direct product, p-group, metabelian, nilpotent (class 3), monomial
Aliases: C2×C42.78C22, C42.354D4, C42.705C23, (C4×C8)⋊71C22, C4⋊C4.83C23, (C2×C8).490C23, (C2×C4).328C24, (C2×D4).97C23, C23.872(C2×D4), (C22×C4).565D4, (C2×Q8).85C23, Q8⋊C4⋊61C22, C4.20(C4.4D4), C22.97(C4○D8), C42.C2⋊33C22, (C22×C8).519C22, C22.588(C22×D4), D4⋊C4.143C22, (C2×C42).1123C22, (C22×C4).1550C23, C4.4D4.132C22, C22.82(C4.4D4), (C22×D4).365C22, (C22×Q8).298C22, (C2×C4×C8)⋊21C2, C4.37(C2×C4○D4), C2.28(C2×C4○D8), (C2×C4).693(C2×D4), (C2×Q8⋊C4)⋊19C2, (C2×C42.C2)⋊33C2, C2.39(C2×C4.4D4), (C2×D4⋊C4).19C2, (C2×C4).707(C4○D4), (C2×C4⋊C4).620C22, (C2×C4.4D4).38C2, SmallGroup(128,1862)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 404 in 206 conjugacy classes, 100 normal (14 characteristic)
C1, C2, C2 [×6], C2 [×2], C4 [×4], C4 [×10], C22, C22 [×6], C22 [×10], C8 [×4], C2×C4 [×2], C2×C4 [×8], C2×C4 [×16], D4 [×6], Q8 [×6], C23, C23 [×8], C42 [×4], C22⋊C4 [×8], C4⋊C4 [×4], C4⋊C4 [×10], C2×C8 [×4], C2×C8 [×4], C22×C4, C22×C4 [×2], C22×C4 [×3], C2×D4 [×2], C2×D4 [×5], C2×Q8 [×2], C2×Q8 [×5], C24, C4×C8 [×4], D4⋊C4 [×8], Q8⋊C4 [×8], C2×C42, C2×C22⋊C4 [×2], C2×C4⋊C4 [×2], C2×C4⋊C4 [×2], C4.4D4 [×4], C4.4D4 [×2], C42.C2 [×4], C42.C2 [×2], C22×C8 [×2], C22×D4, C22×Q8, C2×C4×C8, C2×D4⋊C4 [×2], C2×Q8⋊C4 [×2], C42.78C22 [×8], C2×C4.4D4, C2×C42.C2, C2×C42.78C22
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×4], C24, C4.4D4 [×4], C4○D8 [×4], C22×D4, C2×C4○D4 [×2], C42.78C22 [×4], C2×C4.4D4, C2×C4○D8 [×2], C2×C42.78C22
Generators and relations
G = < a,b,c,d,e | a2=b4=c4=d2=1, e2=c, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd=b-1c2, be=eb, dcd=c-1, ce=ec, ede-1=b2cd >
►On 64 points
(1 53)(2 54)(3 55)(4 56)(5 49)(6 50)(7 51)(8 52)(9 59)(10 60)(11 61)(12 62)(13 63)(14 64)(15 57)(16 58)(17 27)(18 28)(19 29)(20 30)(21 31)(22 32)(23 25)(24 26)(33 42)(34 43)(35 44)(36 45)(37 46)(38 47)(39 48)(40 41)
(1 64 17 38)(2 57 18 39)(3 58 19 40)(4 59 20 33)(5 60 21 34)(6 61 22 35)(7 62 23 36)(8 63 24 37)(9 30 42 56)(10 31 43 49)(11 32 44 50)(12 25 45 51)(13 26 46 52)(14 27 47 53)(15 28 48 54)(16 29 41 55)
(1 3 5 7)(2 4 6 8)(9 11 13 15)(10 12 14 16)(17 19 21 23)(18 20 22 24)(25 27 29 31)(26 28 30 32)(33 35 37 39)(34 36 38 40)(41 43 45 47)(42 44 46 48)(49 51 53 55)(50 52 54 56)(57 59 61 63)(58 60 62 64)
(1 31)(2 56)(3 29)(4 54)(5 27)(6 52)(7 25)(8 50)(9 35)(10 60)(11 33)(12 58)(13 39)(14 64)(15 37)(16 62)(17 49)(18 30)(19 55)(20 28)(21 53)(22 26)(23 51)(24 32)(34 43)(36 41)(38 47)(40 45)(42 61)(44 59)(46 57)(48 63)
(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)
G:=sub<Sym(64)| (1,53)(2,54)(3,55)(4,56)(5,49)(6,50)(7,51)(8,52)(9,59)(10,60)(11,61)(12,62)(13,63)(14,64)(15,57)(16,58)(17,27)(18,28)(19,29)(20,30)(21,31)(22,32)(23,25)(24,26)(33,42)(34,43)(35,44)(36,45)(37,46)(38,47)(39,48)(40,41), (1,64,17,38)(2,57,18,39)(3,58,19,40)(4,59,20,33)(5,60,21,34)(6,61,22,35)(7,62,23,36)(8,63,24,37)(9,30,42,56)(10,31,43,49)(11,32,44,50)(12,25,45,51)(13,26,46,52)(14,27,47,53)(15,28,48,54)(16,29,41,55), (1,3,5,7)(2,4,6,8)(9,11,13,15)(10,12,14,16)(17,19,21,23)(18,20,22,24)(25,27,29,31)(26,28,30,32)(33,35,37,39)(34,36,38,40)(41,43,45,47)(42,44,46,48)(49,51,53,55)(50,52,54,56)(57,59,61,63)(58,60,62,64), (1,31)(2,56)(3,29)(4,54)(5,27)(6,52)(7,25)(8,50)(9,35)(10,60)(11,33)(12,58)(13,39)(14,64)(15,37)(16,62)(17,49)(18,30)(19,55)(20,28)(21,53)(22,26)(23,51)(24,32)(34,43)(36,41)(38,47)(40,45)(42,61)(44,59)(46,57)(48,63), (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)>;
