p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.360C23, C4⋊C4.353D4, Q8⋊D4⋊6C2, C4⋊SD16⋊8C2, Q8.Q8⋊21C2, (C4×SD16)⋊32C2, C4⋊C4.79C23, C4⋊C8.59C22, (C2×C8).53C23, (C4×C8).263C22, (C2×C4).324C24, Q8.14(C4○D4), C22⋊C4.154D4, (C4×D4).86C22, (C2×D4).95C23, C23.263(C2×D4), SD16⋊C4⋊18C2, (C4×Q8).82C22, C2.D8.93C22, C8⋊C4.16C22, C2.32(D4○SD16), C4⋊1D4.61C22, C4⋊D4.32C22, C22⋊C8.37C22, (C2×Q8).383C23, C4.Q8.156C22, D4⋊C4.39C22, C23.19D4⋊21C2, C42.7C22⋊9C2, (C22×C4).297C23, Q8⋊C4.39C22, C22.584(C22×D4), C42.C2.14C22, (C2×SD16).147C22, (C22×Q8).295C22, C23.32C23⋊10C2, C42⋊C2.135C22, C22.34C24.1C2, C2.125(C22.19C24), C4.209(C2×C4○D4), (C2×C4).508(C2×D4), SmallGroup(128,1858)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 364 in 191 conjugacy classes, 88 normal (20 characteristic)
C1, C2, C2 [×2], C2 [×3], C4 [×2], C4 [×13], C22, C22 [×9], C8 [×4], C2×C4 [×2], C2×C4 [×4], C2×C4 [×14], D4 [×8], Q8 [×4], Q8 [×6], C23, C23 [×2], C42 [×2], C42 [×5], C22⋊C4 [×2], C22⋊C4 [×5], C4⋊C4 [×6], C4⋊C4 [×6], C2×C8 [×4], SD16 [×8], C22×C4, C22×C4 [×3], C2×D4 [×2], C2×D4 [×4], C2×Q8 [×2], C2×Q8 [×5], C4×C8, C8⋊C4, C22⋊C8 [×2], D4⋊C4 [×4], Q8⋊C4 [×4], C4⋊C8 [×2], C4.Q8 [×2], C2.D8 [×2], C42⋊C2 [×2], C42⋊C2 [×2], C4×D4 [×2], C4×Q8 [×4], C4×Q8 [×2], C4⋊D4 [×2], C4⋊D4 [×2], C22.D4 [×2], C42.C2, C4⋊1D4, C2×SD16 [×4], C22×Q8, C42.7C22, C4×SD16 [×2], SD16⋊C4 [×2], Q8⋊D4 [×2], C4⋊SD16 [×2], Q8.Q8 [×2], C23.19D4 [×2], C23.32C23, C22.34C24, C42.360C23
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×4], C24, C22×D4, C2×C4○D4 [×2], C22.19C24, D4○SD16 [×2], C42.360C23
Generators and relations
G = < a,b,c,d,e | a4=b4=c2=d2=1, e2=b2, ab=ba, ac=ca, dad=ab2, ae=ea, cbc=ebe-1=b-1, bd=db, dcd=a2b2c, ece-1=bc, de=ed >
►On 64 points
(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 28 20 23)(2 25 17 24)(3 26 18 21)(4 27 19 22)(5 9 15 63)(6 10 16 64)(7 11 13 61)(8 12 14 62)(29 39 34 41)(30 40 35 42)(31 37 36 43)(32 38 33 44)(45 54 50 60)(46 55 51 57)(47 56 52 58)(48 53 49 59)
(1 49)(2 50)(3 51)(4 52)(5 36)(6 33)(7 34)(8 35)(9 37)(10 38)(11 39)(12 40)(13 29)(14 30)(15 31)(16 32)(17 45)(18 46)(19 47)(20 48)(21 57)(22 58)(23 59)(24 60)(25 54)(26 55)(27 56)(28 53)(41 61)(42 62)(43 63)(44 64)
(1 45)(2 51)(3 47)(4 49)(5 40)(6 43)(7 38)(8 41)(9 35)(10 31)(11 33)(12 29)(13 44)(14 39)(15 42)(16 37)(17 46)(18 52)(19 48)(20 50)(21 58)(22 53)(23 60)(24 55)(25 57)(26 56)(27 59)(28 54)(30 63)(32 61)(34 62)(36 64)
