p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.359C23, C4⋊C4.352D4, D4⋊2Q8⋊8C2, Q8⋊Q8⋊8C2, D8⋊C4⋊12C2, (C4×SD16)⋊31C2, D4⋊D4.3C2, C4⋊C4.78C23, C4⋊C8.58C22, (C2×C8).52C23, Q16⋊C4⋊11C2, D4.15(C4○D4), D4.2D4⋊22C2, D4.7D4⋊21C2, (C2×C4).323C24, (C4×C8).262C22, Q8.13(C4○D4), Q8.D4⋊22C2, C22⋊C4.153D4, (C4×D4).85C22, (C2×D8).61C22, C23.262(C2×D4), C4⋊Q8.108C22, (C2×Q8).83C23, (C4×Q8).81C22, C4.Q8.22C22, C8⋊C4.15C22, C2.31(D4○SD16), (C2×D4).413C23, C23.47D4⋊7C2, C23.46D4⋊7C2, C4⋊D4.31C22, C22⋊C8.36C22, (C2×Q16).62C22, C22⋊Q8.31C22, D4⋊C4.38C22, C42.7C22⋊8C2, (C22×C4).296C23, C4.4D4.30C22, C22.583(C22×D4), C22.36C24⋊3C2, Q8⋊C4.176C22, (C2×SD16).146C22, C42⋊C2.134C22, C23.33C23⋊14C2, C2.124(C22.19C24), C4.208(C2×C4○D4), (C2×C4).507(C2×D4), (C2×C4⋊C4).617C22, (C2×C4○D4).146C22, SmallGroup(128,1857)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
C1 — C22 — C42⋊C2 — C42.359C23 |
Generators and relations for C42.359C23
G = < a,b,c,d,e | a4=b4=e2=1, c2=d2=b2, ab=ba, ac=ca, dad-1=ab2, ae=ea, cbc-1=ebe=b-1, bd=db, dcd-1=a2b2c, ece=bc, de=ed >
Subgroups: 364 in 192 conjugacy classes, 88 normal (84 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C23, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, D8, SD16, Q16, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C4○D4, C4×C8, C8⋊C4, C22⋊C8, D4⋊C4, Q8⋊C4, C4⋊C8, C4.Q8, C2×C4⋊C4, C2×C4⋊C4, C42⋊C2, C42⋊C2, C4×D4, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8, C22⋊Q8, C22.D4, C4.4D4, C4.4D4, C42⋊2C2, C4⋊Q8, C2×D8, C2×SD16, C2×Q16, C2×C4○D4, C42.7C22, C4×SD16, Q16⋊C4, D8⋊C4, D4⋊D4, D4.7D4, D4.2D4, Q8.D4, Q8⋊Q8, D4⋊2Q8, C23.46D4, C23.47D4, C23.33C23, C22.36C24, C42.359C23
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C22×D4, C2×C4○D4, C22.19C24, D4○SD16, C42.359C23
(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 20 26 23)(2 17 27 24)(3 18 28 21)(4 19 25 22)(5 11 62 14)(6 12 63 15)(7 9 64 16)(8 10 61 13)(29 34 41 39)(30 35 42 40)(31 36 43 37)(32 33 44 38)(45 49 57 54)(46 50 58 55)(47 51 59 56)(48 52 60 53)
(1 48 26 60)(2 45 27 57)(3 46 28 58)(4 47 25 59)(5 40 62 35)(6 37 63 36)(7 38 64 33)(8 39 61 34)(9 44 16 32)(10 41 13 29)(11 42 14 30)(12 43 15 31)(17 54 24 49)(18 55 21 50)(19 56 22 51)(20 53 23 52)
(1 45 26 57)(2 58 27 46)(3 47 28 59)(4 60 25 48)(5 29 62 41)(6 42 63 30)(7 31 64 43)(8 44 61 32)(9 36 16 37)(10 38 13 33)(11 34 14 39)(12 40 15 35)(17 55 24 50)(18 51 21 56)(19 53 22 52)(20 49 23 54)
(1 44)(2 41)(3 42)(4 43)(5 58)(6 59)(7 60)(8 57)(9 52)(10 49)(11 50)(12 51)(13 54)(14 55)(15 56)(16 53)(17 34)(18 35)(19 36)(20 33)(21 40)(22 37)(23 38)(24 39)(25 31)(26 32)(27 29)(28 30)(45 61)(46 62)(47 63)(48 64)