G:=Group( (1,53)(2,54)(3,55)(4,56)(5,49)(6,50)(7,51)(8,52)(9,59)(10,60)(11,61)(12,62)(13,63)(14,64)(15,57)(16,58)(17,27)(18,28)(19,29)(20,30)(21,31)(22,32)(23,25)(24,26)(33,42)(34,43)(35,44)(36,45)(37,46)(38,47)(39,48)(40,41), (1,64,17,38)(2,57,18,39)(3,58,19,40)(4,59,20,33)(5,60,21,34)(6,61,22,35)(7,62,23,36)(8,63,24,37)(9,30,42,56)(10,31,43,49)(11,32,44,50)(12,25,45,51)(13,26,46,52)(14,27,47,53)(15,28,48,54)(16,29,41,55), (1,3,5,7)(2,4,6,8)(9,11,13,15)(10,12,14,16)(17,19,21,23)(18,20,22,24)(25,27,29,31)(26,28,30,32)(33,35,37,39)(34,36,38,40)(41,43,45,47)(42,44,46,48)(49,51,53,55)(50,52,54,56)(57,59,61,63)(58,60,62,64), (1,31)(2,56)(3,29)(4,54)(5,27)(6,52)(7,25)(8,50)(9,35)(10,60)(11,33)(12,58)(13,39)(14,64)(15,37)(16,62)(17,49)(18,30)(19,55)(20,28)(21,53)(22,26)(23,51)(24,32)(34,43)(36,41)(38,47)(40,45)(42,61)(44,59)(46,57)(48,63), (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) );
G=PermutationGroup([(1,53),(2,54),(3,55),(4,56),(5,49),(6,50),(7,51),(8,52),(9,59),(10,60),(11,61),(12,62),(13,63),(14,64),(15,57),(16,58),(17,27),(18,28),(19,29),(20,30),(21,31),(22,32),(23,25),(24,26),(33,42),(34,43),(35,44),(36,45),(37,46),(38,47),(39,48),(40,41)], [(1,64,17,38),(2,57,18,39),(3,58,19,40),(4,59,20,33),(5,60,21,34),(6,61,22,35),(7,62,23,36),(8,63,24,37),(9,30,42,56),(10,31,43,49),(11,32,44,50),(12,25,45,51),(13,26,46,52),(14,27,47,53),(15,28,48,54),(16,29,41,55)], [(1,3,5,7),(2,4,6,8),(9,11,13,15),(10,12,14,16),(17,19,21,23),(18,20,22,24),(25,27,29,31),(26,28,30,32),(33,35,37,39),(34,36,38,40),(41,43,45,47),(42,44,46,48),(49,51,53,55),(50,52,54,56),(57,59,61,63),(58,60,62,64)], [(1,31),(2,56),(3,29),(4,54),(5,27),(6,52),(7,25),(8,50),(9,35),(10,60),(11,33),(12,58),(13,39),(14,64),(15,37),(16,62),(17,49),(18,30),(19,55),(20,28),(21,53),(22,26),(23,51),(24,32),(34,43),(36,41),(38,47),(40,45),(42,61),(44,59),(46,57),(48,63)], [(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)])
Matrix representation ►G ⊆ GL5(𝔽17)
| 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 |
| 16 | 0 | 0 | 0 | 0 |
| 0 | 7 | 8 | 0 | 0 |
| 0 | 15 | 10 | 0 | 0 |
| 0 | 0 | 0 | 0 | 13 |
| 0 | 0 | 0 | 4 | 0 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 |
| 0 | 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 |
| 0 | 6 | 1 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 6 | 2 | 0 | 0 |
| 0 | 8 | 11 | 0 | 0 |
| 0 | 0 | 0 | 3 | 14 |
| 0 | 0 | 0 | 3 | 3 |
G:=sub<GL(5,GF(17))| [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],[16,0,0,0,0,0,7,15,0,0,0,8,10,0,0,0,0,0,0,4,0,0,0,13,0],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,16,0],[1,0,0,0,0,0,16,6,0,0,0,0,1,0,0,0,0,0,16,0,0,0,0,0,1],[1,0,0,0,0,0,6,8,0,0,0,2,11,0,0,0,0,0,3,3,0,0,0,14,3] >;
44 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 4A | ··· | 4L | 4M | ··· | 4R | 8A | ··· | 8P |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
| size | 1 | 1 | ··· | 1 | 8 | 8 | 2 | ··· | 2 | 8 | ··· | 8 | 2 | ··· | 2 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | C4○D4 | C4○D8 |
| kernel | C2×C42.78C22 | C2×C4×C8 | C2×D4⋊C4 | C2×Q8⋊C4 | C42.78C22 | C2×C4.4D4 | C2×C42.C2 | C42 | C22×C4 | C2×C4 | C22 |
| # reps | 1 | 1 | 2 | 2 | 8 | 1 | 1 | 2 | 2 | 8 | 16 |
In GAP, Magma, Sage, TeX
C_2\times C_4^2._{78}C_2^2 % in TeX
G:=Group("C2xC4^2.78C2^2"); // GroupNames label
G:=SmallGroup(128,1862);
// by ID
G=gap.SmallGroup(128,1862);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,680,758,100,2804,172,4037,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^4=c^4=d^2=1,e^2=c,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,d*b*d=b^-1*c^2,b*e=e*b,d*c*d=c^-1,c*e=e*c,e*d*e^-1=b^2*c*d>;
// generators/relations