direct product, p-group, metabelian, nilpotent (class 2), monomial
Aliases: C2×C23.23D4, C24.138D4, C25.14C22, C23.169C24, C24.532C23, C24⋊8(C2×C4), (C24×C4)⋊1C2, (C22×C4)⋊51D4, (C22×D4)⋊19C4, C23⋊4(C22×C4), (D4×C23).5C2, C23⋊6(C22⋊C4), (C23×C4)⋊55C22, C22.107(C4×D4), C23.598(C2×D4), C22.60(C23×C4), C22.67(C22×D4), C22.107C22≀C2, C23.358(C4○D4), (C22×C4).447C23, C22.156(C4⋊D4), C2.C42⋊59C22, (C22×D4).462C22, C22.98(C22.D4), C2.6(C2×C4×D4), (C2×C4)⋊20(C2×D4), (C2×D4)⋊36(C2×C4), (C2×C4)⋊5(C22×C4), C2.2(C2×C4⋊D4), (C22×C4)⋊16(C2×C4), C2.3(C2×C22≀C2), C22⋊2(C2×C22⋊C4), (C22×C22⋊C4)⋊4C2, C22.61(C2×C4○D4), C2.8(C22×C22⋊C4), (C2×C22⋊C4)⋊69C22, C2.3(C2×C22.D4), (C2×C2.C42)⋊16C2, SmallGroup(128,1019)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 1452 in 788 conjugacy classes, 236 normal (16 characteristic)
C1, C2 [×3], C2 [×12], C2 [×12], C4 [×16], C22 [×3], C22 [×40], C22 [×92], C2×C4 [×12], C2×C4 [×88], D4 [×32], C23, C23 [×46], C23 [×108], C22⋊C4 [×24], C22×C4 [×22], C22×C4 [×80], C2×D4 [×16], C2×D4 [×48], C24, C24 [×20], C24 [×20], C2.C42 [×8], C2×C22⋊C4 [×12], C2×C22⋊C4 [×12], C23×C4, C23×C4 [×8], C23×C4 [×12], C22×D4 [×12], C22×D4 [×8], C25 [×2], C2×C2.C42 [×2], C23.23D4 [×8], C22×C22⋊C4, C22×C22⋊C4 [×2], C24×C4, D4×C23, C2×C23.23D4
Quotients:
C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], D4 [×16], C23 [×15], C22⋊C4 [×16], C22×C4 [×14], C2×D4 [×24], C4○D4 [×4], C24, C2×C22⋊C4 [×12], C4×D4 [×8], C22≀C2 [×4], C4⋊D4 [×8], C22.D4 [×4], C23×C4, C22×D4 [×4], C2×C4○D4 [×2], C23.23D4 [×8], C22×C22⋊C4, C2×C4×D4 [×2], C2×C22≀C2, C2×C4⋊D4 [×2], C2×C22.D4, C2×C23.23D4
Generators and relations
G = < a,b,c,d,e,f | a2=b2=c2=d2=e4=f2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, fbf=bc=cb, bd=db, be=eb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef=de-1 >
(1 20)(2 17)(3 18)(4 19)(5 29)(6 30)(7 31)(8 32)(9 41)(10 42)(11 43)(12 44)(13 58)(14 59)(15 60)(16 57)(21 50)(22 51)(23 52)(24 49)(25 64)(26 61)(27 62)(28 63)(33 54)(34 55)(35 56)(36 53)(37 45)(38 46)(39 47)(40 48)
(1 39)(2 40)(3 37)(4 38)(5 55)(6 56)(7 53)(8 54)(9 49)(10 50)(11 51)(12 52)(13 28)(14 25)(15 26)(16 27)(17 48)(18 45)(19 46)(20 47)(21 42)(22 43)(23 44)(24 41)(29 34)(30 35)(31 36)(32 33)(57 62)(58 63)(59 64)(60 61)
(1 23)(2 24)(3 21)(4 22)(5 26)(6 27)(7 28)(8 25)(9 48)(10 45)(11 46)(12 47)(13 53)(14 54)(15 55)(16 56)(17 49)(18 50)(19 51)(20 52)(29 61)(30 62)(31 63)(32 64)(33 59)(34 60)(35 57)(36 58)(37 42)(38 43)(39 44)(40 41)
(1 34)(2 35)(3 36)(4 33)(5 47)(6 48)(7 45)(8 46)(9 27)(10 28)(11 25)(12 26)(13 50)(14 51)(15 52)(16 49)(17 56)(18 53)(19 54)(20 55)(21 58)(22 59)(23 60)(24 57)(29 39)(30 40)(31 37)(32 38)(41 62)(42 63)(43 64)(44 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 15)(2 51)(3 13)(4 49)(5 39)(6 32)(7 37)(8 30)(9 43)(10 63)(11 41)(12 61)(14 35)(16 33)(17 22)(18 58)(19 24)(20 60)(21 53)(23 55)(25 62)(26 44)(27 64)(28 42)(29 47)(31 45)(34 52)(36 50)(38 48)(40 46)(54 57)(56 59)
