p-group, metabelian, nilpotent (class 2), monomial
Aliases: C24.524C23, C23.156C24, C22.152- 1+4, C22.272+ 1+4, (C2×C4)⋊3C42, C42⋊12(C2×C4), C4.29(C2×C42), C42⋊C2⋊21C4, C2.9(C22×C42), (C2×C42).7C22, (C23×C4).31C22, C22.28(C23×C4), C22.15(C2×C42), C23.114(C22×C4), (C22×C4).1232C23, C2.2(C22.11C24), C2.C42.567C22, C2.1(C23.32C23), C2.1(C23.33C23), C4⋊C4○2(C4⋊C4), (C2×C4⋊C4)⋊31C4, (C4×C4⋊C4)⋊11C2, C4⋊C4⋊49(C2×C4), C22⋊C4○2(C4⋊C4), (C4×C22⋊C4).3C2, (C22×C4⋊C4).18C2, C22⋊C4.83(C2×C4), (C2×C4⋊C4).970C22, (C22×C4).130(C2×C4), (C2×C4).564(C22×C4), (C2×C42⋊C2).21C2, (C2×C22⋊C4).552C22, C4⋊C4○(C2×C22⋊C4), SmallGroup(128,1006)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C24.524C23
G = < a,b,c,d,e,f,g | a2=b2=c2=g2=1, d2=b, e2=c, f2=a, ab=ba, ac=ca, ede-1=gdg=ad=da, fef-1=ae=ea, af=fa, ag=ga, bc=cb, bd=db, be=eb, bf=fb, bg=gb, cd=dc, ce=ec, cf=fc, cg=gc, df=fd, eg=ge, fg=gf >
Subgroups: 476 in 348 conjugacy classes, 260 normal (10 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C2×C4, C2×C4, C23, C23, C23, C42, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C24, C2.C42, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C42⋊C2, C23×C4, C23×C4, C4×C22⋊C4, C4×C4⋊C4, C22×C4⋊C4, C2×C42⋊C2, C24.524C23
Quotients: C1, C2, C4, C22, C2×C4, C23, C42, C22×C4, C24, C2×C42, C23×C4, 2+ 1+4, 2- 1+4, C22×C42, C22.11C24, C23.32C23, C23.33C23, C24.524C23
►On 64 points
Generators in S64
(1 39)(2 40)(3 37)(4 38)(5 17)(6 18)(7 19)(8 20)(9 41)(10 42)(11 43)(12 44)(13 45)(14 46)(15 47)(16 48)(21 49)(22 50)(23 51)(24 52)(25 53)(26 54)(27 55)(28 56)(29 57)(30 58)(31 59)(32 60)(33 62)(34 63)(35 64)(36 61)
(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 59 51 15)(2 32 52 48)(3 57 49 13)(4 30 50 46)(5 41 36 25)(6 10 33 54)(7 43 34 27)(8 12 35 56)(9 61 53 17)(11 63 55 19)(14 38 58 22)(16 40 60 24)(18 42 62 26)(20 44 64 28)(21 45 37 29)(23 47 39 31)
(1 43 39 11)(2 44 40 12)(3 41 37 9)(4 42 38 10)(5 13 17 45)(6 14 18 46)(7 15 19 47)(8 16 20 48)(21 53 49 25)(22 54 50 26)(23 55 51 27)(24 56 52 28)(29 36 57 61)(30 33 58 62)(31 34 59 63)(32 35 60 64)
(1 51)(2 24)(3 49)(4 22)(5 36)(6 62)(7 34)(8 64)(9 53)(10 26)(11 55)(12 28)(13 57)(14 30)(15 59)(16 32)(17 61)(18 33)(19 63)(20 35)(21 37)(23 39)(25 41)(27 43)(29 45)(31 47)(38 50)(40 52)(42 54)(44 56)(46 58)(48 60)
G:=sub<Sym(64)| (1,39)(2,40)(3,37)(4,38)(5,17)(6,18)(7,19)(8,20)(9,41)(10,42)(11,43)(12,44)(13,45)(14,46)(15,47)(16,48)(21,49)(22,50)(23,51)(24,52)(25,53)(26,54)(27,55)(28,56)(29,57)(30,58)(31,59)(32,60)(33,62)(34,63)(35,64)(36,61), (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), (1,59,51,15)(2,32,52,48)(3,57,49,13)(4,30,50,46)(5,41,36,25)(6,10,33,54)(7,43,34,27)(8,12,35,56)(9,61,53,17)(11,63,55,19)(14,38,58,22)(16,40,60,24)(18,42,62,26)(20,44,64,28)(21,45,37,29)(23,47,39,31), (1,43,39,11)(2,44,40,12)(3,41,37,9)(4,42,38,10)(5,13,17,45)(6,14,18,46)(7,15,19,47)(8,16,20,48)(21,53,49,25)(22,54,50,26)(23,55,51,27)(24,56,52,28)(29,36,57,61)(30,33,58,62)(31,34,59,63)(32,35,60,64), (1,51)(2,24)(3,49)(4,22)(5,36)(6,62)(7,34)(8,64)(9,53)(10,26)(11,55)(12,28)(13,57)(14,30)(15,59)(16,32)(17,61)(18,33)(19,63)(20,35)(21,37)(23,39)(25,41)(27,43)(29,45)(31,47)(38,50)(40,52)(42,54)(44,56)(46,58)(48,60)>;
