p-group, metabelian, nilpotent (class 2), monomial
Aliases: C42.301C23, C4.1752+ 1+4, (C8xD4):49C2, C4:Q8.34C4, C8:6D4:44C2, C8:9D4:45C2, C4.21(C8oD4), C4:D4.29C4, C4:1D4.21C4, (C4xC8).29C22, C4:C8.367C22, C42.226(C2xC4), (C2xC4).678C24, (C2xC8).439C23, C4.4D4.22C4, C8:C4.98C22, C42.6C4:53C2, (C4xD4).303C22, C23.45(C22xC4), C42.12C4:55C2, C2.32(Q8oM4(2)), C22:C8.146C22, (C22xC8).451C22, (C2xC42).785C22, (C22xC4).945C23, C22.202(C23xC4), C2.52(C22.11C24), (C2xM4(2)).248C22, C22.26C24.29C2, C2.31(C2xC8oD4), C4:C4.121(C2xC4), (C2xD4).145(C2xC4), C22:C4.22(C2xC4), (C2xC4).83(C22xC4), (C2xQ8).127(C2xC4), (C22xC8):C2:37C2, (C22xC4).358(C2xC4), (C2xC4oD4).98C22, SmallGroup(128,1713)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.301C23
G = < a,b,c,d,e | a4=b4=d2=e2=1, c2=b, ab=ba, cac-1=a-1b2, dad=a-1, ae=ea, bc=cb, bd=db, be=eb, dcd=ece=a2b2c, ede=b2d >
Subgroups: 332 in 201 conjugacy classes, 126 normal (28 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2xC4, C2xC4, C2xC4, D4, Q8, C23, C23, C42, C22:C4, C4:C4, C2xC8, C2xC8, M4(2), C22xC4, C22xC4, C2xD4, C2xQ8, C4oD4, C4xC8, C8:C4, C22:C8, C4:C8, C2xC42, C4xD4, C4:D4, C4.4D4, C4:1D4, C4:Q8, C22xC8, C2xM4(2), C2xC4oD4, (C22xC8):C2, C42.12C4, C42.6C4, C8xD4, C8:9D4, C8:6D4, C22.26C24, C42.301C23
Quotients: C1, C2, C4, C22, C2xC4, C23, C22xC4, C24, C8oD4, C23xC4, 2+ 1+4, C22.11C24, C2xC8oD4, Q8oM4(2), C42.301C23
(1 44 51 37)(2 34 52 41)(3 46 53 39)(4 36 54 43)(5 48 55 33)(6 38 56 45)(7 42 49 35)(8 40 50 47)(9 18 58 28)(10 25 59 23)(11 20 60 30)(12 27 61 17)(13 22 62 32)(14 29 63 19)(15 24 64 26)(16 31 57 21)
(1 3 5 7)(2 4 6 8)(9 11 13 15)(10 12 14 16)(17 19 21 23)(18 20 22 24)(25 27 29 31)(26 28 30 32)(33 35 37 39)(34 36 38 40)(41 43 45 47)(42 44 46 48)(49 51 53 55)(50 52 54 56)(57 59 61 63)(58 60 62 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 23)(2 30)(3 17)(4 32)(5 19)(6 26)(7 21)(8 28)(9 47)(10 37)(11 41)(12 39)(13 43)(14 33)(15 45)(16 35)(18 50)(20 52)(22 54)(24 56)(25 51)(27 53)(29 55)(31 49)(34 60)(36 62)(38 64)(40 58)(42 57)(44 59)(46 61)(48 63)
(1 17)(2 32)(3 19)(4 26)(5 21)(6 28)(7 23)(8 30)(9 38)(10 42)(11 40)(12 44)(13 34)(14 46)(15 36)(16 48)(18 56)(20 50)(22 52)(24 54)(25 49)(27 51)(29 53)(31 55)(33 57)(35 59)(37 61)(39 63)(41 62)(43 64)(45 58)(47 60)
G:=sub<Sym(64)| (1,44,51,37)(2,34,52,41)(3,46,53,39)(4,36,54,43)(5,48,55,33)(6,38,56,45)(7,42,49,35)(8,40,50,47)(9,18,58,28)(10,25,59,23)(11,20,60,30)(12,27,61,17)(13,22,62,32)(14,29,63,19)(15,24,64,26)(16,31,57,21), (1,3,5,7)(2,4,6,8)(9,11,13,15)(10,12,14,16)(17,19,21,23)(18,20,22,24)(25,27,29,31)(26,28,30,32)(33,35,37,39)(34,36,38,40)(41,43,45,47)(42,44,46,48)(49,51,53,55)(50,52,54,56)(57,59,61,63)(58,60,62,64), (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), (1,23)(2,30)(3,17)(4,32)(5,19)(6,26)(7,21)(8,28)(9,47)(10,37)(11,41)(12,39)(13,43)(14,33)(15,45)(16,35)(18,50)(20,52)(22,54)(24,56)(25,51)(27,53)(29,55)(31,49)(34,60)(36,62)(38,64)(40,58)(42,57)(44,59)(46,61)(48,63), (1,17)(2,32)(3,19)(4,26)(5,21)(6,28)(7,23)(8,30)(9,38)(10,42)(11,40)(12,44)(13,34)(14,46)(15,36)(16,48)(18,56)(20,50)(22,52)(24,54)(25,49)(27,51)(29,53)(31,55)(33,57)(35,59)(37,61)(39,63)(41,62)(43,64)(45,58)(47,60)>;
G:=Group( (1,44,51,37)(2,34,52,41)(3,46,53,39)(4,36,54,43)(5,48,55,33)(6,38,56,45)(7,42,49,35)(8,40,50,47)(9,18,58,28)(10,25,59,23)(11,20,60,30)(12,27,61,17)(13,22,62,32)(14,29,63,19)(15,24,64,26)(16,31,57,21), (1,3,5,7)(2,4,6,8)(9,11,13,15)(10,12,14,16)(17,19,21,23)(18,20,22,24)(25,27,29,31)(26,28,30,32)(33,35,37,39)(34,36,38,40)(41,43,45,47)(42,44,46,48)(49,51,53,55)(50,52,54,56)(57,59,61,63)(58,60,62,64), (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), (1,23)(2,30)(3,17)(4,32)(5,19)(6,26)(7,21)(8,28)(9,47)(10,37)(11,41)(12,39)(13,43)(14,33)(15,45)(16,35)(18,50)(20,52)(22,54)(24,56)(25,51)(27,53)(29,55)(31,49)(34,60)(36,62)(38,64)(40,58)(42,57)(44,59)(46,61)(48,63), (1,17)(2,32)(3,19)(4,26)(5,21)(6,28)(7,23)(8,30)(9,38)(10,42)(11,40)(12,44)(13,34)(14,46)(15,36)(16,48)(18,56)(20,50)(22,52)(24,54)(25,49)(27,51)(29,53)(31,55)(33,57)(35,59)(37,61)(39,63)(41,62)(43,64)(45,58)(47,60) );
