p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.527C23, C4.1442+ (1+4), (C8×Q8)⋊16C2, C4⋊C4.285D4, (C4×Q16)⋊26C2, (C4×D8).11C2, C4⋊Q16⋊13C2, C8.94(C4○D4), C4.50(C4○D8), (C2×Q8).189D4, C2.70(Q8○D8), D4.D4⋊47C2, C4⋊C8.330C22, C4⋊C4.444C23, (C4×C8).126C22, (C2×C4).585C24, (C2×C8).219C23, Q8.D4⋊10C2, C8.12D4.4C2, C4⋊Q8.213C22, C2.39(Q8⋊6D4), (C2×D8).149C22, (C4×D4).219C22, (C2×D4).280C23, (C4×Q8).210C22, (C2×Q16).39C22, (C2×Q8).264C23, C2.D8.237C22, C4.4D4.86C22, C22.845(C22×D4), D4⋊C4.175C22, Q8⋊C4.190C22, (C2×SD16).103C22, C22.50C24⋊13C2, C2.79(C2×C4○D8), C4.163(C2×C4○D4), (C2×C4).181(C2×D4), SmallGroup(128,2125)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 328 in 180 conjugacy classes, 88 normal (24 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×2], C4 [×11], C22, C22 [×6], C8 [×2], C8 [×3], C2×C4 [×3], C2×C4 [×4], C2×C4 [×10], D4 [×4], Q8 [×10], C23 [×2], C42, C42 [×2], C42 [×4], C22⋊C4 [×10], C4⋊C4, C4⋊C4 [×4], C4⋊C4 [×8], C2×C8 [×2], C2×C8 [×2], D8 [×2], SD16 [×4], Q16 [×6], C22×C4 [×2], C2×D4 [×2], C2×Q8, C2×Q8 [×4], C4×C8, C4×C8 [×2], D4⋊C4 [×2], Q8⋊C4 [×4], C4⋊C8, C4⋊C8 [×2], C2.D8, C42⋊C2 [×2], C4×D4 [×2], C4×Q8, C4×Q8 [×4], C22⋊Q8 [×2], C4.4D4 [×4], C42⋊2C2 [×4], C4⋊Q8 [×2], C2×D8, C2×SD16 [×4], C2×Q16 [×4], C4×D8, C4×Q16 [×2], C8×Q8, D4.D4 [×2], Q8.D4 [×4], C4⋊Q16, C8.12D4 [×2], C22.50C24 [×2], C42.527C23
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×2], C24, C4○D8 [×2], C22×D4, C2×C4○D4, 2+ (1+4), Q8⋊6D4, C2×C4○D8, Q8○D8, C42.527C23
Generators and relations
G = < a,b,c,d,e | a4=b4=c2=1, d2=e2=a2b2, ab=ba, cac=eae-1=a-1b2, ad=da, cbc=dbd-1=b-1, be=eb, dcd-1=bc, ce=ec, 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 53 47 58)(2 54 48 59)(3 55 45 60)(4 56 46 57)(5 42 61 52)(6 43 62 49)(7 44 63 50)(8 41 64 51)(9 34 21 37)(10 35 22 38)(11 36 23 39)(12 33 24 40)(13 28 18 31)(14 25 19 32)(15 26 20 29)(16 27 17 30)
(2 46)(4 48)(5 52)(6 41)(7 50)(8 43)(9 34)(10 40)(11 36)(12 38)(14 17)(16 19)(21 37)(22 33)(23 39)(24 35)(25 27)(26 29)(28 31)(30 32)(42 61)(44 63)(49 64)(51 62)(53 58)(54 56)(55 60)(57 59)
(1 44 45 52)(2 41 46 49)(3 42 47 50)(4 43 48 51)(5 58 63 55)(6 59 64 56)(7 60 61 53)(8 57 62 54)(9 18 23 15)(10 19 24 16)(11 20 21 13)(12 17 22 14)(25 40 30 35)(26 37 31 36)(27 38 32 33)(28 39 29 34)
(1 20 45 13)(2 14 46 17)(3 18 47 15)(4 16 48 19)(5 36 63 37)(6 38 64 33)(7 34 61 39)(8 40 62 35)(9 42 23 50)(10 51 24 43)(11 44 21 52)(12 49 22 41)(25 57 30 54)(26 55 31 58)(27 59 32 56)(28 53 29 60)
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), (1,53,47,58)(2,54,48,59)(3,55,45,60)(4,56,46,57)(5,42,61,52)(6,43,62,49)(7,44,63,50)(8,41,64,51)(9,34,21,37)(10,35,22,38)(11,36,23,39)(12,33,24,40)(13,28,18,31)(14,25,19,32)(15,26,20,29)(16,27,17,30), (2,46)(4,48)(5,52)(6,41)(7,50)(8,43)(9,34)(10,40)(11,36)(12,38)(14,17)(16,19)(21,37)(22,33)(23,39)(24,35)(25,27)(26,29)(28,31)(30,32)(42,61)(44,63)(49,64)(51,62)(53,58)(54,56)(55,60)(57,59), (1,44,45,52)(2,41,46,49)(3,42,47,50)(4,43,48,51)(5,58,63,55)(6,59,64,56)(7,60,61,53)(8,57,62,54)(9,18,23,15)(10,19,24,16)(11,20,21,13)(12,17,22,14)(25,40,30,35)(26,37,31,36)(27,38,32,33)(28,39,29,34), (1,20,45,13)(2,14,46,17)(3,18,47,15)(4,16,48,19)(5,36,63,37)(6,38,64,33)(7,34,61,39)(8,40,62,35)(9,42,23,50)(10,51,24,43)(11,44,21,52)(12,49,22,41)(25,57,30,54)(26,55,31,58)(27,59,32,56)(28,53,29,60)>;