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), (1,20,26,23)(2,17,27,24)(3,18,28,21)(4,19,25,22)(5,11,62,14)(6,12,63,15)(7,9,64,16)(8,10,61,13)(29,34,41,39)(30,35,42,40)(31,36,43,37)(32,33,44,38)(45,49,57,54)(46,50,58,55)(47,51,59,56)(48,52,60,53), (1,48,26,60)(2,45,27,57)(3,46,28,58)(4,47,25,59)(5,40,62,35)(6,37,63,36)(7,38,64,33)(8,39,61,34)(9,44,16,32)(10,41,13,29)(11,42,14,30)(12,43,15,31)(17,54,24,49)(18,55,21,50)(19,56,22,51)(20,53,23,52), (1,45,26,57)(2,58,27,46)(3,47,28,59)(4,60,25,48)(5,29,62,41)(6,42,63,30)(7,31,64,43)(8,44,61,32)(9,36,16,37)(10,38,13,33)(11,34,14,39)(12,40,15,35)(17,55,24,50)(18,51,21,56)(19,53,22,52)(20,49,23,54), (1,44)(2,41)(3,42)(4,43)(5,58)(6,59)(7,60)(8,57)(9,52)(10,49)(11,50)(12,51)(13,54)(14,55)(15,56)(16,53)(17,34)(18,35)(19,36)(20,33)(21,40)(22,37)(23,38)(24,39)(25,31)(26,32)(27,29)(28,30)(45,61)(46,62)(47,63)(48,64)>;
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), (1,20,26,23)(2,17,27,24)(3,18,28,21)(4,19,25,22)(5,11,62,14)(6,12,63,15)(7,9,64,16)(8,10,61,13)(29,34,41,39)(30,35,42,40)(31,36,43,37)(32,33,44,38)(45,49,57,54)(46,50,58,55)(47,51,59,56)(48,52,60,53), (1,48,26,60)(2,45,27,57)(3,46,28,58)(4,47,25,59)(5,40,62,35)(6,37,63,36)(7,38,64,33)(8,39,61,34)(9,44,16,32)(10,41,13,29)(11,42,14,30)(12,43,15,31)(17,54,24,49)(18,55,21,50)(19,56,22,51)(20,53,23,52), (1,45,26,57)(2,58,27,46)(3,47,28,59)(4,60,25,48)(5,29,62,41)(6,42,63,30)(7,31,64,43)(8,44,61,32)(9,36,16,37)(10,38,13,33)(11,34,14,39)(12,40,15,35)(17,55,24,50)(18,51,21,56)(19,53,22,52)(20,49,23,54), (1,44)(2,41)(3,42)(4,43)(5,58)(6,59)(7,60)(8,57)(9,52)(10,49)(11,50)(12,51)(13,54)(14,55)(15,56)(16,53)(17,34)(18,35)(19,36)(20,33)(21,40)(22,37)(23,38)(24,39)(25,31)(26,32)(27,29)(28,30)(45,61)(46,62)(47,63)(48,64) );
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)], [(1,20,26,23),(2,17,27,24),(3,18,28,21),(4,19,25,22),(5,11,62,14),(6,12,63,15),(7,9,64,16),(8,10,61,13),(29,34,41,39),(30,35,42,40),(31,36,43,37),(32,33,44,38),(45,49,57,54),(46,50,58,55),(47,51,59,56),(48,52,60,53)], [(1,48,26,60),(2,45,27,57),(3,46,28,58),(4,47,25,59),(5,40,62,35),(6,37,63,36),(7,38,64,33),(8,39,61,34),(9,44,16,32),(10,41,13,29),(11,42,14,30),(12,43,15,31),(17,54,24,49),(18,55,21,50),(19,56,22,51),(20,53,23,52)], [(1,45,26,57),(2,58,27,46),(3,47,28,59),(4,60,25,48),(5,29,62,41),(6,42,63,30),(7,31,64,43),(8,44,61,32),(9,36,16,37),(10,38,13,33),(11,34,14,39),(12,40,15,35),(17,55,24,50),(18,51,21,56),(19,53,22,52),(20,49,23,54)], [(1,44),(2,41),(3,42),(4,43),(5,58),(6,59),(7,60),(8,57),(9,52),(10,49),(11,50),(12,51),(13,54),(14,55),(15,56),(16,53),(17,34),(18,35),(19,36),(20,33),(21,40),(22,37),(23,38),(24,39),(25,31),(26,32),(27,29),(28,30),(45,61),(46,62),(47,63),(48,64)]])
