p-group, metabelian, nilpotent (class 2), monomial
Aliases: C24.5Q8, C24.51D4, C25.4C22, C24.629C23, C23⋊3(C4⋊C4), C24.42(C2×C4), C22.122(C4×D4), C23.719(C2×D4), (C22×C4).121D4, (C23×C4).7C22, C23.129(C2×Q8), C22.66C22≀C2, C2.2(C23⋊2D4), C2.1(C23⋊Q8), C23.342(C4○D4), C22.98(C4⋊D4), C22.24(C4⋊1D4), C23.298(C22×C4), C22.60(C22⋊Q8), C2.1(C23.Q8), C2.1(C23.4Q8), C2.5(C23.7Q8), C2.4(C23.8Q8), C2.6(C23.23D4), C2.1(C23.11D4), C2.2(C23.10D4), C22.48(C4.4D4), C22.23(C42⋊2C2), C22.72(C42⋊C2), C2.5(C24.C22), C2.5(C24.3C22), C22.71(C22.D4), (C22×C4⋊C4)⋊1C2, (C2×C22⋊C4)⋊17C4, (C2×C4)⋊5(C22⋊C4), C22.76(C2×C4⋊C4), (C22×C4).168(C2×C4), (C2×C2.C42)⋊3C2, (C22×C22⋊C4).3C2, C22.139(C2×C22⋊C4), SmallGroup(128,171)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C24.5Q8
G = < a,b,c,d,e,f | a2=b2=c2=d2=e4=1, f2=be2, ab=ba, faf-1=ac=ca, eae-1=ad=da, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=e-1 >
Subgroups: 812 in 390 conjugacy classes, 116 normal (34 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C2×C4, C2×C4, C23, C23, C23, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C24, C24, C24, C2.C42, C2×C22⋊C4, C2×C22⋊C4, C2×C4⋊C4, C23×C4, C23×C4, C25, C2×C2.C42, C2×C2.C42, C22×C22⋊C4, C22×C22⋊C4, C22×C4⋊C4, C24.5Q8
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×Q8, C4○D4, C2×C22⋊C4, C2×C4⋊C4, C42⋊C2, C4×D4, C22≀C2, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C42⋊2C2, C4⋊1D4, C23.7Q8, C23.8Q8, C23.23D4, C24.C22, C24.3C22, C23⋊2D4, C23⋊Q8, C23.10D4, C23.Q8, C23.11D4, C23.4Q8, C24.5Q8
►On 64 points
Generators in S64
(1 43)(2 52)(3 41)(4 50)(5 34)(6 29)(7 36)(8 31)(9 54)(10 59)(11 56)(12 57)(13 49)(14 42)(15 51)(16 44)(17 58)(18 55)(19 60)(20 53)(21 35)(22 30)(23 33)(24 32)(25 40)(26 45)(27 38)(28 47)(37 63)(39 61)(46 64)(48 62)
(1 41)(2 42)(3 43)(4 44)(5 30)(6 31)(7 32)(8 29)(9 46)(10 47)(11 48)(12 45)(13 51)(14 52)(15 49)(16 50)(17 38)(18 39)(19 40)(20 37)(21 33)(22 34)(23 35)(24 36)(25 60)(26 57)(27 58)(28 59)(53 63)(54 64)(55 61)(56 62)
(1 6)(2 7)(3 8)(4 5)(9 25)(10 26)(11 27)(12 28)(13 23)(14 24)(15 21)(16 22)(17 62)(18 63)(19 64)(20 61)(29 43)(30 44)(31 41)(32 42)(33 49)(34 50)(35 51)(36 52)(37 55)(38 56)(39 53)(40 54)(45 59)(46 60)(47 57)(48 58)
(1 15)(2 16)(3 13)(4 14)(5 24)(6 21)(7 22)(8 23)(9 17)(10 18)(11 19)(12 20)(25 62)(26 63)(27 64)(28 61)(29 35)(30 36)(31 33)(32 34)(37 45)(38 46)(39 47)(40 48)(41 49)(42 50)(43 51)(44 52)(53 57)(54 58)(55 59)(56 60)
(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 43 37)(2 17 44 40)(3 20 41 39)(4 19 42 38)(5 64 32 56)(6 63 29 55)(7 62 30 54)(8 61 31 53)(9 52 48 16)(10 51 45 15)(11 50 46 14)(12 49 47 13)(21 26 35 59)(22 25 36 58)(23 28 33 57)(24 27 34 60)
