direct product, p-group, metabelian, nilpotent (class 3), monomial
Aliases: C2×C23.24D4, C24.139D4, (C23×C8)⋊6C2, C4.3(C23×C4), C4⋊C4.340C23, (C2×C8).467C23, (C2×C4).173C24, (C22×C8)⋊63C22, D4.17(C22×C4), C4.138(C22×D4), (C22×C4).819D4, C23.376(C2×D4), Q8.17(C22×C4), D4⋊C4⋊97C22, Q8⋊C4⋊99C22, (C2×D4).358C23, C22.81(C4○D8), C4○(C23.24D4), (C2×Q8).331C23, C42⋊C2⋊73C22, (C23×C4).690C22, C22.123(C22×D4), C23.130(C22⋊C4), (C22×C4).1497C23, (C22×D4).551C22, (C22×Q8).455C22, C4○(C2×D4⋊C4), C4○(C2×Q8⋊C4), C2.1(C2×C4○D8), (C2×C4○D4)⋊18C4, C4○D4⋊12(C2×C4), (C2×C4)○2(D4⋊C4), (C2×D4⋊C4)⋊58C2, (C2×C4)○2(Q8⋊C4), (C2×Q8⋊C4)⋊59C2, (C2×D4).226(C2×C4), (C2×C4).1563(C2×D4), C4.121(C2×C22⋊C4), (C2×Q8).204(C2×C4), (C2×C42⋊C2)⋊40C2, (C22×C4)○(D4⋊C4), (C2×C4⋊C4).899C22, (C2×C4).458(C22×C4), (C22×C4).415(C2×C4), (C22×C4)○(Q8⋊C4), (C22×C4○D4).18C2, C22.20(C2×C22⋊C4), C2.35(C22×C22⋊C4), (C2×C4).284(C22⋊C4), (C2×C4○D4).272C22, (C2×C4)○(C23.24D4), (C2×C4)○(C2×D4⋊C4), (C2×C4)○(C2×Q8⋊C4), (C22×C4)○(C2×D4⋊C4), (C22×C4)○(C2×Q8⋊C4), SmallGroup(128,1624)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C2×C23.24D4
G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e4=d, f2=c, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, fbf-1=bd=db, be=eb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=cde3 >
Subgroups: 668 in 396 conjugacy classes, 180 normal (18 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C23, C23, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C4○D4, C4○D4, C24, C24, D4⋊C4, Q8⋊C4, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C42⋊C2, C42⋊C2, C22×C8, C22×C8, C23×C4, C23×C4, C22×D4, C22×D4, C22×Q8, C2×C4○D4, C2×C4○D4, C2×D4⋊C4, C2×Q8⋊C4, C23.24D4, C2×C42⋊C2, C23×C8, C22×C4○D4, C2×C23.24D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, C24, C2×C22⋊C4, C4○D8, C23×C4, C22×D4, C23.24D4, C22×C22⋊C4, C2×C4○D8, C2×C23.24D4
►On 64 points
(1 15)(2 16)(3 9)(4 10)(5 11)(6 12)(7 13)(8 14)(17 48)(18 41)(19 42)(20 43)(21 44)(22 45)(23 46)(24 47)(25 62)(26 63)(27 64)(28 57)(29 58)(30 59)(31 60)(32 61)(33 49)(34 50)(35 51)(36 52)(37 53)(38 54)(39 55)(40 56)
(1 22)(2 23)(3 24)(4 17)(5 18)(6 19)(7 20)(8 21)(9 47)(10 48)(11 41)(12 42)(13 43)(14 44)(15 45)(16 46)(25 56)(26 49)(27 50)(28 51)(29 52)(30 53)(31 54)(32 55)(33 63)(34 64)(35 57)(36 58)(37 59)(38 60)(39 61)(40 62)
(1 35)(2 36)(3 37)(4 38)(5 39)(6 40)(7 33)(8 34)(9 53)(10 54)(11 55)(12 56)(13 49)(14 50)(15 51)(16 52)(17 60)(18 61)(19 62)(20 63)(21 64)(22 57)(23 58)(24 59)(25 42)(26 43)(27 44)(28 45)(29 46)(30 47)(31 48)(32 41)
(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)
(1 34 35 8)(2 7 36 33)(3 40 37 6)(4 5 38 39)(9 56 53 12)(10 11 54 55)(13 52 49 16)(14 15 50 51)(17 22 60 57)(18 64 61 21)(19 20 62 63)(23 24 58 59)(25 26 42 43)(27 32 44 41)(28 48 45 31)(29 30 46 47)
G:=sub<Sym(64)| (1,15)(2,16)(3,9)(4,10)(5,11)(6,12)(7,13)(8,14)(17,48)(18,41)(19,42)(20,43)(21,44)(22,45)(23,46)(24,47)(25,62)(26,63)(27,64)(28,57)(29,58)(30,59)(31,60)(32,61)(33,49)(34,50)(35,51)(36,52)(37,53)(38,54)(39,55)(40,56), (1,22)(2,23)(3,24)(4,17)(5,18)(6,19)(7,20)(8,21)(9,47)(10,48)(11,41)(12,42)(13,43)(14,44)(15,45)(16,46)(25,56)(26,49)(27,50)(28,51)(29,52)(30,53)(31,54)(32,55)(33,63)(34,64)(35,57)(36,58)(37,59)(38,60)(39,61)(40,62), (1,35)(2,36)(3,37)(4,38)(5,39)(6,40)(7,33)(8,34)(9,53)(10,54)(11,55)(12,56)(13,49)(14,50)(15,51)(16,52)(17,60)(18,61)(19,62)(20,63)(21,64)(22,57)(23,58)(24,59)(25,42)(26,43)(27,44)(28,45)(29,46)(30,47)(31,48)(32,41), (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), (1,34,35,8)(2,7,36,33)(3,40,37,6)(4,5,38,39)(9,56,53,12)(10,11,54,55)(13,52,49,16)(14,15,50,51)(17,22,60,57)(18,64,61,21)(19,20,62,63)(23,24,58,59)(25,26,42,43)(27,32,44,41)(28,48,45,31)(29,30,46,47)>;
