p-group, metabelian, nilpotent (class 3), monomial
Aliases: C24.74D4, C4⋊C4⋊31D4, C4.62(C4×D4), C4⋊D4⋊11C4, C2.5(D4⋊D4), C23.767(C2×D4), (C22×C4).289D4, C4.128(C4⋊D4), C22.4Q16⋊13C2, C22.92C22≀C2, C22.54(C4○D8), (C22×C8).34C22, C22.75(C8⋊C22), C23.80(C22⋊C4), (C23×C4).257C22, (C22×D4).16C22, (C22×C4).1362C23, C2.3(C23.19D4), C2.25(C23.24D4), C2.20(C23.37D4), C2.14(C23.23D4), C22.87(C22.D4), C4⋊C4.70(C2×C4), (C2×D4⋊C4)⋊4C2, (C2×C22⋊C8)⋊15C2, (C2×D4).71(C2×C4), (C2×C4).994(C2×D4), (C2×C4⋊D4).7C2, (C2×C42⋊C2)⋊1C2, (C2×C4).758(C4○D4), (C2×C4⋊C4).764C22, (C2×C4).380(C22×C4), (C22×C4).279(C2×C4), (C2×C4).192(C22⋊C4), C22.266(C2×C22⋊C4), SmallGroup(128,607)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C24.74D4
G = < a,b,c,d,e,f | a2=b2=c2=d2=f2=1, e4=d, ab=ba, eae-1=ac=ca, ad=da, faf=acd, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef=be3 >
Subgroups: 476 in 206 conjugacy classes, 64 normal (20 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, C23, C23, C23, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C24, C24, C22⋊C8, D4⋊C4, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C42⋊C2, C4⋊D4, C4⋊D4, C22×C8, C23×C4, C22×D4, C22×D4, C22.4Q16, C2×C22⋊C8, C2×D4⋊C4, C2×C42⋊C2, C2×C4⋊D4, C24.74D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, C4○D4, C2×C22⋊C4, C4×D4, C22≀C2, C4⋊D4, C22.D4, C4○D8, C8⋊C22, C23.23D4, C23.24D4, C23.37D4, D4⋊D4, C23.19D4, C24.74D4
►On 64 points
(1 63)(2 45)(3 57)(4 47)(5 59)(6 41)(7 61)(8 43)(9 34)(10 20)(11 36)(12 22)(13 38)(14 24)(15 40)(16 18)(17 27)(19 29)(21 31)(23 25)(26 39)(28 33)(30 35)(32 37)(42 55)(44 49)(46 51)(48 53)(50 64)(52 58)(54 60)(56 62)
(1 9)(2 10)(3 11)(4 12)(5 13)(6 14)(7 15)(8 16)(17 42)(18 43)(19 44)(20 45)(21 46)(22 47)(23 48)(24 41)(25 53)(26 54)(27 55)(28 56)(29 49)(30 50)(31 51)(32 52)(33 62)(34 63)(35 64)(36 57)(37 58)(38 59)(39 60)(40 61)
(1 49)(2 50)(3 51)(4 52)(5 53)(6 54)(7 55)(8 56)(9 29)(10 30)(11 31)(12 32)(13 25)(14 26)(15 27)(16 28)(17 40)(18 33)(19 34)(20 35)(21 36)(22 37)(23 38)(24 39)(41 60)(42 61)(43 62)(44 63)(45 64)(46 57)(47 58)(48 59)
(1 5)(2 6)(3 7)(4 8)(9 13)(10 14)(11 15)(12 16)(17 21)(18 22)(19 23)(20 24)(25 29)(26 30)(27 31)(28 32)(33 37)(34 38)(35 39)(36 40)(41 45)(42 46)(43 47)(44 48)(49 53)(50 54)(51 55)(52 56)(57 61)(58 62)(59 63)(60 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)
(2 12)(3 7)(4 10)(6 16)(8 14)(11 15)(17 40)(18 64)(19 38)(20 62)(21 36)(22 60)(23 34)(24 58)(26 56)(27 31)(28 54)(30 52)(32 50)(33 45)(35 43)(37 41)(39 47)(42 61)(44 59)(46 57)(48 63)(51 55)