G=PermutationGroup([[(1,44,51,37),(2,34,52,41),(3,46,53,39),(4,36,54,43),(5,48,55,33),(6,38,56,45),(7,42,49,35),(8,40,50,47),(9,18,58,28),(10,25,59,23),(11,20,60,30),(12,27,61,17),(13,22,62,32),(14,29,63,19),(15,24,64,26),(16,31,57,21)], [(1,3,5,7),(2,4,6,8),(9,11,13,15),(10,12,14,16),(17,19,21,23),(18,20,22,24),(25,27,29,31),(26,28,30,32),(33,35,37,39),(34,36,38,40),(41,43,45,47),(42,44,46,48),(49,51,53,55),(50,52,54,56),(57,59,61,63),(58,60,62,64)], [(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)], [(1,23),(2,30),(3,17),(4,32),(5,19),(6,26),(7,21),(8,28),(9,47),(10,37),(11,41),(12,39),(13,43),(14,33),(15,45),(16,35),(18,50),(20,52),(22,54),(24,56),(25,51),(27,53),(29,55),(31,49),(34,60),(36,62),(38,64),(40,58),(42,57),(44,59),(46,61),(48,63)], [(1,17),(2,32),(3,19),(4,26),(5,21),(6,28),(7,23),(8,30),(9,38),(10,42),(11,40),(12,44),(13,34),(14,46),(15,36),(16,48),(18,56),(20,50),(22,52),(24,54),(25,49),(27,51),(29,53),(31,55),(33,57),(35,59),(37,61),(39,63),(41,62),(43,64),(45,58),(47,60)]])
44 conjugacy classes
class | 1 | 2A | 2B | 2C | 2D | ··· | 2H | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | ··· | 4O | 8A | ··· | 8H | 8I | ··· | 8T |
order | 1 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 8 | ··· | 8 | 8 | ··· | 8 |
size | 1 | 1 | 1 | 1 | 4 | ··· | 4 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
44 irreducible representations
dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 4 | 4 |
type | + | + | + | + | + | + | + | + | + | ||||||
image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C4 | C8oD4 | 2+ 1+4 | Q8oM4(2) |
kernel | C42.301C23 | (C22xC8):C2 | C42.12C4 | C42.6C4 | C8xD4 | C8:9D4 | C8:6D4 | C22.26C24 | C4:D4 | C4.4D4 | C4:1D4 | C4:Q8 | C4 | C4 | C2 |
# reps | 1 | 4 | 1 | 1 | 2 | 4 | 2 | 1 | 8 | 4 | 2 | 2 | 8 | 2 | 2 |
Matrix representation of C42.301C23 ►in GL6(F17)
0 | 13 | 0 | 0 | 0 | 0 |
13 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 16 |
0 | 0 | 0 | 0 | 16 | 0 |
0 | 0 | 0 | 1 | 0 | 0 |
0 | 0 | 1 | 0 | 0 | 0 |
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 |
2 | 0 | 0 | 0 | 0 | 0 |
0 | 2 | 0 | 0 | 0 | 0 |
0 | 0 | 16 | 0 | 0 | 0 |
0 | 0 | 0 | 1 | 0 | 0 |
0 | 0 | 0 | 0 | 16 | 0 |
0 | 0 | 0 | 0 | 0 | 1 |
0 | 4 | 0 | 0 | 0 | 0 |
13 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 1 | 0 | 0 |
0 | 0 | 1 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 16 |
0 | 0 | 0 | 0 | 16 | 0 |
0 | 1 | 0 | 0 | 0 | 0 |
1 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 1 | 0 | 0 |
0 | 0 | 1 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 1 |
0 | 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(6,GF(17))| [0,13,0,0,0,0,13,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,16,0,0,0,0,16,0,0,0],[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],[2,0,0,0,0,0,0,2,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,1],[0,13,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,16,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;
C42.301C23 in GAP, Magma, Sage, TeX
C_4^2._{301}C_2^3
% in TeX
G:=Group("C4^2.301C2^3");
// GroupNames label
G:=SmallGroup(128,1713);
// by ID
G=gap.SmallGroup(128,1713);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,891,675,1018,521,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^4=d^2=e^2=1,c^2=b,a*b=b*a,c*a*c^-1=a^-1*b^2,d*a*d=a^-1,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,d*c*d=e*c*e=a^2*b^2*c,e*d*e=b^2*d>;
// generators/relations