(1 33 20 32)(2 34 17 29)(3 35 18 30)(4 36 19 31)(5 58 15 56)(6 59 16 53)(7 60 13 54)(8 57 14 55)(9 52 63 47)(10 49 64 48)(11 50 61 45)(12 51 62 46)(21 42 26 40)(22 43 27 37)(23 44 28 38)(24 41 25 39)
G:=sub<Sym(64)| (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), (1,28,20,23)(2,25,17,24)(3,26,18,21)(4,27,19,22)(5,9,15,63)(6,10,16,64)(7,11,13,61)(8,12,14,62)(29,39,34,41)(30,40,35,42)(31,37,36,43)(32,38,33,44)(45,54,50,60)(46,55,51,57)(47,56,52,58)(48,53,49,59), (1,49)(2,50)(3,51)(4,52)(5,36)(6,33)(7,34)(8,35)(9,37)(10,38)(11,39)(12,40)(13,29)(14,30)(15,31)(16,32)(17,45)(18,46)(19,47)(20,48)(21,57)(22,58)(23,59)(24,60)(25,54)(26,55)(27,56)(28,53)(41,61)(42,62)(43,63)(44,64), (1,45)(2,51)(3,47)(4,49)(5,40)(6,43)(7,38)(8,41)(9,35)(10,31)(11,33)(12,29)(13,44)(14,39)(15,42)(16,37)(17,46)(18,52)(19,48)(20,50)(21,58)(22,53)(23,60)(24,55)(25,57)(26,56)(27,59)(28,54)(30,63)(32,61)(34,62)(36,64), (1,33,20,32)(2,34,17,29)(3,35,18,30)(4,36,19,31)(5,58,15,56)(6,59,16,53)(7,60,13,54)(8,57,14,55)(9,52,63,47)(10,49,64,48)(11,50,61,45)(12,51,62,46)(21,42,26,40)(22,43,27,37)(23,44,28,38)(24,41,25,39)>;
G:=Group( (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), (1,28,20,23)(2,25,17,24)(3,26,18,21)(4,27,19,22)(5,9,15,63)(6,10,16,64)(7,11,13,61)(8,12,14,62)(29,39,34,41)(30,40,35,42)(31,37,36,43)(32,38,33,44)(45,54,50,60)(46,55,51,57)(47,56,52,58)(48,53,49,59), (1,49)(2,50)(3,51)(4,52)(5,36)(6,33)(7,34)(8,35)(9,37)(10,38)(11,39)(12,40)(13,29)(14,30)(15,31)(16,32)(17,45)(18,46)(19,47)(20,48)(21,57)(22,58)(23,59)(24,60)(25,54)(26,55)(27,56)(28,53)(41,61)(42,62)(43,63)(44,64), (1,45)(2,51)(3,47)(4,49)(5,40)(6,43)(7,38)(8,41)(9,35)(10,31)(11,33)(12,29)(13,44)(14,39)(15,42)(16,37)(17,46)(18,52)(19,48)(20,50)(21,58)(22,53)(23,60)(24,55)(25,57)(26,56)(27,59)(28,54)(30,63)(32,61)(34,62)(36,64), (1,33,20,32)(2,34,17,29)(3,35,18,30)(4,36,19,31)(5,58,15,56)(6,59,16,53)(7,60,13,54)(8,57,14,55)(9,52,63,47)(10,49,64,48)(11,50,61,45)(12,51,62,46)(21,42,26,40)(22,43,27,37)(23,44,28,38)(24,41,25,39) );
G=PermutationGroup([(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)], [(1,28,20,23),(2,25,17,24),(3,26,18,21),(4,27,19,22),(5,9,15,63),(6,10,16,64),(7,11,13,61),(8,12,14,62),(29,39,34,41),(30,40,35,42),(31,37,36,43),(32,38,33,44),(45,54,50,60),(46,55,51,57),(47,56,52,58),(48,53,49,59)], [(1,49),(2,50),(3,51),(4,52),(5,36),(6,33),(7,34),(8,35),(9,37),(10,38),(11,39),(12,40),(13,29),(14,30),(15,31),(16,32),(17,45),(18,46),(19,47),(20,48),(21,57),(22,58),(23,59),(24,60),(25,54),(26,55),(27,56),(28,53),(41,61),(42,62),(43,63),