direct product, p-group, metabelian, nilpotent (class 3), monomial
Aliases: C4×C8.C22, C42.442D4, C42.274C23, Q16⋊4(C2×C4), (C4×Q16)⋊27C2, SD16⋊2(C2×C4), C4.134(C4×D4), C8.1(C22×C4), (C4×SD16)⋊12C2, (C4×M4(2))⋊2C2, M4(2)⋊8(C2×C4), C4.22(C23×C4), D4.5(C22×C4), C22.47(C4×D4), C4○2(Q16⋊C4), Q16⋊C4⋊33C2, Q8.5(C22×C4), C4⋊C4.362C23, (C2×C4).202C24, (C2×C8).413C23, (C4×C8).173C22, C23.644(C2×D4), (C22×C4).711D4, C4○3(SD16⋊C4), SD16⋊C4⋊55C2, (C2×D4).371C23, (C4×D4).292C22, (C2×Q8).344C23, (C4×Q8).275C22, C4○3(M4(2)⋊C4), M4(2)⋊C4⋊47C2, C8⋊C4.112C22, C4.Q8.126C22, C2.D8.212C22, C4○2(C23.38D4), C4○3(C23.36D4), C23.38D4⋊39C2, C2.5(D8⋊C22), (C2×C42).767C22, (C22×C4).923C23, (C2×Q16).152C22, C22.146(C22×D4), D4⋊C4.196C22, Q8⋊C4.196C22, (C2×SD16).108C22, C23.36D4.15C2, (C22×Q8).463C22, C42⋊C2.298C22, (C2×M4(2)).351C22, (C2×C4×Q8)⋊33C2, C2.62(C2×C4×D4), (C2×Q8)⋊29(C2×C4), C4.10(C2×C4○D4), C4○D4.20(C2×C4), (C4×C4○D4).14C2, (C2×C4).692(C2×D4), C2.6(C2×C8.C22), (C2×C4).69(C22×C4), (C2×C4).263(C4○D4), (C2×C4⋊C4).913C22, (C2×C8.C22).13C2, (C2×C4)○(M4(2)⋊C4), (C2×C4○D4).292C22, SmallGroup(128,1677)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 372 in 242 conjugacy classes, 142 normal (44 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×4], C4 [×13], C22, C22 [×2], C22 [×6], C8 [×4], C8 [×2], C2×C4 [×6], C2×C4 [×4], C2×C4 [×24], D4 [×2], D4 [×5], Q8 [×6], Q8 [×7], C23, C23, C42 [×4], C42 [×7], C22⋊C4 [×3], C4⋊C4 [×2], C4⋊C4 [×2], C4⋊C4 [×7], C2×C8 [×4], M4(2) [×4], M4(2) [×2], SD16 [×8], Q16 [×8], C22×C4 [×3], C22×C4 [×5], C2×D4, C2×D4, C2×Q8, C2×Q8 [×6], C2×Q8 [×3], C4○D4 [×4], C4○D4 [×2], C4×C8 [×2], C8⋊C4 [×2], D4⋊C4 [×2], Q8⋊C4 [×6], C4.Q8 [×2], C2.D8 [×2], C2×C42, C2×C42 [×2], C2×C4⋊C4, C2×C4⋊C4, C42⋊C2, C42⋊C2, C4×D4 [×2], C4×D4 [×2], C4×Q8 [×2], C4×Q8 [×4], C4×Q8 [×2], C2×M4(2) [×2], C2×SD16 [×2], C2×Q16 [×2], C8.C22 [×8], C22×Q8, C2×C4○D4, C4×M4(2), C23.36D4, C23.38D4, M4(2)⋊C4, C4×SD16 [×2], C4×Q16 [×2], SD16⋊C4 [×2], Q16⋊C4 [×2], C2×C4×Q8, C4×C4○D4, C2×C8.C22, C4×C8.C22
Quotients:
C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], D4 [×4], C23 [×15], C22×C4 [×14], C2×D4 [×6], C4○D4 [×2], C24, C4×D4 [×4], C8.C22 [×2], C23×C4, C22×D4, C2×C4○D4, C2×C4×D4, C2×C8.C22, D8⋊C22, C4×C8.C22
Generators and relations
G = < a,b,c,d | a4=b8=c2=d2=1, ab=ba, ac=ca, ad=da, cbc=b3, dbd=b5, dcd=b4c >
►On 64 points
(1 61 53 48)(2 62 54 41)(3 63 55 42)(4 64 56 43)(5 57 49 44)(6 58 50 45)(7 59 51 46)(8 60 52 47)(9 21 39 32)(10 22 40 25)(11 23 33 26)(12 24 34 27)(13 17 35 28)(14 18 36 29)(15 19 37 30)(16 20 38 31)
(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 53)(2 56)(3 51)(4 54)(5 49)(6 52)(7 55)(8 50)(9 35)(10 38)(11 33)(12 36)(13 39)(14 34)(15 37)(16 40)(17 32)(18 27)(19 30)(20 25)(21 28)(22 31)(23 26)(24 29)(41 64)(42 59)(43 62)(44 57)(45 60)(46 63)(47 58)(48 61)
(1 28)(2 25)(3 30)(4 27)(5 32)(6 29)(7 26)(8 31)(9 57)(10 62)(11 59)(12 64)(13 61)(14 58)(15 63)(16 60)(17 53)(18 50)(19 55)(20 52)(21 49)(22 54)(23 51)(24 56)(33 46)(34 43)(35 48)(36 45)(37 42)(38 47)(39 44)(40 41)
G:=sub<Sym(64)| (1,61,53,48)(2,62,54,41)(3,63,55,42)(4,64,56,43)(5,57,49,44)(6,58,50,45)(7,59,51,46)(8,60,52,47)(9,21,39,32)(10,22,40,25)(11,23,33,26)(12,24,34,27)(13,17,35,28)(14,18,36,29)(15,19,37,30)(16,20,38,31), (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), (1,53)(2,56)(3,51)(4,54)(5,49)(6,52)(7,55)(8,50)(9,35)(10,38)(11,33)(12,36)(13,39)(14,34)(15,37)(16,40)(17,32)(18,27)(19,30)(20,25)(21,28)(22,31)(23,26)(24,29)(41,64)(42,59)(43,62)(44,57)(45,60)(46,63)(47,58)(48,61), (1,28)(2,25)(3,30)(4,27)(5,32)(6,29)(7,26)(8,31)(9,57)(10,62)(11,59)(12,64)(13,61)(14,58)(15,63)(16,60)(17,53)(18,50)(19,55)(20,52)(21,49)(22,54)(23,51)(24,56)(33,46)(34,43)(35,48)(36,45)(37,42)(38,47)(39,44)(40,41)>;
