),(24,38),(29,33),(30,34),(31,35),(32,36),(57,64),(58,61),(59,62),(60,63)], [(1,61),(2,62),(3,63),(4,64),(5,35),(6,36),(7,33),(8,34),(9,39),(10,40),(11,37),(12,38),(13,19),(14,20),(15,17),(16,18),(21,52),(22,49),(23,50),(24,51),(25,58),(26,59),(27,60),(28,57),(29,56),(30,53),(31,54),(32,55),(41,45),(42,46),(43,47),(44,48)], [(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)], [(1,2,3,4),(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)], [(1,17,25,42),(2,43,26,18),(3,19,27,44),(4,41,28,20),(5,24,54,38),(6,39,55,21),(7,22,56,40),(8,37,53,23),(9,32,52,36),(10,33,49,29),(11,30,50,34),(12,35,51,31),(13,60,48,63),(14,64,45,57),(15,58,46,61),(16,62,47,59)], [(1,51,25,12),(2,21,26,39),(3,49,27,10),(4,23,28,37),(5,13,54,48),(6,20,55,41),(7,15,56,46),(8,18,53,43),(9,62,52,59),(11,64,50,57),(14,32,45,36),(16,30,47,34),(17,29,42,33),(19,31,44,35),(22,60,40,63),(24,58,38,61)], [(1,34,61,8),(2,35,62,5),(3,36,63,6),(4,33,64,7),(9,13,39,19),(10,14,40,20),(11,15,37,17),(12,16,38,18),(21,44,52,48),(22,41,49,45),(23,42,50,46),(24,43,51,47),(25,30,58,53),(26,31,59,54),(27,32,60,55),(28,29,57,56)]])
32 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 4A | 4B | 4C | 4D | 4E | ··· | 4P | 4Q | ··· | 4W |
| order | 1 | 2 | ··· | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 |
| size | 1 | 1 | ··· | 1 | 8 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 8 | ··· | 8 |
32 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 | C23.554C24 | C4×C4⋊C4 | C23.63C23 | C24.C22 | C23.65C23 | C23.Q8 | C23.11D4 | C23.81C23 | C23.4Q8 | C23.83C23 | C2×C4 | C22 | C22 |
| # reps | 1 | 1 | 2 | 3 | 1 | 1 | 2 | 2 | 1 | 2 | 12 | 2 | 2 |
Matrix representation of C23.554C24 ►in GL8(𝔽5)
| 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 |
| 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 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 4 |
| 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 | 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 |
| 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 2 | 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 | 3 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 3 |
| 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 |
| 3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 3 |
| 0 | 0 | 0 | 0 | 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 0 | 0 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 1 | 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 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 4 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(8,GF(5))| [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],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[2,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,2,2,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,3],[0,3,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,4,4,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,3,0,0,0,0,0,0,3,0,0,0,0,0,0,3,0,0,0],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,4,3,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0] >;
C23.554C24 in GAP, Magma, Sage, TeX
C_2^3._{554}C_2^4 % in TeX
G:=Group("C2^3.554C2^4"); // GroupNames label
G:=SmallGroup(128,1386);
// by ID
G=gap.SmallGroup(128,1386);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,120,758,723,100,185,136]);
// Polycyclic
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=1,d^2=c,e^2=f^2=a,g^2=b,a*b=b*a,a*c=c*a,e*d*e^-1=a*d=d*a,a*e=e*a,g*f*g^-1=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,d*g=g*d,e*g=g*e>;
// generators/relations