(44,64)], [(1,45),(2,51),(3,47),(4,49),(5,40),(6,43),(7,38),(8,41),(9,35),(10,31),(11,33),(12,29),(13,44),(14,39),(15,42),(16,37),(17,46),(18,52),(19,48),(20,50),(21,58),(22,53),(23,60),(24,55),(25,57),(26,56),(27,59),(28,54),(30,63),(32,61),(34,62),(36,64)], [(1,33,20,32),(2,34,17,29),(3,35,18,30),(4,36,19,31),(5,58,15,56),(6,59,16,53),(7,60,13,54),(8,57,14,55),(9,52,63,47),(10,49,64,48),(11,50,61,45),(12,51,62,46),(21,42,26,40),(22,43,27,37),(23,44,28,38),(24,41,25,39)])
Matrix representation ►G ⊆ GL6(𝔽17)
| 13 | 0 | 0 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 6 | 16 | 0 | 0 | 0 | 0 |
| 1 | 11 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 13 | 6 | 0 |
| 0 | 0 | 13 | 0 | 0 | 11 |
| 0 | 0 | 6 | 0 | 0 | 13 |
| 0 | 0 | 0 | 11 | 13 | 0 |
| 7 | 13 | 0 | 0 | 0 | 0 |
| 12 | 10 | 0 | 0 | 0 | 0 |
| 0 | 0 | 11 | 0 | 0 | 4 |
| 0 | 0 | 0 | 11 | 13 | 0 |
| 0 | 0 | 0 | 13 | 6 | 0 |
| 0 | 0 | 4 | 0 | 0 | 6 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 13 | 13 | 3 | 14 |
| 0 | 0 | 13 | 4 | 14 | 14 |
| 0 | 0 | 3 | 14 | 13 | 13 |
| 0 | 0 | 14 | 14 | 13 | 4 |
G:=sub<GL(6,GF(17))| [13,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0],[6,1,0,0,0,0,16,11,0,0,0,0,0,0,0,13,6,0,0,0,13,0,0,11,0,0,6,0,0,13,0,0,0,11,13,0],[7,12,0,0,0,0,13,10,0,0,0,0,0,0,11,0,0,4,0,0,0,11,13,0,0,0,0,13,6,0,0,0,4,0,0,6],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,13,13,3,14,0,0,13,4,14,14,0,0,3,14,13,13,0,0,14,14,13,4] >;
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | ··· | 4F | 4G | ··· | 4Q | 4R | 4S | 8A | 8B | 8C | 8D | 8E | 8F |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | 8 |
| size | 1 | 1 | 1 | 1 | 4 | 8 | 8 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | 8 | 4 | 4 | 4 | 4 | 8 | 8 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | C4○D4 | D4○SD16 |
| kernel | C42.360C23 | C42.7C22 | C4×SD16 | SD16⋊C4 | Q8⋊D4 | C4⋊SD16 | Q8.Q8 | C23.19D4 | C23.32C23 | C22.34C24 | C22⋊C4 | C4⋊C4 | Q8 | C2 |
| # reps | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 1 | 1 | 2 | 2 | 8 | 4 |
In GAP, Magma, Sage, TeX
C_4^2._{360}C_2^3 % in TeX
G:=Group("C4^2.360C2^3"); // GroupNames label
G:=SmallGroup(128,1858);
// by ID
G=gap.SmallGroup(128,1858);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,758,352,1018,304,4037,1027,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^4=c^2=d^2=1,e^2=b^2,a*b=b*a,a*c=c*a,d*a*d=a*b^2,a*e=e*a,c*b*c=e*b*e^-1=b^-1,b*d=d*b,d*c*d=a^2*b^2*c,e*c*e^-1=b*c,d*e=e*d>;
// generators/relations