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, C4.50(C4○D8), C8.94(C4○D4), (C2×Q8).189D4, C2.70(Q8○D8), D4.D4⋊47C2, C4⋊C4.444C23, C4⋊C8.330C22, (C4×C8).126C22, (C2×C8).219C23, (C2×C4).585C24, Q8.D4⋊10C2, C8.12D4.4C2, C4⋊Q8.213C22, C2.39(Q8⋊6D4), (C4×D4).219C22, (C2×D8).149C22, (C2×D4).280C23, (C2×Q16).39C22, (C4×Q8).210C22, (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
Generators and relations for C42.527C23
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 >
Subgroups: 328 in 180 conjugacy classes, 88 normal (24 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C8, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C42, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, D8, SD16, Q16, C22×C4, C2×D4, C2×Q8, C2×Q8, C4×C8, C4×C8, D4⋊C4, Q8⋊C4, C4⋊C8, C4⋊C8, C2.D8, C42⋊C2, C4×D4, C4×Q8, C4×Q8, C22⋊Q8, C4.4D4, C42⋊2C2, C4⋊Q8, C2×D8, C2×SD16, C2×Q16, C4×D8, C4×Q16, C8×Q8, D4.D4, Q8.D4, C4⋊Q16, C8.12D4, C22.50C24, C42.527C23
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C4○D8, C22×D4, C2×C4○D4, 2+ 1+4, Q8⋊6D4, C2×C4○D8, Q8○D8, C42.527C23
►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 12 26)(2 54 9 27)(3 55 10 28)(4 56 11 25)(5 43 39 23)(6 44 40 24)(7 41 37 21)(8 42 38 22)(13 31 50 60)(14 32 51 57)(15 29 52 58)(16 30 49 59)(17 62 48 33)(18 63 45 34)(19 64 46 35)(20 61 47 36)
(2 11)(4 9)(5 23)(6 42)(7 21)(8 44)(14 49)(16 51)(17 35)(18 63)(19 33)(20 61)(22 40)(24 38)(25 27)(26 53)(28 55)(29 58)(30 32)(31 60)(34 45)(36 47)(37 41)(39 43)(46 62)(48 64)(54 56)(57 59)
(1 18 10 47)(2 19 11 48)(3 20 12 45)(4 17 9 46)(5 58 37 31)(6 59 38 32)(7 60 39 29)(8 57 40 30)(13 43 52 21)(14 44 49 22)(15 41 50 23)(16 42 51 24)(25 62 54 35)(26 63 55 36)(27 64 56 33)(28 61 53 34)
(1 15 10 50)(2 51 11 16)(3 13 12 52)(4 49 9 14)(5 63 37 36)(6 33 38 64)(7 61 39 34)(8 35 40 62)(17 22 46 44)(18 41 47 23)(19 24 48 42)(20 43 45 21)(25 30 54 57)(26 58 55 31)(27 32 56 59)(28 60 53 29)
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), (1,53,12,26)(2,54,9,27)(3,55,10,28)(4,56,11,25)(5,43,39,23)(6,44,40,24)(7,41,37,21)(8,42,38,22)(13,31,50,60)(14,32,51,57)(15,29,52,58)(16,30,49,59)(17,62,48,33)(18,63,45,34)(19,64,46,35)(20,61,47,36), (2,11)(4,9)(5,23)(6,42)(7,21)(8,44)(14,49)(16,51)(17,35)(18,63)(19,33)(20,61)(22,40)(24,38)(25,27)(26,53)(28,55)(29,58)(30,32)(31,60)(34,45)(36,47)(37,41)(39,43)(46,62)(48,64)(54,56)(57,59), (1,18,10,47)(2,19,11,48)(3,20,12,45)(4,17,9,46)(5,58,37,31)(6,59,38,32)(7,60,39,29)(8,57,40,30)(13,43,52,21)(14,44,49,22)(15,41,50,23)(16,42,51,24)(25,62,54,35)(26,63,55,36)(27,64,56,33)(28,61,53,34), (1,15,10,50)(2,51,11,16)(3,13,12,52)(4,49,9,14)(5,63,37,36)(6,33,38,64)(7,61,39,34)(8,35,40,62)(17,22,46,44)(18,41,47,23)(19,24,48,42)(20,43,45,21)(25,30,54,57)(26,58,55,31)(27,32,56,59)(28,60,53,29)>;
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), (1,53,12,26)(2,54,9,27)(3,55,10,28)(4,56,11,25)(5,43,39,23)(6,44,40,24)(7,41,37,21)(8,42,38,22)(13,31,50,60)(14,32,51,57)(15,29,52,58)(16,30,49,59)(17,62,48,33)(18,63,45,34)(19,64,46,35)(20,61,47,36), (2,11)(4,9)(5,23)(6,42)(7,21)(8,44)(14,49)(16,51)(17,35)(18,63)(19,33)(20,61)(22,40)(24,38)(25,27)(26,53)(28,55)(29,58)(30,32)(31,60)(34,45)(36,47)(37,41)(39,43)(46,62)(48,64)(54,56)(57,59), (1,18,10,47)(2,19,11,48)(3,20,12,45)(4,17,9,46)(5,58,37,31)(6,59,38,32)(7,60,39,29)(8,57,40,30)(13,43,52,21)(14,44,49,22)(15,41,50,23)(16,42,51,24)(25,62,54,35)(26,63,55,36)(27,64,56,33)(28,61,53,34), (1,15,10,50)(2,51,11,16)(3,13,12,52)(4,49,9,14)(5,63,37,36)(6,33,38,64)(7,61,39,34)(8,35,40,62)(17,22,46,44)(18,41,47,23)(19,24,48,42)(20,43,45,21)(25,30,54,57)(26,58,55,31)(27,32,56,59)(28,60,53,29) );
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)], [(1,53,12,26),(2,54,9,27),(3,55,10,28),(4,56,11,25),(5,43,39,23),(6,44,40,24),(7,41,37,21),(8,42,38,22),(13,31,50,60),(14,32,51,57),(15,29,52,58),(16,30,49,59),(17,62,48,33),(18,63,45,34),(19,64,46,35),(20,61,47,36)], [(2,11),(4,9),(5,23),(6,42),(7,21),(8,44),(14,49),(16,51),(17,35),(18,63),(19,33),(20,61),(22,40),(24,38),(25,27),(26,53),(28,55),(29,58),(30,32),(31,60),(34,45),(36,47),(37,41),(39,43),(46,62),(48,64),(54,56),(57,59)], [(1,18,10,47),(2,19,11,48),(3,20,12,45),(4,17,9,46),(5,58,37,31),(6,59,38,32),(7,60,39,29),(8,57,40,30),(13,43,52,21),(14,44,49,22),(15,41,50,23),(16,42,51,24),(25,62,54,35),(26,63,55,36),(27,64,56,33),(28,61,53,34)], [(1,15,10,50),(2,51,11,16),(3,13,12,52),(4,49,9,14),(5,63,37,36),(6,33,38,64),(7,61,39,34),(8,35,40,62),(17,22,46,44),(18,41,47,23),(19,24,48,42),(20,43,45,21),(25,30,54,57),(26,58,55,31),(27,32,56,59),(28,60,53,29)]])
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 |
Matrix representation of C42.527C23 ►in 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] >;
C42.527C23 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