G:=sub<Sym(64)| (1,43)(2,52)(3,41)(4,50)(5,34)(6,29)(7,36)(8,31)(9,54)(10,59)(11,56)(12,57)(13,49)(14,42)(15,51)(16,44)(17,58)(18,55)(19,60)(20,53)(21,35)(22,30)(23,33)(24,32)(25,40)(26,45)(27,38)(28,47)(37,63)(39,61)(46,64)(48,62), (1,41)(2,42)(3,43)(4,44)(5,30)(6,31)(7,32)(8,29)(9,46)(10,47)(11,48)(12,45)(13,51)(14,52)(15,49)(16,50)(17,38)(18,39)(19,40)(20,37)(21,33)(22,34)(23,35)(24,36)(25,60)(26,57)(27,58)(28,59)(53,63)(54,64)(55,61)(56,62), (1,6)(2,7)(3,8)(4,5)(9,25)(10,26)(11,27)(12,28)(13,23)(14,24)(15,21)(16,22)(17,62)(18,63)(19,64)(20,61)(29,43)(30,44)(31,41)(32,42)(33,49)(34,50)(35,51)(36,52)(37,55)(38,56)(39,53)(40,54)(45,59)(46,60)(47,57)(48,58), (1,15)(2,16)(3,13)(4,14)(5,24)(6,21)(7,22)(8,23)(9,17)(10,18)(11,19)(12,20)(25,62)(26,63)(27,64)(28,61)(29,35)(30,36)(31,33)(32,34)(37,45)(38,46)(39,47)(40,48)(41,49)(42,50)(43,51)(44,52)(53,57)(54,58)(55,59)(56,60), (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), (1,18,43,37)(2,17,44,40)(3,20,41,39)(4,19,42,38)(5,64,32,56)(6,63,29,55)(7,62,30,54)(8,61,31,53)(9,52,48,16)(10,51,45,15)(11,50,46,14)(12,49,47,13)(21,26,35,59)(22,25,36,58)(23,28,33,57)(24,27,34,60)>;
G:=Group( (1,43)(2,52)(3,41)(4,50)(5,34)(6,29)(7,36)(8,31)(9,54)(10,59)(11,56)(12,57)(13,49)(14,42)(15,51)(16,44)(17,58)(18,55)(19,60)(20,53)(21,35)(22,30)(23,33)(24,32)(25,40)(26,45)(27,38)(28,47)(37,63)(39,61)(46,64)(48,62), (1,41)(2,42)(3,43)(4,44)(5,30)(6,31)(7,32)(8,29)(9,46)(10,47)(11,48)(12,45)(13,51)(14,52)(15,49)(16,50)(17,38)(18,39)(19,40)(20,37)(21,33)(22,34)(23,35)(24,36)(25,60)(26,57)(27,58)(28,59)(53,63)(54,64)(55,61)(56,62), (1,6)(2,7)(3,8)(4,5)(9,25)(10,26)(11,27)(12,28)(13,23)(14,24)(15,21)(16,22)(17,62)(18,63)(19,64)(20,61)(29,43)(30,44)(31,41)(32,42)(33,49)(34,50)(35,51)(36,52)(37,55)(38,56)(39,53)(40,54)(45,59)(46,60)(47,57)(48,58), (1,15)(2,16)(3,13)(4,14)(5,24)(6,21)(7,22)(8,23)(9,17)(10,18)(11,19)(12,20)(25,62)(26,63)(27,64)(28,61)(29,35)(30,36)(31,33)(32,34)(37,45)(38,46)(39,47)(40,48)(41,49)(42,50)(43,51)(44,52)(53,57)(54,58)(55,59)(56,60), (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), (1,18,43,37)(2,17,44,40)(3,20,41,39)(4,19,42,38)(5,64,32,56)(6,63,29,55)(7,62,30,54)(8,61,31,53)(9,52,48,16)(10,51,45,15)(11,50,46,14)(12,49,47,13)(21,26,35,59)(22,25,36,58)(23,28,33,57)(24,27,34,60) );
