direct product, p-group, metabelian, nilpotent (class 2), monomial
Aliases: C2×C23.34D4, C24.164D4, C25.84C22, C23.161C24, C24.640C23, (C23×C4)⋊15C4, (C24×C4).5C2, C24.121(C2×C4), C23.821(C2×D4), C22.52(C23×C4), C22.61(C22×D4), C23.354(C4○D4), (C23×C4).642C22, C23.206(C22×C4), (C22×C4).439C23, C23.205(C22⋊C4), C2.C42⋊49C22, C22.64(C42⋊C2), C22.96(C22.D4), (C22×C4)⋊50(C2×C4), C2.7(C2×C42⋊C2), C22.54(C2×C4○D4), C2.6(C22×C22⋊C4), (C2×C2.C42)⋊7C2, (C2×C4).484(C22×C4), C22.72(C2×C22⋊C4), C2.1(C2×C22.D4), (C22×C22⋊C4).10C2, (C2×C22⋊C4).410C22, SmallGroup(128,1011)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 972 in 572 conjugacy classes, 220 normal (8 characteristic)
C1, C2, C2 [×14], C2 [×8], C4 [×16], C22, C22 [×42], C22 [×56], C2×C4 [×8], C2×C4 [×96], C23, C23 [×42], C23 [×56], C22⋊C4 [×16], C22×C4 [×36], C22×C4 [×80], C24, C24 [×14], C24 [×8], C2.C42 [×16], C2×C22⋊C4 [×8], C2×C22⋊C4 [×8], C23×C4 [×18], C23×C4 [×8], C25, C2×C2.C42 [×4], C23.34D4 [×8], C22×C22⋊C4 [×2], C24×C4, C2×C23.34D4
Quotients:
C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], D4 [×8], C23 [×15], C22⋊C4 [×16], C22×C4 [×14], C2×D4 [×12], C4○D4 [×8], C24, C2×C22⋊C4 [×12], C42⋊C2 [×8], C22.D4 [×16], C23×C4, C22×D4 [×2], C2×C4○D4 [×4], C23.34D4 [×8], C22×C22⋊C4, C2×C42⋊C2 [×2], C2×C22.D4 [×4], C2×C23.34D4
Generators and relations
G = < a,b,c,d,e,f | a2=b2=c2=d2=e4=1, f2=dc=cd, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ebe-1=fbf-1=bc=cb, bd=db, ce=ec, cf=fc, de=ed, df=fd, fef-1=de-1 >
(1 29)(2 30)(3 31)(4 32)(5 61)(6 62)(7 63)(8 64)(9 19)(10 20)(11 17)(12 18)(13 22)(14 23)(15 24)(16 21)(25 49)(26 50)(27 51)(28 52)(33 54)(34 55)(35 56)(36 53)(37 43)(38 44)(39 41)(40 42)(45 59)(46 60)(47 57)(48 58)
(2 26)(4 28)(6 38)(8 40)(10 35)(12 33)(13 48)(15 46)(18 54)(20 56)(22 58)(24 60)(30 50)(32 52)(42 64)(44 62)
(1 25)(2 26)(3 27)(4 28)(5 37)(6 38)(7 39)(8 40)(9 34)(10 35)(11 36)(12 33)(13 48)(14 45)(15 46)(16 47)(17 53)(18 54)(19 55)(20 56)(21 57)(22 58)(23 59)(24 60)(29 49)(30 50)(31 51)(32 52)(41 63)(42 64)(43 61)(44 62)
(1 53)(2 54)(3 55)(4 56)(5 21)(6 22)(7 23)(8 24)(9 51)(10 52)(11 49)(12 50)(13 62)(14 63)(15 64)(16 61)(17 25)(18 26)(19 27)(20 28)(29 36)(30 33)(31 34)(32 35)(37 57)(38 58)(39 59)(40 60)(41 45)(42 46)(43 47)(44 48)
(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 24 17 40)(2 7 18 59)(3 22 19 38)(4 5 20 57)(6 27 58 55)(8 25 60 53)(9 44 31 13)(10 47 32 61)(11 42 29 15)(12 45 30 63)(14 50 41 33)(16 52 43 35)(21 28 37 56)(23 26 39 54)(34 62 51 48)(36 64 49 46)
G:=sub<Sym(64)| (1,29)(2,30)(3,31)(4,32)(5,61)(6,62)(7,63)(8,64)(9,19)(10,20)(11,17)(12,18)(13,22)(14,23)(15,24)(16,21)(25,49)(26,50)(27,51)(28,52)(33,54)(34,55)(35,56)(36,53)(37,43)(38,44)(39,41)(40,42)(45,59)(46,60)(47,57)(48,58), (2,26)(4,28)(6,38)(8,40)(10,35)(12,33)(13,48)(15,46)(18,54)(20,56)(22,58)(24,60)(30,50)(32,52)(42,64)(44,62), (1,25)(2,26)(3,27)(4,28)(5,37)(6,38)(7,39)(8,40)(9,34)(10,35)(11,36)(12,33)(13,48)(14,45)(15,46)(16,47)(17,53)(18,54)(19,55)(20,56)(21,57)(22,58)(23,59)(24,60)(29,49)(30,50)(31,51)(32,52)(41,63)(42,64)(43,61)(44,62), (1,53)(2,54)(3,55)(4,56)(5,21)(6,22)(7,23)(8,24)(9,51)(10,52)(11,49)(12,50)(13,62)(14,63)(15,64)(16,61)(17,25)(18,26)(19,27)(20,28)(29,36)(30,33)(31,34)(32,35)(37,57)(38,58)(39,59)(40,60)(41,45)(42,46)(43,47)(44,48), (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), (1,24,17,40)(2,7,18,59)(3,22,19,38)(4,5,20,57)(6,27,58,55)(8,25,60,53)(9,44,31,13)(10,47,32,61)(11,42,29,15)(12,45,30,63)(14,50,41,33)(16,52,43,35)(21,28,37,56)(23,26,39,54)(34,62,51,48)(36,64,49,46)>;