G:=Group( (1,15)(2,16)(3,9)(4,10)(5,11)(6,12)(7,13)(8,14)(17,48)(18,41)(19,42)(20,43)(21,44)(22,45)(23,46)(24,47)(25,62)(26,63)(27,64)(28,57)(29,58)(30,59)(31,60)(32,61)(33,49)(34,50)(35,51)(36,52)(37,53)(38,54)(39,55)(40,56), (1,22)(2,23)(3,24)(4,17)(5,18)(6,19)(7,20)(8,21)(9,47)(10,48)(11,41)(12,42)(13,43)(14,44)(15,45)(16,46)(25,56)(26,49)(27,50)(28,51)(29,52)(30,53)(31,54)(32,55)(33,63)(34,64)(35,57)(36,58)(37,59)(38,60)(39,61)(40,62), (1,35)(2,36)(3,37)(4,38)(5,39)(6,40)(7,33)(8,34)(9,53)(10,54)(11,55)(12,56)(13,49)(14,50)(15,51)(16,52)(17,60)(18,61)(19,62)(20,63)(21,64)(22,57)(23,58)(24,59)(25,42)(26,43)(27,44)(28,45)(29,46)(30,47)(31,48)(32,41), (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), (1,34,35,8)(2,7,36,33)(3,40,37,6)(4,5,38,39)(9,56,53,12)(10,11,54,55)(13,52,49,16)(14,15,50,51)(17,22,60,57)(18,64,61,21)(19,20,62,63)(23,24,58,59)(25,26,42,43)(27,32,44,41)(28,48,45,31)(29,30,46,47) );
G=PermutationGroup([[(1,15),(2,16),(3,9),(4,10),(5,11),(6,12),(7,13),(8,14),(17,48),(18,41),(19,42),(20,43),(21,44),(22,45),(23,46),(24,47),(25,62),(26,63),(27,64),(28,57),(29,58),(30,59),(31,60),(32,61),(33,49),(34,50),(35,51),(36,52),(37,53),(38,54),(39,55),(40,56)], [(1,22),(2,23),(3,24),(4,17),(5,18),(6,19),(7,20),(8,21),(9,47),(10,48),(11,41),(12,42),(13,43),(14,44),(15,45),(16,46),(25,56),(26,49),(27,50),(28,51),(29,52),(30,53),(31,54),(32,55),(33,63),(34,64),(35,57),(36,58),(37,59),(38,60),(39,61),(40,62)], [(1,35),(2,36),(3,37),(4,38),(5,39),(6,40),(7,33),(8,34),(9,53),(10,54),(11,55),(12,56),(13,49),(14,50),(15,51),(16,52),(17,60),(18,61),(19,62),(20,63),(21,64),(22,57),(23,58),(24,59),(25,42),(26,43),(27,44),(28,45),(29,46),(30,47),(31,48),(32,41)], [(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)], [(1,34,35,8),(2,7,36,33),(3,40,37,6),(4,5,38,39),(9,56,53,12),(10,11,54,55),(13,52,49,16),(14,15,50,51),(17,22,60,57),(18,64,61,21),(19,20,62,63),(23,24,58,59),(25,26,42,43),(27,32,44,41),(28,48,45,31),(29,30,46,47)]])
56 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 2L | 2M | 2N | 2O | 4A | ··· | 4H | 4I | 4J | 4K | 4L | 4M | ··· | 4X | 8A | ··· | 8P |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 2 | ··· | 2 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | D4 | C4○D8 |
| kernel | C2×C23.24D4 | C2×D4⋊C4 | C2×Q8⋊C4 | C23.24D4 | C2×C42⋊C2 | C23×C8 | C22×C4○D4 | C2×C4○D4 | C22×C4 | C24 | C22 |
| # reps | 1 | 2 | 2 | 8 | 1 | 1 | 1 | 16 | 7 | 1 | 16 |
Matrix representation of C2×C23.24D4 ►in GL6(𝔽17)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 |
| 0 | 0 | 0 | 0 | 13 | 0 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 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 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 14 | 3 |
| 0 | 0 | 0 | 0 | 14 | 14 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 14 | 3 |
| 0 | 0 | 0 | 0 | 3 | 3 |
G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,13,0,0,0,0,4,0],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[0,16,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,16,0,0,0,0,0,0,0,14,14,0,0,0,0,3,14],[0,1,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,16,0,0,0,0,0,0,0,14,3,0,0,0,0,3,3] >;
C2×C23.24D4 in GAP, Magma, Sage, TeX
C_2\times C_2^3._{24}D_4 % in TeX
G:=Group("C2xC2^3.24D4"); // GroupNames label
G:=SmallGroup(128,1624);
// by ID
G=gap.SmallGroup(128,1624);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,352,2804,1411,172]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^4=d,f^2=c,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,f*b*f^-1=b*d=d*b,b*e=e*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=c*d*e^3>;
// generators/relations