G=PermutationGroup([[(1,43),(2,52),(3,41),(4,50),(5,34),(6,29),(7,36),(8,31),(9,54),(10,59),(11,56),(12,57),(13,49),(14,42),(15,51),(16,44),(17,58),(18,55),(19,60),(20,53),(21,35),(22,30),(23,33),(24,32),(25,40),(26,45),(27,38),(28,47),(37,63),(39,61),(46,64),(48,62)], [(1,41),(2,42),(3,43),(4,44),(5,30),(6,31),(7,32),(8,29),(9,46),(10,47),(11,48),(12,45),(13,51),(14,52),(15,49),(16,50),(17,38),(18,39),(19,40),(20,37),(21,33),(22,34),(23,35),(24,36),(25,60),(26,57),(27,58),(28,59),(53,63),(54,64),(55,61),(56,62)], [(1,6),(2,7),(3,8),(4,5),(9,25),(10,26),(11,27),(12,28),(13,23),(14,24),(15,21),(16,22),(17,62),(18,63),(19,64),(20,61),(29,43),(30,44),(31,41),(32,42),(33,49),(34,50),(35,51),(36,52),(37,55),(38,56),(39,53),(40,54),(45,59),(46,60),(47,57),(48,58)], [(1,15),(2,16),(3,13),(4,14),(5,24),(6,21),(7,22),(8,23),(9,17),(10,18),(11,19),(12,20),(25,62),(26,63),(27,64),(28,61),(29,35),(30,36),(31,33),(32,34),(37,45),(38,46),(39,47),(40,48),(41,49),(42,50),(43,51),(44,52),(53,57),(54,58),(55,59),(56,60)], [(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)], [(1,18,43,37),(2,17,44,40),(3,20,41,39),(4,19,42,38),(5,64,32,56),(6,63,29,55),(7,62,30,54),(8,61,31,53),(9,52,48,16),(10,51,45,15),(11,50,46,14),(12,49,47,13),(21,26,35,59),(22,25,36,58),(23,28,33,57),(24,27,34,60)]])
44 conjugacy classes
| class | 1 | 2A | ··· | 2O | 2P | 2Q | 2R | 2S | 4A | ··· | 4X |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
| size | 1 | 1 | ··· | 1 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C4 | D4 | D4 | Q8 | C4○D4 |
| kernel | C24.5Q8 | C2×C2.C42 | C22×C22⋊C4 | C22×C4⋊C4 | C2×C22⋊C4 | C22×C4 | C24 | C24 | C23 |
| # reps | 1 | 3 | 3 | 1 | 8 | 12 | 2 | 2 | 12 |
Matrix representation of C24.5Q8 ►in GL7(𝔽5)
| 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 4 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 4 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 4 |
| 4 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 3 | 4 | 0 | 0 | 0 | 0 |
| 0 | 3 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 4 | 0 |
| 3 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 2 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 2 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 3 |
| 0 | 0 | 0 | 0 | 0 | 3 | 0 |
G:=sub<GL(7,GF(5))| [1,0,0,0,0,0,0,0,4,0,0,0,0,0,0,2,1,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,1],[4,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,0,3,3,0,0,0,0,0,4,2,0,0,0,0,0,0,0,0,4,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,1,0],[3,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,3,0,0,0,0,0,3,0] >;
C24.5Q8 in GAP, Magma, Sage, TeX
C_2^4._5Q_8
% in TeX
G:=Group("C2^4.5Q8"); // GroupNames label
G:=SmallGroup(128,171);
// by ID
G=gap.SmallGroup(128,171);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,2,448,141,64,422,387]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=e^4=1,f^2=b*e^2,a*b=b*a,f*a*f^-1=a*c=c*a,e*a*e^-1=a*d=d*a,b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=e^-1>;
// generators/relations