p-group, metabelian, nilpotent (class 2), monomial
Aliases: C24.375C23, C23.549C24, C22.3242+ 1+4, C22.2402- 1+4, C23.71(C4○D4), (C2×C42).81C22, C23.8Q8⋊90C2, C23.Q8⋊44C2, C23.4Q8⋊33C2, C23.11D4⋊68C2, (C22×C4).159C23, (C23×C4).144C22, C23.10D4.34C2, C23.23D4.47C2, (C22×D4).203C22, C24.C22⋊108C2, C23.83C23⋊67C2, C2.47(C22.32C24), C23.63C23⋊116C2, C23.65C23⋊107C2, C2.C42.267C22, C2.56(C22.36C24), C2.33(C22.34C24), C2.46(C22.33C24), C2.101(C23.36C23), (C4×C22⋊C4)⋊96C2, (C2×C4).174(C4○D4), (C2×C4⋊C4).375C22, C22.421(C2×C4○D4), (C2×C22⋊C4).472C22, SmallGroup(128,1381)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C24.375C23
G = < a,b,c,d,e,f,g | a2=b2=c2=e2=f2=1, d2=ca=ac, g2=b, ab=ba, ede=ad=da, ae=ea, gfg-1=af=fa, ag=ga, bc=cb, fdf=bd=db, be=eb, bf=fb, bg=gb, cd=dc, fef=ce=ec, cf=fc, cg=gc, dg=gd, eg=ge >
Subgroups: 452 in 217 conjugacy classes, 88 normal (82 characteristic)
C1, C2, C2, C4, C22, C22, C2×C4, C2×C4, D4, C23, C23, C23, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C24, C2.C42, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C23×C4, C22×D4, C4×C22⋊C4, C23.8Q8, C23.23D4, C23.63C23, C24.C22, C23.65C23, C23.10D4, C23.Q8, C23.11D4, C23.4Q8, C23.83C23, C24.375C23
Quotients: C1, C2, C22, C23, C4○D4, C24, C2×C4○D4, 2+ 1+4, 2- 1+4, C23.36C23, C22.32C24, C22.33C24, C22.34C24, C22.36C24, C24.375C23
►On 64 points
Generators in S64
(1 10)(2 11)(3 12)(4 9)(5 37)(6 38)(7 39)(8 40)(13 52)(14 49)(15 50)(16 51)(17 46)(18 47)(19 48)(20 45)(21 43)(22 44)(23 41)(24 42)(25 54)(26 55)(27 56)(28 53)(29 60)(30 57)(31 58)(32 59)(33 64)(34 61)(35 62)(36 63)
(1 26)(2 27)(3 28)(4 25)(5 23)(6 24)(7 21)(8 22)(9 54)(10 55)(11 56)(12 53)(13 60)(14 57)(15 58)(16 59)(17 62)(18 63)(19 64)(20 61)(29 52)(30 49)(31 50)(32 51)(33 48)(34 45)(35 46)(36 47)(37 41)(38 42)(39 43)(40 44)
(1 12)(2 9)(3 10)(4 11)(5 39)(6 40)(7 37)(8 38)(13 50)(14 51)(15 52)(16 49)(17 48)(18 45)(19 46)(20 47)(21 41)(22 42)(23 43)(24 44)(25 56)(26 53)(27 54)(28 55)(29 58)(30 59)(31 60)(32 57)(33 62)(34 63)(35 64)(36 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 47)(2 19)(3 45)(4 17)(5 58)(6 32)(7 60)(8 30)(9 46)(10 18)(11 48)(12 20)(13 21)(14 44)(15 23)(16 42)(22 49)(24 51)(25 62)(26 36)(27 64)(28 34)(29 39)(31 37)(33 56)(35 54)(38 59)(40 57)(41 50)(43 52)(53 61)(55 63)
(2 27)(4 25)(5 7)(6 22)(8 24)(9 54)(11 56)(13 52)(14 30)(15 50)(16 32)(17 33)(18 45)(19 35)(20 47)(21 23)(29 60)(31 58)(34 63)(36 61)(37 39)(38 44)(40 42)(41 43)(46 64)(48 62)(49 57)(51 59)
(1 31 26 50)(2 32 27 51)(3 29 28 52)(4 30 25 49)(5 63 23 18)(6 64 24 19)(7 61 21 20)(8 62 22 17)(9 57 54 14)(10 58 55 15)(11 59 56 16)(12 60 53 13)(33 42 48 38)(34 43 45 39)(35 44 46 40)(36 41 47 37)
G:=sub<Sym(64)| (1,10)(2,11)(3,12)(4,9)(5,37)(6,38)(7,39)(8,40)(13,52)(14,49)(15,50)(16,51)(17,46)(18,47)(19,48)(20,45)(21,43)(22,44)(23,41)(24,42)(25,54)(26,55)(27,56)(28,53)(29,60)(30,57)(31,58)(32,59)(33,64)(34,61)(35,62)(36,63), (1,26)(2,27)(3,28)(4,25)(5,23)(6,24)(7,21)(8,22)(9,54)(10,55)(11,56)(12,53)(13,60)(14,57)(15,58)(16,59)(17,62)(18,63)(19,64)(20,61)(29,52)(30,49)(31,50)(32,51)(33,48)(34,45)(35,46)(36,47)(37,41)(38,42)(39,43)(40,44), (1,12)(2,9)(3,10)(4,11)(5,39)(6,