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), (C22×C4).439C23, C23.206(C22×C4), (C23×C4).642C22, 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
Generators and relations for C2×C23.34D4
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 >
Subgroups: 972 in 572 conjugacy classes, 220 normal (8 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C2×C4, C2×C4, C23, C23, C23, C22⋊C4, C22×C4, C22×C4, C24, C24, C24, C2.C42, C2×C22⋊C4, C2×C22⋊C4, C23×C4, C23×C4, C25, C2×C2.C42, C23.34D4, C22×C22⋊C4, C24×C4, C2×C23.34D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, C4○D4, C24, C2×C22⋊C4, C42⋊C2, C22.D4, C23×C4, C22×D4, C2×C4○D4, C23.34D4, C22×C22⋊C4, C2×C42⋊C2, C2×C22.D4, C2×C23.34D4
►On 64 points
Generators in S64
(1 29)(2 30)(3 31)(4 32)(5 63)(6 64)(7 61)(8 62)(9 17)(10 18)(11 19)(12 20)(13 60)(14 57)(15 58)(16 59)(21 47)(22 48)(23 45)(24 46)(25 51)(26 52)(27 49)(28 50)(33 56)(34 53)(35 54)(36 55)(37 41)(38 42)(39 43)(40 44)
(2 28)(4 26)(6 38)(8 40)(10 35)(12 33)(13 46)(15 48)(18 54)(20 56)(22 58)(24 60)(30 50)(32 52)(42 64)(44 62)
(1 27)(2 28)(3 25)(4 26)(5 37)(6 38)(7 39)(8 40)(9 34)(10 35)(11 36)(12 33)(13 46)(14 47)(15 48)(16 45)(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 23)(6 24)(7 21)(8 22)(9 49)(10 50)(11 51)(12 52)(13 42)(14 43)(15 44)(16 41)(17 27)(18 28)(19 25)(20 26)(29 34)(30 35)(31 36)(32 33)(37 59)(38 60)(39 57)(40 58)(45 63)(46 64)(47 61)(48 62)
(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 38)(2 5 18 59)(3 22 19 40)(4 7 20 57)(6 27 60 53)(8 25 58 55)(9 42 29 46)(10 16 30 63)(11 44 31 48)(12 14 32 61)(13 34 64 49)(15 36 62 51)(21 26 39 56)(23 28 37 54)(33 47 52 43)(35 45 50 41)
G:=sub<Sym(64)| (1,29)(2,30)(3,31)(4,32)(5,63)(6,64)(7,61)(8,62)(9,17)(10,18)(11,19)(12,20)(13,60)(14,57)(15,58)(16,59)(21,47)(22,48)(23,45)(24,46)(25,51)(26,52)(27,49)(28,50)(33,56)(34,53)(35,54)(36,55)(37,41)(38,42)(39,43)(40,44), (2,28)(4,26)(6,38)(8,40)(10,35)(12,33)(13,46)(15,48)(18,54)(20,56)(22,58)(24,60)(30,50)(32,52)(42,64)(44,62), (1,27)(2,28)(3,25)(4,26)(5,37)(6,38)(7,39)(8,40)(9,34)(10,35)(11,36)(12,33)(13,46)(14,47)(15,48)(16,45)(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)(5,23)(6,24)(7,21)(8,22)(9,49)(10,50)(11,51)(12,52)(13,42)(14,43)(15,44)(16,41)(17,27)(18,28)(19,25)(20,26)(29,34)(30,35)(31,36)(32,33)(37,59)(38,60)(39,57)(40,58)(45,63)(46,64)(47,61)(48,62), (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), (1,24,17,38)(2,5,18,59)(3,22,19,40)(4,7,20,57)(6,27,60,53)(8,25,58,55)(9,42,29,46)(10,16,30,63)(11,44,31,48)(12,14,32,61)(13,34,64,49)(15,36,62,51)(21,26,39,56)(23,28,37,54)(33,47,52,43)(35,45,50,41)>;
G:=Group( (1,29)(2,30)(3,31)(4,32)(5,63)(6,64)(7,61)(8,62)(9,17)(10,18)(11,19)(12,20)(13,60)(14,57)(15,58)(16,59)(21,47)(22,48)(23,45)(24,46)(25,51)(26,52)(27,49)(28,50)(33,56)(34,53)(35,54)(36,55)(37,41)(38,42)(39,43)(40,44), (2,28)(4,26)(6,38)(8,40)(10,35)(12,33)(13,46)(15,48)(18,54)(20,56)(22,58)(24,60)(30,50)(32,52)(42,64)(44,62), (1,27)(2,28)(3,25)(4,26)(5,37)(6,38)(7,39)(8,40)(9,34)(10,35)(11,36)(12,33)(13,46)(14,47)(15,48)(16,45)(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)(5,23)(6,24)(7,21)(8,22)(9,49)(10,50)(11,51)(12,52)(13,42)(14,43)(15,44)(16,41)(17,27)(18,28)(19,25)(20,26)(29,34)(30,35)(31,36)(32,33)(37,59)(38,60)(39,57)(40,58)(45,63)(46,64)(47,61)(48,62), (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), (1,24,17,38)(2,5,18,59)(3,22,19,40)(4,7,20,57)(6,27,60,53)(8,25,58,55)(9,42,29,46)(10,16,30,63)(11,44,31,48)(12,14,32,61)(13,34,64,49)(15,36,62,51)(21,26,39,56)(23,28,37,54)(33,47,52,43)(35,45,50,41) );
G=PermutationGroup([[(1,29),(2,30),(3,31),(4,32),(5,63),(6,64),(7,61),(8,62),(9,17),(10,18),(11,19),(12,20),(13,60),(14,57),(15,58),(16,59),(21,47),(22,48),(23,45),(24,46),(25,51),(26,52),(27,49),(28,50),(33,56),(34,53),(35,54),(36,55),(37,41),(38,42),(39,43),(40,44)], [(2,28),(4,26),(6,38),(8,40),(10,35),(12,33),(13,46),(15,48),(18,54),(20,56),(22,58),(24,60),(30,50),(32,52),(42,64),(44,62)], [(1,27),(2,28),(3,25),(4,26),(5,37),(6,38),(7,39),(8,40),(9,34),(10,35),(11,36),(12,33),(13,46),(14,47),(15,48),(16,45),(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),(5,23),(6,24),(7,21),(8,22),(9,49),(10,50),(11,51),(12,52),(13,42),(14,43),(15,44),(16,41),(17,27),(18,28),(19,25),(20,26),(29,34),(30,35),(31,36),(32,33),(37,59),(38,60),(39,57),(40,58),(45,63),(46,64),(47,61),(48,62)], [(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)], [(1,24,17,38),(2,5,18,59),(3,22,19,40),(4,7,20,57),(6,27,60,53),(8,25,58,55),(9,42,29,46),(10,16,30,63),(11,44,31,48),(12,14,32,61),(13,34,64,49),(15,36,62,51),(21,26,39,56),(23,28,37,54),(33,47,52,43),(35,45,50,41)]])
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 |
Matrix representation of C2×C23.34D4 ►in 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] >;
C2×C23.34D4 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