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), (1,53,47,58)(2,54,48,59)(3,55,45,60)(4,56,46,57)(5,42,61,52)(6,43,62,49)(7,44,63,50)(8,41,64,51)(9,34,21,37)(10,35,22,38)(11,36,23,39)(12,33,24,40)(13,28,18,31)(14,25,19,32)(15,26,20,29)(16,27,17,30), (2,46)(4,48)(5,52)(6,41)(7,50)(8,43)(9,34)(10,40)(11,36)(12,38)(14,17)(16,19)(21,37)(22,33)(23,39)(24,35)(25,27)(26,29)(28,31)(30,32)(42,61)(44,63)(49,64)(51,62)(53,58)(54,56)(55,60)(57,59), (1,44,45,52)(2,41,46,49)(3,42,47,50)(4,43,48,51)(5,58,63,55)(6,59,64,56)(7,60,61,53)(8,57,62,54)(9,18,23,15)(10,19,24,16)(11,20,21,13)(12,17,22,14)(25,40,30,35)(26,37,31,36)(27,38,32,33)(28,39,29,34), (1,20,45,13)(2,14,46,17)(3,18,47,15)(4,16,48,19)(5,36,63,37)(6,38,64,33)(7,34,61,39)(8,40,62,35)(9,42,23,50)(10,51,24,43)(11,44,21,52)(12,49,22,41)(25,57,30,54)(26,55,31,58)(27,59,32,56)(28,53,29,60) );
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)], [(1,53,47,58),(2,54,48,59),(3,55,45,60),(4,56,46,57),(5,42,61,52),(6,43,62,49),(7,44,63,50),(8,41,64,51),(9,34,21,37),(10,35,22,38),(11,36,23,39),(12,33,24,40),(13,28,18,31),(14,25,19,32),(15,26,20,29),(16,27,17,30)], [(2,46),(4,48),(5,52),(6,41),(7,50),(8,43),(9,34),(10,40),(11,36),(12,38),(14,17),(16,19),(21,37),(22,33),(23,39),(24,35),(25,27),(26,29),(28,31),(30,32),(42,61),(44,63),(49,64),(51,62),(53,58),(54,56),(55,60),(57,59)], [(1,44,45,52),(2,41,46,49),(3,42,47,50),(4,43,48,51),(5,58,63,55),(6,59,64,56),(7,60,61,53),(8,57,62,54),(9,18,23,15),(10,19,24,16),(11,20,21,13),(12,17,22,14),(25,40,30,35),(26,37,31,36),(27,38,32,33),(28,39,29,34)], [(1,20,45,13),(2,14,46,17),(3,18,47,15),(4,16,48,19),(5,36,63,37),(6,38,64,33),(7,34,61,39),(8,40,62,35),(9,42,23,50),(10,51,24,43),(11,44,21,52),(12,49,22,41),(25,57,30,54),(26,55,31,58),(27,59,32,56),(28,53,29,60)])
Matrix representation ►G ⊆ GL4(𝔽17) generated by
| 13 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 |
| 0 | 0 | 16 | 15 |
| 0 | 0 | 1 | 1 |
| 0 | 1 | 0 | 0 |
| 16 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 16 | 16 |
| 3 | 3 | 0 | 0 |
| 3 | 14 | 0 | 0 |
| 0 | 0 | 13 | 0 |
| 0 | 0 | 0 | 13 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 13 | 13 |
G:=sub<GL(4,GF(17))| [13,0,0,0,0,13,0,0,0,0,16,1,0,0,15,1],[0,16,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,16,0,0,0,0,1,16,0,0,0,16],[3,3,0,0,3,14,0,0,0,0,13,0,0,0,0,13],[1,0,0,0,0,1,0,0,0,0,4,13,0,0,0,13] >;
35 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | ··· | 4H | 4I | ··· | 4O | 4P | 4Q | 4R | 4S | 8A | 8B | 8C | 8D | 8E | ··· | 8J |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 8 | 8 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 8 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
35 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | C4○D4 | C4○D8 | 2+ (1+4) | Q8○D8 |
| kernel | C42.527C23 | C4×D8 | C4×Q16 | C8×Q8 | D4.D4 | Q8.D4 | C4⋊Q16 | C8.12D4 | C22.50C24 | C4⋊C4 | C2×Q8 | C8 | C4 | C4 | C2 |
| # reps | 1 | 1 | 2 | 1 | 2 | 4 | 1 | 2 | 2 | 3 | 1 | 4 | 8 | 1 | 2 |
In GAP, Magma, Sage, TeX
C_4^2._{527}C_2^3 % in TeX
G:=Group("C4^2.527C2^3"); // GroupNames label
G:=SmallGroup(128,2125);
// by ID
G=gap.SmallGroup(128,2125);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,568,758,100,346,80,4037,1027,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^4=c^2=1,d^2=e^2=a^2*b^2,a*b=b*a,c*a*c=e*a*e^-1=a^-1*b^2,a*d=d*a,c*b*c=d*b*d^-1=b^-1,b*e=e*b,d*c*d^-1=b*c,c*e=e*c,d*e=e*d>;
// generators/relations