G:=sub<Sym(64)| (1,20)(2,17)(3,18)(4,19)(5,29)(6,30)(7,31)(8,32)(9,41)(10,42)(11,43)(12,44)(13,58)(14,59)(15,60)(16,57)(21,50)(22,51)(23,52)(24,49)(25,64)(26,61)(27,62)(28,63)(33,54)(34,55)(35,56)(36,53)(37,45)(38,46)(39,47)(40,48), (1,39)(2,40)(3,37)(4,38)(5,55)(6,56)(7,53)(8,54)(9,49)(10,50)(11,51)(12,52)(13,28)(14,25)(15,26)(16,27)(17,48)(18,45)(19,46)(20,47)(21,42)(22,43)(23,44)(24,41)(29,34)(30,35)(31,36)(32,33)(57,62)(58,63)(59,64)(60,61), (1,23)(2,24)(3,21)(4,22)(5,26)(6,27)(7,28)(8,25)(9,48)(10,45)(11,46)(12,47)(13,53)(14,54)(15,55)(16,56)(17,49)(18,50)(19,51)(20,52)(29,61)(30,62)(31,63)(32,64)(33,59)(34,60)(35,57)(36,58)(37,42)(38,43)(39,44)(40,41), (1,34)(2,35)(3,36)(4,33)(5,47)(6,48)(7,45)(8,46)(9,27)(10,28)(11,25)(12,26)(13,50)(14,51)(15,52)(16,49)(17,56)(18,53)(19,54)(20,55)(21,58)(22,59)(23,60)(24,57)(29,39)(30,40)(31,37)(32,38)(41,62)(42,63)(43,64)(44,61), (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), (1,15)(2,51)(3,13)(4,49)(5,39)(6,32)(7,37)(8,30)(9,43)(10,63)(11,41)(12,61)(14,35)(16,33)(17,22)(18,58)(19,24)(20,60)(21,53)(23,55)(25,62)(26,44)(27,64)(28,42)(29,47)(31,45)(34,52)(36,50)(38,48)(40,46)(54,57)(56,59)>;
G:=Group( (1,20)(2,17)(3,18)(4,19)(5,29)(6,30)(7,31)(8,32)(9,41)(10,42)(11,43)(12,44)(13,58)(14,59)(15,60)(16,57)(21,50)(22,51)(23,52)(24,49)(25,64)(26,61)(27,62)(28,63)(33,54)(34,55)(35,56)(36,53)(37,45)(38,46)(39,47)(40,48), (1,39)(2,40)(3,37)(4,38)(5,55)(6,56)(7,53)(8,54)(9,49)(10,50)(11,51)(12,52)(13,28)(14,25)(15,26)(16,27)(17,48)(18,45)(19,46)(20,47)(21,42)(22,43)(23,44)(24,41)(29,34)(30,35)(31,36)(32,33)(57,62)(58,63)(59,64)(60,61), (1,23)(2,24)(3,21)(4,22)(5,26)(6,27)(7,28)(8,25)(9,48)(10,45)(11,46)(12,47)(13,53)(14,54)(15,55)(16,56)(17,49)(18,50)(19,51)(20,52)(29,61)(30,62)(31,63)(32,64)(33,59)(34,60)(35,57)(36,58)(37,42)(38,43)(39,44)(40,41), (1,34)(2,35)(3,36)(4,33)(5,47)(6,48)(7,45)(8,46)(9,27)(10,28)(11,25)(12,26)(13,50)(14,51)(15,52)(16,49)(17,56)(18,53)(19,54)(20,55)(21,58)(22,59)(23,60)(24,57)(29,39)(30,40)(31,37)(32,38)(41,62)(42,63)(43,64)(44,61), (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), (1,15)(2,51)(3,13)(4,49)(5,39)(6,32)(7,37)(8,30)(9,43)(10,63)(11,41)(12,61)(14,35)(16,33)(17,22)(18,58)(19,24)(20,60)(21,53)(23,55)(25,62)(26,44)(27,64)(28,42)(29,47)(31,45)(34,52)(36,50)(38,48)(40,46)(54,57)(56,59) );