G:=sub<Sym(64)| (1,63)(2,45)(3,57)(4,47)(5,59)(6,41)(7,61)(8,43)(9,34)(10,20)(11,36)(12,22)(13,38)(14,24)(15,40)(16,18)(17,27)(19,29)(21,31)(23,25)(26,39)(28,33)(30,35)(32,37)(42,55)(44,49)(46,51)(48,53)(50,64)(52,58)(54,60)(56,62), (1,9)(2,10)(3,11)(4,12)(5,13)(6,14)(7,15)(8,16)(17,42)(18,43)(19,44)(20,45)(21,46)(22,47)(23,48)(24,41)(25,53)(26,54)(27,55)(28,56)(29,49)(30,50)(31,51)(32,52)(33,62)(34,63)(35,64)(36,57)(37,58)(38,59)(39,60)(40,61), (1,49)(2,50)(3,51)(4,52)(5,53)(6,54)(7,55)(8,56)(9,29)(10,30)(11,31)(12,32)(13,25)(14,26)(15,27)(16,28)(17,40)(18,33)(19,34)(20,35)(21,36)(22,37)(23,38)(24,39)(41,60)(42,61)(43,62)(44,63)(45,64)(46,57)(47,58)(48,59), (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16)(17,21)(18,22)(19,23)(20,24)(25,29)(26,30)(27,31)(28,32)(33,37)(34,38)(35,39)(36,40)(41,45)(42,46)(43,47)(44,48)(49,53)(50,54)(51,55)(52,56)(57,61)(58,62)(59,63)(60,64), (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), (2,12)(3,7)(4,10)(6,16)(8,14)(11,15)(17,40)(18,64)(19,38)(20,62)(21,36)(22,60)(23,34)(24,58)(26,56)(27,31)(28,54)(30,52)(32,50)(33,45)(35,43)(37,41)(39,47)(42,61)(44,59)(46,57)(48,63)(51,55)>;
G:=Group( (1,63)(2,45)(3,57)(4,47)(5,59)(6,41)(7,61)(8,43)(9,34)(10,20)(11,36)(12,22)(13,38)(14,24)(15,40)(16,18)(17,27)(19,29)(21,31)(23,25)(26,39)(28,33)(30,35)(32,37)(42,55)(44,49)(46,51)(48,53)(50,64)(52,58)(54,60)(56,62), (1,9)(2,10)(3,11)(4,12)(5,13)(6,14)(7,15)(8,16)(17,42)(18,43)(19,44)(20,45)(21,46)(22,47)(23,48)(24,41)(25,53)(26,54)(27,55)(28,56)(29,49)(30,50)(31,51)(32,52)(33,62)(34,63)(35,64)(36,57)(37,58)(38,59)(39,60)(40,61), (1,49)(2,50)(3,51)(4,52)(5,53)(6,54)(7,55)(8,56)(9,29)(10,30)(11,31)(12,32)(13,25)(14,26)(15,27)(16,28)(17,40)(18,33)(19,34)(20,35)(21,36)(22,37)(23,38)(24,39)(41,60)(42,61)(43,62)(44,63)(45,64)(46,57)(47,58)(48,59), (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16)(17,21)(18,22)(19,23)(20,24)(25,29)(26,30)(27,31)(28,32)(33,37)(34,38)(35,39)(36,40)(41,45)(42,46)(43,47)(44,48)(49,53)(50,54)(51,55)(52,56)(57,61)(58,62)(59,63)(60,64), (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), (2,12)(3,7)(4,10)(6,16)(8,14)(11,15)(17,40)(18,64)(19,38)(20,62)(21,36)(22,60)(23,34)(24,58)(26,56)(27,31)(28,54)(30,52)(32,50)(33,45)(35,43)(37,41)(39,47)(42,61)(44,59)(46,57)(48,63)(51,55) );
G=PermutationGroup([[(1,63),(2,45),(3,57),(4,47),(5,59),(6,41),(7,61),(8,43),(9,34),(10,20),(11,36),(12,22),(13,38),(14,24),(15,40),(16,18),(17,27),(19,29),(21,31),(23,25),(26,39),(28,33),(30,35),(32,37),(42,55),(44,49),(46,51),(48,53),(50,64),(52,58),(54,60),(56,62)], [(1,9),(2,10),(3,11),(4,12),(5,13),(6,14),(7,15),(8,16),(17,42),(18,43),(19,44),(20,45),(21,46),(22,47),(23,48),(24,41),(25,53),(26,54),(27,55),(28,56),(29,49),(30,50),(31,51),(32,52),(33,62),(34,63),(35,64),(36,57),(37,58),(38,59),(39,60),(40,61)], [(1,49),(2,50),(3,51),(4,52),(5,53