G:=Group( (1,61,53,48)(2,62,54,41)(3,63,55,42)(4,64,56,43)(5,57,49,44)(6,58,50,45)(7,59,51,46)(8,60,52,47)(9,21,39,32)(10,22,40,25)(11,23,33,26)(12,24,34,27)(13,17,35,28)(14,18,36,29)(15,19,37,30)(16,20,38,31), (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), (1,53)(2,56)(3,51)(4,54)(5,49)(6,52)(7,55)(8,50)(9,35)(10,38)(11,33)(12,36)(13,39)(14,34)(15,37)(16,40)(17,32)(18,27)(19,30)(20,25)(21,28)(22,31)(23,26)(24,29)(41,64)(42,59)(43,62)(44,57)(45,60)(46,63)(47,58)(48,61), (1,28)(2,25)(3,30)(4,27)(5,32)(6,29)(7,26)(8,31)(9,57)(10,62)(11,59)(12,64)(13,61)(14,58)(15,63)(16,60)(17,53)(18,50)(19,55)(20,52)(21,49)(22,54)(23,51)(24,56)(33,46)(34,43)(35,48)(36,45)(37,42)(38,47)(39,44)(40,41) );
G=PermutationGroup([(1,61,53,48),(2,62,54,41),(3,63,55,42),(4,64,56,43),(5,57,49,44),(6,58,50,45),(7,59,51,46),(8,60,52,47),(9,21,39,32),(10,22,40,25),(11,23,33,26),(12,24,34,27),(13,17,35,28),(14,18,36,29),(15,19,37,30),(16,20,38,31)], [(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)], [(1,53),(2,56),(3,51),(4,54),(5,49),(6,52),(7,55),(8,50),(9,35),(10,38),(11,33),(12,36),(13,39),(14,34),(15,37),(16,40),(17,32),(18,27),(19,30),(20,25),(21,28),(22,31),(23,26),(24,29),(41,64),(42,59),(43,62),(44,57),(45,60),(46,63),(47,58),(48,61)], [(1,28),(2,25),(3,30),(4,27),(5,32),(6,29),(7,26),(8,31),(9,57),(10,62),(11,59),(12,64),(13,61),(14,58),(15,63),(16,60),(17,53),(18,50),(19,55),(20,52),(21,49),(22,54),(23,51),(24,56),(33,46),(34,43),(35,48),(36,45),(37,42),(38,47),(39,44),(40,41)])
Matrix representation ►G ⊆ GL6(𝔽17)
| 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 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 16 | 2 | 0 | 0 | 0 | 0 |
| 16 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 10 | 12 | 12 |
| 0 | 0 | 7 | 7 | 5 | 12 |
| 0 | 0 | 2 | 2 | 10 | 7 |
| 0 | 0 | 15 | 2 | 10 | 10 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 16 | 1 | 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 | 16 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 13 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 13 |
| 0 | 0 | 0 | 1 | 4 | 0 |
G:=sub<GL(6,GF(17))| [4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[16,16,0,0,0,0,2,1,0,0,0,0,0,0,7,7,2,15,0,0,10,7,2,2,0,0,12,5,10,10,0,0,12,12,7,10],[16,16,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,13,1,0,0,0,4,0,0,1,0,0,0,0,0,4,0,0,0,0,13,0] >;
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | ··· | 4N | 4O | ··· | 4AB | 8A | ··· | 8H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | D4 | C4○D4 | C8.C22 | D8⋊C22 |
| kernel | C4×C8.C22 | C4×M4(2) | C23.36D4 | C23.38D4 | M4(2)⋊C4 | C4×SD16 | C4×Q16 | SD16⋊C4 | Q16⋊C4 | C2×C4×Q8 | C4×C4○D4 | C2×C8.C22 | C8.C22 | C42 | C22×C4 | C2×C4 | C4 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 16 | 2 | 2 | 4 | 2 | 2 |
In GAP, Magma, Sage, TeX
C_4\times C_8.C_2^2
% in TeX
G:=Group("C4xC8.C2^2"); // GroupNames label
G:=SmallGroup(128,1677);
// by ID
G=gap.SmallGroup(128,1677);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,456,184,2019,2804,1411,172]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^8=c^2=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^3,d*b*d=b^5,d*c*d=b^4*c>;
// generators/relations