G=PermutationGroup([(1,20),(2,17),(3,18),(4,19),(5,29),(6,30),(7,31),(8,32),(9,41),(10,42),(11,43),(12,44),(13,58),(14,59),(15,60),(16,57),(21,50),(22,51),(23,52),(24,49),(25,64),(26,61),(27,62),(28,63),(33,54),(34,55),(35,56),(36,53),(37,45),(38,46),(39,47),(40,48)], [(1,39),(2,40),(3,37),(4,38),(5,55),(6,56),(7,53),(8,54),(9,49),(10,50),(11,51),(12,52),(13,28),(14,25),(15,26),(16,27),(17,48),(18,45),(19,46),(20,47),(21,42),(22,43),(23,44),(24,41),(29,34),(30,35),(31,36),(32,33),(57,62),(58,63),(59,64),(60,61)], [(1,23),(2,24),(3,21),(4,22),(5,26),(6,27),(7,28),(8,25),(9,48),(10,45),(11,46),(12,47),(13,53),(14,54),(15,55),(16,56),(17,49),(18,50),(19,51),(20,52),(29,61),(30,62),(31,63),(32,64),(33,59),(34,60),(35,57),(36,58),(37,42),(38,43),(39,44),(40,41)], [(1,34),(2,35),(3,36),(4,33),(5,47),(6,48),(7,45),(8,46),(9,27),(10,28),(11,25),(12,26),(13,50),(14,51),(15,52),(16,49),(17,56),(18,53),(19,54),(20,55),(21,58),(22,59),(23,60),(24,57),(29,39),(30,40),(31,37),(32,38),(41,62),(42,63),(43,64),(44,61)], [(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)], [(1,15),(2,51),(3,13),(4,49),(5,39),(6,32),(7,37),(8,30),(9,43),(10,63),(11,41),(12,61),(14,35),(16,33),(17,22),(18,58),(19,24),(20,60),(21,53),(23,55),(25,62),(26,44),(27,64),(28,42),(29,47),(31,45),(34,52),(36,50),(38,48),(40,46),(54,57),(56,59)])
Matrix representation ►G ⊆ GL6(𝔽5)
| 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 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 2 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 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 |
| 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 |
| 1 | 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 | 0 | 0 | 4 | 2 |
| 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 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 4 | 1 |
G:=sub<GL(6,GF(5))| [4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[4,0,0,0,0,0,0,1,0,0,0,0,0,0,0,4,0,0,0,0,4,0,0,0,0,0,0,0,4,0,0,0,0,0,2,1],[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],[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],[1,0,0,0,0,0,0,3,0,0,0,0,0,0,2,0,0,0,0,0,0,2,0,0,0,0,0,0,4,0,0,0,0,0,2,1],[1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,4,4,0,0,0,0,0,1] >;
56 conjugacy classes
| class | 1 | 2A | ··· | 2O | 2P | ··· | 2W | 2X | 2Y | 2Z | 2AA | 4A | ··· | 4P | 4Q | ··· | 4AB |
| order | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
| size | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | D4 | C4○D4 |
| kernel | C2×C23.23D4 | C2×C2.C42 | C23.23D4 | C22×C22⋊C4 | C24×C4 | D4×C23 | C22×D4 | C22×C4 | C24 | C23 |
| # reps | 1 | 2 | 8 | 3 | 1 | 1 | 16 | 8 | 8 | 8 |
In GAP, Magma, Sage, TeX
C_2\times C_2^3._{23}D_4 % in TeX
G:=Group("C2xC2^3.23D4"); // GroupNames label
G:=SmallGroup(128,1019);
// by ID
G=gap.SmallGroup(128,1019);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,568,758]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=e^4=f^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,f*b*f=b*c=c*b,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=d*e^-1>;
// generators/relations