),(6,54),(7,55),(8,56),(9,29),(10,30),(11,31),(12,32),(13,25),(14,26),(15,27),(16,28),(17,40),(18,33),(19,34),(20,35),(21,36),(22,37),(23,38),(24,39),(41,60),(42,61),(43,62),(44,63),(45,64),(46,57),(47,58),(48,59)], [(1,5),(2,6),(3,7),(4,8),(9,13),(10,14),(11,15),(12,16),(17,21),(18,22),(19,23),(20,24),(25,29),(26,30),(27,31),(28,32),(33,37),(34,38),(35,39),(36,40),(41,45),(42,46),(43,47),(44,48),(49,53),(50,54),(51,55),(52,56),(57,61),(58,62),(59,63),(60,64)], [(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)], [(2,12),(3,7),(4,10),(6,16),(8,14),(11,15),(17,40),(18,64),(19,38),(20,62),(21,36),(22,60),(23,34),(24,58),(26,56),(27,31),(28,54),(30,52),(32,50),(33,45),(35,43),(37,41),(39,47),(42,61),(44,59),(46,57),(48,63),(51,55)]])
38 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 4A | ··· | 4H | 4I | ··· | 4P | 4Q | 4R | 8A | ··· | 8H |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | 4 | 8 | ··· | 8 |
| size | 1 | 1 | ··· | 1 | 4 | 4 | 8 | 8 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | 8 | 4 | ··· | 4 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | D4 | D4 | C4○D4 | C4○D8 | C8⋊C22 |
| kernel | C24.74D4 | C22.4Q16 | C2×C22⋊C8 | C2×D4⋊C4 | C2×C42⋊C2 | C2×C4⋊D4 | C4⋊D4 | C4⋊C4 | C22×C4 | C24 | C2×C4 | C22 | C22 |
| # reps | 1 | 2 | 1 | 2 | 1 | 1 | 8 | 4 | 3 | 1 | 4 | 8 | 2 |
Matrix representation of C24.74D4 ►in GL5(𝔽17)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 2 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 4 | 8 |
| 0 | 0 | 0 | 13 | 13 |
| 16 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 16 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 16 |
| 4 | 0 | 0 | 0 | 0 |
| 0 | 16 | 15 | 0 | 0 |
| 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 6 |
| 0 | 0 | 0 | 14 | 11 |
| 16 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 16 | 16 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 16 | 16 |
G:=sub<GL(5,GF(17))| [1,0,0,0,0,0,1,0,0,0,0,2,16,0,0,0,0,0,4,13,0,0,0,8,13],[16,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,16,0,0,0,0,0,16],[1,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,16,0,0,0,0,0,16],[4,0,0,0,0,0,16,1,0,0,0,15,1,0,0,0,0,0,0,14,0,0,0,6,11],[16,0,0,0,0,0,1,16,0,0,0,0,16,0,0,0,0,0,1,16,0,0,0,0,16] >;
C24.74D4 in GAP, Magma, Sage, TeX
C_2^4._{74}D_4 % in TeX
G:=Group("C2^4.74D4"); // GroupNames label
G:=SmallGroup(128,607);
// by ID
G=gap.SmallGroup(128,607);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,736,422,352,1018,248,2028,124]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=f^2=1,e^4=d,a*b=b*a,e*a*e^-1=a*c=c*a,a*d=d*a,f*a*f=a*c*d,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=b*e^3>;
// generators/relations