40)(7,37)(8,38)(13,50)(14,51)(15,52)(16,49)(17,48)(18,45)(19,46)(20,47)(21,41)(22,42)(23,43)(24,44)(25,56)(26,53)(27,54)(28,55)(29,58)(30,59)(31,60)(32,57)(33,62)(34,63)(35,64)(36,61), (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), (1,47)(2,19)(3,45)(4,17)(5,58)(6,32)(7,60)(8,30)(9,46)(10,18)(11,48)(12,20)(13,21)(14,44)(15,23)(16,42)(22,49)(24,51)(25,62)(26,36)(27,64)(28,34)(29,39)(31,37)(33,56)(35,54)(38,59)(40,57)(41,50)(43,52)(53,61)(55,63), (2,27)(4,25)(5,7)(6,22)(8,24)(9,54)(11,56)(13,52)(14,30)(15,50)(16,32)(17,33)(18,45)(19,35)(20,47)(21,23)(29,60)(31,58)(34,63)(36,61)(37,39)(38,44)(40,42)(41,43)(46,64)(48,62)(49,57)(51,59), (1,31,26,50)(2,32,27,51)(3,29,28,52)(4,30,25,49)(5,63,23,18)(6,64,24,19)(7,61,21,20)(8,62,22,17)(9,57,54,14)(10,58,55,15)(11,59,56,16)(12,60,53,13)(33,42,48,38)(34,43,45,39)(35,44,46,40)(36,41,47,37)>;
G:=Group( (1,10)(2,11)(3,12)(4,9)(5,37)(6,38)(7,39)(8,40)(13,52)(14,49)(15,50)(16,51)(17,46)(18,47)(19,48)(20,45)(21,43)(22,44)(23,41)(24,42)(25,54)(26,55)(27,56)(28,53)(29,60)(30,57)(31,58)(32,59)(33,64)(34,61)(35,62)(36,63), (1,26)(2,27)(3,28)(4,25)(5,23)(6,24)(7,21)(8,22)(9,54)(10,55)(11,56)(12,53)(13,60)(14,57)(15,58)(16,59)(17,62)(18,63)(19,64)(20,61)(29,52)(30,49)(31,50)(32,51)(33,48)(34,45)(35,46)(36,47)(37,41)(38,42)(39,43)(40,44), (1,12)(2,9)(3,10)(4,11)(5,39)(6,40)(7,37)(8,38)(13,50)(14,51)(15,52)(16,49)(17,48)(18,45)(19,46)(20,47)(21,41)(22,42)(23,43)(24,44)(25,56)(26,53)(27,54)(28,55)(29,58)(30,59)(31,60)(32,57)(33,62)(34,63)(35,64)(36,61), (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), (1,47)(2,19)(3,45)(4,17)(5,58)(6,32)(7,60)(8,30)(9,46)(10,18)(11,48)(12,20)(13,21)(14,44)(15,23)(16,42)(22,49)(24,51)(25,62)(26,36)(27,64)(28,34)(29,39)(31,37)(33,56)(35,54)(38,59)(40,57)(41,50)(43,52)(53,61)(55,63), (2,27)(4,25)(5,7)(6,22)(8,24)(9,54)(11,56)(13,52)(14,30)(15,50)(16,32)(17,33)(18,45)(19,35)(20,47)(21,23)(29,60)(31,58)(34,63)(36,61)(37,39)(38,44)(40,42)(41,43)(46,64)(48,62)(49,57)(51,59), (1,31,26,50)(2,32,27,51)(3,29,28,52)(4,30,25,49)(5,63,23,18)(6,64,24,19)(7,61,21,20)(8,62,22,17)(9,57,54,14)(10,58,55,15)(11,59,56,16)(12,60,53,13)(33,42,48,38)(34,43,45,39)(35,44,46,40)(36,41,47,37) );
G=PermutationGroup([[(1,10),(2,11),(3,12),(4,9),(5,37),(6,38),(7,39),(8,40),(13,52),(14,49),(15,50),(16,51),(17,46),(18,47),(19,48),(20,45),(21,43),(22,44),(23,41),(24,42),(25,54),(26,55),(27,56),(28,53),(29,60),(30,57),(31,58),(32,59),(33,64),(34,61),(35,62),(36,63)], [(1,26),(2,27),(3,28),(4,25),(5,23),(6,24),(7,21),(8,22),(9,54),(10,55),(11,56),(12,53),(13,60),(14,57),(15,58),(16,59),(17,62),(18,63),(19,64),(20,61),(29,52),(30,49),(31,50),(32,51),(33,48),(34,45),(35,46),(36,47),(37,41),(38,42),(39,43),(40,44)], [(1,12),(2,9),(3,10),(4,11),(5,39),(6,40),(7,37),(8,38),(13,50),(14,51),(15,52),(16,49),(17,48),(18,45),(19,46),(20,47),(21,41),(22,42),(23,43),(24,44),(25,56),(26,53),(27,54),(28,55),(29,58),(30,59),(31,60),(32,57),(33,62),(34,63),(35,64),(36,61)], [(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)], [(1,47),(2,19),(3,45),(4,17),(5,58),(6,32),(7,60),(8,30),(9,46),(10,18),(11,48),(12,20),(13,21),(14,44),(15,23),(16,42),(22,49),(24,51),(25,62