G:=Group( (1,39)(2,40)(3,37)(4,38)(5,17)(6,18)(7,19)(8,20)(9,41)(10,42)(11,43)(12,44)(13,45)(14,46)(15,47)(16,48)(21,49)(22,50)(23,51)(24,52)(25,53)(26,54)(27,55)(28,56)(29,57)(30,58)(31,59)(32,60)(33,62)(34,63)(35,64)(36,61), (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), (1,59,51,15)(2,32,52,48)(3,57,49,13)(4,30,50,46)(5,41,36,25)(6,10,33,54)(7,43,34,27)(8,12,35,56)(9,61,53,17)(11,63,55,19)(14,38,58,22)(16,40,60,24)(18,42,62,26)(20,44,64,28)(21,45,37,29)(23,47,39,31), (1,43,39,11)(2,44,40,12)(3,41,37,9)(4,42,38,10)(5,13,17,45)(6,14,18,46)(7,15,19,47)(8,16,20,48)(21,53,49,25)(22,54,50,26)(23,55,51,27)(24,56,52,28)(29,36,57,61)(30,33,58,62)(31,34,59,63)(32,35,60,64), (1,51)(2,24)(3,49)(4,22)(5,36)(6,62)(7,34)(8,64)(9,53)(10,26)(11,55)(12,28)(13,57)(14,30)(15,59)(16,32)(17,61)(18,33)(19,63)(20,35)(21,37)(23,39)(25,41)(27,43)(29,45)(31,47)(38,50)(40,52)(42,54)(44,56)(46,58)(48,60) );
G=PermutationGroup([[(1,39),(2,40),(3,37),(4,38),(5,17),(6,18),(7,19),(8,20),(9,41),(10,42),(11,43),(12,44),(13,45),(14,46),(15,47),(16,48),(21,49),(22,50),(23,51),(24,52),(25,53),(26,54),(27,55),(28,56),(29,57),(30,58),(31,59),(32,60),(33,62),(34,63),(35,64),(36,61)], [(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)], [(1,59,51,15),(2,32,52,48),(3,57,49,13),(4,30,50,46),(5,41,36,25),(6,10,33,54),(7,43,34,27),(8,12,35,56),(9,61,53,17),(11,63,55,19),(14,38,58,22),(16,40,60,24),(18,42,62,26),(20,44,64,28),(21,45,37,29),(23,47,39,31)], [(1,43,39,11),(2,44,40,12),(3,41,37,9),(4,42,38,10),(5,13,17,45),(6,14,18,46),(7,15,19,47),(8,16,20,48),(21,53,49,25),(22,54,50,26),(23,55,51,27),(24,56,52,28),(29,36,57,61),(30,33,58,62),(31,34,59,63),(32,35,60,64)], [(1,51),(2,24),(3,49),(4,22),(5,36),(6,62),(7,34),(8,64),(9,53),(10,26),(11,55),(12,28),(13,57),(14,30),(15,59),(16,32),(17,61),(18,33),(19,63),(20,35),(21,37),(23,39),(25,41),(27,43),(29,45),(31,47),(38,50),(40,52),(42,54),(44,56),(46,58),(48,60)]])
68 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 4A | ··· | 4BD |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 2 | ··· | 2 |
68 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 4 |
| type | + | + | + | + | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | 2+ 1+4 | 2- 1+4 |
| kernel | C24.524C23 | C4×C22⋊C4 | C4×C4⋊C4 | C22×C4⋊C4 | C2×C42⋊C2 | C2×C4⋊C4 | C42⋊C2 | C22 | C22 |
| # reps | 1 | 4 | 8 | 1 | 2 | 16 | 32 | 2 | 2 |
Matrix representation of C24.524C23 ►in GL6(𝔽5)
| 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 | 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 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 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 |
| 3 | 0 | 0 | 0 | 0 | 0 |
| 0 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 4 | 0 | 0 |
| 0 | 0 | 0 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 1 |
| 0 | 0 | 0 | 0 | 0 | 2 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 0 | 0 | 0 |
| 0 | 0 | 3 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 0 | 3 | 3 |
| 4 | 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 |
G:=sub<GL(6,GF(5))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[3,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,2,0,0,0,0,0,0,2,0,0,0,0,0,4,3,0,0,0,0,0,0,3,0,0,0,0,0,1,2],[1,0,0,0,0,0,0,4,0,0,0,0,0,0,2,3,0,0,0,0,0,3,0,0,0,0,0,0,2,3,0,0,0,0,0,3],[4,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;
C24.524C23 in GAP, Magma, Sage, TeX
C_2^4._{524}C_2^3 % in TeX
G:=Group("C2^4.524C2^3"); // GroupNames label
G:=SmallGroup(128,1006);
// by ID
G=gap.SmallGroup(128,1006);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,2,224,253,456,219,100,675]);
// Polycyclic
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=g^2=1,d^2=b,e^2=c,f^2=a,a*b=b*a,a*c=c*a,e*d*e^-1=g*d*g=a*d=d*a,f*e*f^-1=a*e=e*a,a*f=f*a,a*g=g*a,b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,b*g=g*b,c*d=d*c,c*e=e*c,c*f=f*c,c*g=g*c,d*f=f*d,e*g=g*e,f*g=g*f>;
// generators/relations