32 conjugacy classes
class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | ··· | 4F | 4G | ··· | 4O | 4P | 4Q | 4R | 8A | 8B | 8C | 8D | 8E | 8F |
order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | 8 |
size | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 8 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 4 | 4 | 4 | 4 | 8 | 8 |
32 irreducible representations
dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 |
type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||
image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | C4○D4 | C4○D4 | D4○SD16 |
kernel | C42.359C23 | C42.7C22 | C4×SD16 | Q16⋊C4 | D8⋊C4 | D4⋊D4 | D4.7D4 | D4.2D4 | Q8.D4 | Q8⋊Q8 | D4⋊2Q8 | C23.46D4 | C23.47D4 | C23.33C23 | C22.36C24 | C22⋊C4 | C4⋊C4 | D4 | Q8 | C2 |
# reps | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 |
Matrix representation of C42.359C23 ►in GL6(𝔽17)
4 | 0 | 0 | 0 | 0 | 0 |
0 | 4 | 0 | 0 | 0 | 0 |
0 | 0 | 16 | 2 | 0 | 0 |
0 | 0 | 16 | 1 | 0 | 0 |
0 | 0 | 0 | 0 | 16 | 2 |
0 | 0 | 0 | 0 | 16 | 1 |
16 | 0 | 0 | 0 | 0 | 0 |
0 | 16 | 0 | 0 | 0 | 0 |
0 | 0 | 1 | 15 | 0 | 0 |
0 | 0 | 1 | 16 | 0 | 0 |
0 | 0 | 0 | 0 | 16 | 2 |
0 | 0 | 0 | 0 | 16 | 1 |
0 | 13 | 0 | 0 | 0 | 0 |
4 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 10 | 7 |
0 | 0 | 0 | 0 | 5 | 0 |
0 | 0 | 0 | 10 | 0 | 0 |
0 | 0 | 12 | 10 | 0 | 0 |
0 | 1 | 0 | 0 | 0 | 0 |
1 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 7 |
0 | 0 | 0 | 0 | 12 | 0 |
0 | 0 | 0 | 7 | 0 | 0 |
0 | 0 | 12 | 0 | 0 | 0 |
0 | 1 | 0 | 0 | 0 | 0 |
1 | 0 | 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 |
G:=sub<GL(6,GF(17))| [4,0,0,0,0,0,0,4,0,0,0,0,0,0,16,16,0,0,0,0,2,1,0,0,0,0,0,0,16,16,0,0,0,0,2,1],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,1,0,0,0,0,15,16,0,0,0,0,0,0,16,16,0,0,0,0,2,1],[0,4,0,0,0,0,13,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,10,10,0,0,10,5,0,0,0,0,7,0,0,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,7,0,0,0,0,12,0,0,0,0,7,0,0,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0] >;
C42.359C23 in GAP, Magma, Sage, TeX
C_4^2._{359}C_2^3
% in TeX
G:=Group("C4^2.359C2^3");
// GroupNames label
G:=SmallGroup(128,1857);
// by ID
G=gap.SmallGroup(128,1857);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,456,758,1018,304,4037,1027,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^4=e^2=1,c^2=d^2=b^2,a*b=b*a,a*c=c*a,d*a*d^-1=a*b^2,a*e=e*a,c*b*c^-1=e*b*e=b^-1,b*d=d*b,d*c*d^-1=a^2*b^2*c,e*c*e=b*c,d*e=e*d>;
// generators/relations