),(26,36),(27,64),(28,34),(29,39),(31,37),(33,56),(35,54),(38,59),(40,57),(41,50),(43,52),(53,61),(55,63)], [(2,27),(4,25),(5,7),(6,22),(8,24),(9,54),(11,56),(13,52),(14,30),(15,50),(16,32),(17,33),(18,45),(19,35),(20,47),(21,23),(29,60),(31,58),(34,63),(36,61),(37,39),(38,44),(40,42),(41,43),(46,64),(48,62),(49,57),(51,59)], [(1,31,26,50),(2,32,27,51),(3,29,28,52),(4,30,25,49),(5,63,23,18),(6,64,24,19),(7,61,21,20),(8,62,22,17),(9,57,54,14),(10,58,55,15),(11,59,56,16),(12,60,53,13),(33,42,48,38),(34,43,45,39),(35,44,46,40),(36,41,47,37)]])
32 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 4A | 4B | 4C | 4D | 4E | ··· | 4N | 4O | ··· | 4U |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 |
| size | 1 | 1 | ··· | 1 | 4 | 4 | 8 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 8 | ··· | 8 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4○D4 | C4○D4 | 2+ 1+4 | 2- 1+4 |
| kernel | C24.375C23 | C4×C22⋊C4 | C23.8Q8 | C23.23D4 | C23.63C23 | C24.C22 | C23.65C23 | C23.10D4 | C23.Q8 | C23.11D4 | C23.4Q8 | C23.83C23 | C2×C4 | C23 | C22 | C22 |
| # reps | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 2 | 1 | 2 | 1 | 2 | 8 | 4 | 3 | 1 |
Matrix representation of C24.375C23 ►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 |
| 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 | 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 |
| 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 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 |
| 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 |
| 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 3 | 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 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 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 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 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 | 4 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 4 | 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 | 1 |
| 3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 3 | 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 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 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],[4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[0,2,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,2,1,0,0,0,0,0,0,2,3,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,2,0,0,0,0,0,0,3,0,0,0,0,0,0,3,0,0,0],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,3,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1],[3,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0] >;
C24.375C23 in GAP, Magma, Sage, TeX
C_2^4._{375}C_2^3 % in TeX
G:=Group("C2^4.375C2^3"); // GroupNames label
G:=SmallGroup(128,1381);
// by ID
G=gap.SmallGroup(128,1381);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,2,560,253,758,723,185,136]);
// Polycyclic
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=e^2=f^2=1,d^2=c*a=a*c,g^2=b,a*b=b*a,e*d*e=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=b*d=d*b,b*e=e*b,b*f=f*b,b*g=g*b,c*d=d*c,f*e*f=c*e=e*c,c*f=f*c,c*g=g*c,d*g=g*d,e*g=g*e>;
// generators/relations