G:=Group( (1,29)(2,30)(3,31)(4,32)(5,61)(6,62)(7,63)(8,64)(9,19)(10,20)(11,17)(12,18)(13,22)(14,23)(15,24)(16,21)(25,49)(26,50)(27,51)(28,52)(33,54)(34,55)(35,56)(36,53)(37,43)(38,44)(39,41)(40,42)(45,59)(46,60)(47,57)(48,58), (2,26)(4,28)(6,38)(8,40)(10,35)(12,33)(13,48)(15,46)(18,54)(20,56)(22,58)(24,60)(30,50)(32,52)(42,64)(44,62), (1,25)(2,26)(3,27)(4,28)(5,37)(6,38)(7,39)(8,40)(9,34)(10,35)(11,36)(12,33)(13,48)(14,45)(15,46)(16,47)(17,53)(18,54)(19,55)(20,56)(21,57)(22,58)(23,59)(24,60)(29,49)(30,50)(31,51)(32,52)(41,63)(42,64)(43,61)(44,62), (1,53)(2,54)(3,55)(4,56)(5,21)(6,22)(7,23)(8,24)(9,51)(10,52)(11,49)(12,50)(13,62)(14,63)(15,64)(16,61)(17,25)(18,26)(19,27)(20,28)(29,36)(30,33)(31,34)(32,35)(37,57)(38,58)(39,59)(40,60)(41,45)(42,46)(43,47)(44,48), (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), (1,24,17,40)(2,7,18,59)(3,22,19,38)(4,5,20,57)(6,27,58,55)(8,25,60,53)(9,44,31,13)(10,47,32,61)(11,42,29,15)(12,45,30,63)(14,50,41,33)(16,52,43,35)(21,28,37,56)(23,26,39,54)(34,62,51,48)(36,64,49,46) );
G=PermutationGroup([(1,29),(2,30),(3,31),(4,32),(5,61),(6,62),(7,63),(8,64),(9,19),(10,20),(11,17),(12,18),(13,22),(14,23),(15,24),(16,21),(25,49),(26,50),(27,51),(28,52),(33,54),(34,55),(35,56),(36,53),(37,43),(38,44),(39,41),(40,42),(45,59),(46,60),(47,57),(48,58)], [(2,26),(4,28),(6,38),(8,40),(10,35),(12,33),(13,48),(15,46),(18,54),(20,56),(22,58),(24,60),(30,50),(32,52),(42,64),(44,62)], [(1,25),(2,26),(3,27),(4,28),(5,37),(6,38),(7,39),(8,40),(9,34),(10,35),(11,36),(12,33),(13,48),(14,45),(15,46),(16,47),(17,53),(18,54),(19,55),(20,56),(21,57),(22,58),(23,59),(24,60),(29,49),(30,50),(31,51),(32,52),(41,63),(42,64),(43,61),(44,62)], [(1,53),(2,54),(3,55),(4,56),(5,21),(6,22),(7,23),(8,24),(9,51),(10,52),(11,49),(12,50),(13,62),(14,63),(15,64),(16,61),(17,25),(18,26),(19,27),(20,28),(29,36),(30,33),(31,34),(32,35),(37,57),(38,58),(39,59),(40,60),(41,45),(42,46),(43,47),(44,48)], [(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)], [(1,24,17,40),(2,7,18,59),(3,22,19,38),(4,5,20,57),(6,27,58,55),(8,25,60,53),(9,44,31,13),(10,47,32,61),(11,42,29,15),(12,45,30,63),(14,50,41,33),(16,52,43,35),(21,28,37,56),(23,26,39,54),(34,62,51,48),(36,64,49,46)])
Matrix representation ►G ⊆ GL6(𝔽5)
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 4 | 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 | 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 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 |
| 3 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 2 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 | 4 |
| 2 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 0 | 0 | 0 |
| 0 | 0 | 1 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 3 |
| 0 | 0 | 0 | 0 | 1 | 4 |
G:=sub<GL(6,GF(5))| [4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,1,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[4,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[3,0,0,0,0,0,0,1,0,0,0,0,0,0,4,2,0,0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,0,4],[2,0,0,0,0,0,0,1,0,0,0,0,0,0,2,1,0,0,0,0,0,3,0,0,0,0,0,0,1,1,0,0,0,0,3,4] >;
56 conjugacy classes
| class | 1 | 2A | ··· | 2O | 2P | ··· | 2W | 4A | ··· | 4P | 4Q | ··· | 4AF |
| order | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
| size | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C4 | D4 | C4○D4 |
| kernel | C2×C23.34D4 | C2×C2.C42 | C23.34D4 | C22×C22⋊C4 | C24×C4 | C23×C4 | C24 | C23 |
| # reps | 1 | 4 | 8 | 2 | 1 | 16 | 8 | 16 |
In GAP, Magma, Sage, TeX
C_2\times C_2^3._{34}D_4 % in TeX
G:=Group("C2xC2^3.34D4"); // GroupNames label
G:=SmallGroup(128,1011);
// by ID
G=gap.SmallGroup(128,1011);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,758,100]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=e^4=1,f^2=d*c=c*d,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,e*b*e^-1=f*b*f^-1=b*c=c*b,b*d=d*b,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=d*e^-1>;
// generators/relations