p-group, metabelian, nilpotent (class 2), monomial
Aliases: C4:12+ 1+4, C22.85C25, C23.41C24, C42.573C23, C24.134C23, D42:13C2, D4:15(C2xD4), (C2xD4):39D4, C23:3(C2xD4), D4o2(C4:D4), C4:Q8:90C22, D4:6D4:21C2, D4:5D4:18C2, (C4xD4):43C22, C23:3D4:7C2, (C2xC4).76C24, C2.31(D4xC23), C22wrC2:7C22, C4:D4:26C22, C4:C4.293C23, C4:1D4:50C22, (C2xC42):58C22, (C23xC4):42C22, C4.120(C22xD4), C22:Q8:31C22, (C2xD4).469C23, C4.4D4:81C22, (C22xD4):36C22, C22:C4.20C23, (C2xQ8).445C23, C22.13(C22xD4), (C22xC4).358C23, (C2x2+ 1+4):10C2, C22.D4:6C22, C2.31(C2x2+ 1+4), C2.22(C2.C25), C22.26C24:34C2, C22.31C24:15C2, (C2xC4xD4):90C2, (C2xC4):4(C2xD4), (C2xC4:D4):67C2, (C2xC4:C4):73C22, (C2xC4oD4):29C22, (C2xC22:C4):48C22, SmallGroup(128,2228)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C4:2+ 1+4
G = < a,b,c,d,e | a4=b4=c2=e2=1, d2=b2, bab-1=a-1, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=b2d >
Subgroups: 1460 in 830 conjugacy classes, 430 normal (16 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C2xC4, C2xC4, C2xC4, D4, D4, Q8, C23, C23, C23, C42, C22:C4, C4:C4, C22xC4, C22xC4, C22xC4, C2xD4, C2xD4, C2xQ8, C4oD4, C24, C2xC42, C2xC22:C4, C2xC4:C4, C2xC4:C4, C4xD4, C22wrC2, C4:D4, C22:Q8, C22.D4, C4.4D4, C4:1D4, C4:Q8, C23xC4, C22xD4, C22xD4, C2xC4oD4, 2+ 1+4, C2xC4xD4, C2xC4:D4, C22.26C24, C23:3D4, C22.31C24, D42, D4:5D4, D4:6D4, C2x2+ 1+4, C4:2+ 1+4
Quotients: C1, C2, C22, D4, C23, C2xD4, C24, C22xD4, 2+ 1+4, C25, D4xC23, C2x2+ 1+4, C2.C25, C4:2+ 1+4
(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)
(1 18 15 5)(2 17 16 8)(3 20 13 7)(4 19 14 6)(9 30 22 28)(10 29 23 27)(11 32 24 26)(12 31 21 25)
(1 6)(2 7)(3 8)(4 5)(9 31)(10 32)(11 29)(12 30)(13 17)(14 18)(15 19)(16 20)(21 28)(22 25)(23 26)(24 27)
(1 31 15 25)(2 32 16 26)(3 29 13 27)(4 30 14 28)(5 12 18 21)(6 9 19 22)(7 10 20 23)(8 11 17 24)
(1 25)(2 26)(3 27)(4 28)(5 21)(6 22)(7 23)(8 24)(9 19)(10 20)(11 17)(12 18)(13 29)(14 30)(15 31)(16 32)
G:=sub<Sym(32)| (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), (1,18,15,5)(2,17,16,8)(3,20,13,7)(4,19,14,6)(9,30,22,28)(10,29,23,27)(11,32,24,26)(12,31,21,25), (1,6)(2,7)(3,8)(4,5)(9,31)(10,32)(11,29)(12,30)(13,17)(14,18)(15,19)(16,20)(21,28)(22,25)(23,26)(24,27), (1,31,15,25)(2,32,16,26)(3,29,13,27)(4,30,14,28)(5,12,18,21)(6,9,19,22)(7,10,20,23)(8,11,17,24), (1,25)(2,26)(3,27)(4,28)(5,21)(6,22)(7,23)(8,24)(9,19)(10,20)(11,17)(12,18)(13,29)(14,30)(15,31)(16,32)>;
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), (1,18,15,5)(2,17,16,8)(3,20,13,7)(4,19,14,6)(9,30,22,28)(10,29,23,27)(11,32,24,26)(12,31,21,25), (1,6)(2,7)(3,8)(4,5)(9,31)(10,32)(11,29)(12,30)(13,17)(14,18)(15,19)(16,20)(21,28)(22,25)(23,26)(24,27), (1,31,15,25)(2,32,16,26)(3,29,13,27)(4,30,14,28)(5,12,18,21)(6,9,19,22)(7,10,20,23)(8,11,17,24), (1,25)(2,26)(3,27)(4,28)(5,21)(6,22)(7,23)(8,24)(9,19)(10,20)(11,17)(12,18)(13,29)(14,30)(15,31)(16,32) );
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)], [(1,18,15,5),(2,17,16,8),(3,20,13,7),(4,19,14,6),(9,30,22,28),(10,29,23,27),(11,32,24,26),(12,31,21,25)], [(1,6),(2,7),(3,8),(4,5),(9,31),(10,32),(11,29),(12,30),(13,17),(14,18),(15,19),(16,20),(21,28),(22,25),(23,26),(24,27)], [(1,31,15,25),(2,32,16,26),(3,29,13,27),(4,30,14,28),(5,12,18,21),(6,9,19,22),(7,10,20,23),(8,11,17,24)], [(1,25),(2,26),(3,27),(4,28),(5,21),(6,22),(7,23),(8,24),(9,19),(10,20),(11,17),(12,18),(13,29),(14,30),(15,31),(16,32)]])
44 conjugacy classes
class | 1 | 2A | 2B | 2C | 2D | ··· | 2M | 2N | ··· | 2U | 4A | ··· | 4H | 4I | ··· | 4V |
order | 1 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
44 irreducible representations
dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 4 | 4 |
type | + | + | + | + | + | + | + | + | + | + | + | + | |
image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | 2+ 1+4 | C2.C25 |
kernel | C4:2+ 1+4 | C2xC4xD4 | C2xC4:D4 | C22.26C24 | C23:3D4 | C22.31C24 | D42 | D4:5D4 | D4:6D4 | C2x2+ 1+4 | C2xD4 | C4 | C2 |
# reps | 1 | 1 | 4 | 2 | 4 | 2 | 4 | 8 | 4 | 2 | 8 | 2 | 2 |
Matrix representation of C4:2+ 1+4 ►in GL6(Z)
0 | 1 | 0 | 0 | 0 | 0 |
-1 | 0 | 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 |
0 | 1 | 0 | 0 | 0 | 0 |
1 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | -1 | 2 | 0 | 0 |
0 | 0 | -1 | 1 | 0 | 0 |
0 | 0 | 0 | 1 | 0 | 1 |
0 | 0 | 1 | -1 | -1 | 0 |
-1 | 0 | 0 | 0 | 0 | 0 |
0 | -1 | 0 | 0 | 0 | 0 |
0 | 0 | -1 | 2 | 0 | 0 |
0 | 0 | 0 | 1 | 0 | 0 |
0 | 0 | 0 | 1 | 0 | 1 |
0 | 0 | 0 | -1 | 1 | 0 |
1 | 0 | 0 | 0 | 0 | 0 |
0 | 1 | 0 | 0 | 0 | 0 |
0 | 0 | -1 | 0 | 2 | 0 |
0 | 0 | 0 | 0 | 1 | 1 |
0 | 0 | -1 | 0 | 1 | 0 |
0 | 0 | 1 | -1 | -1 | 0 |
-1 | 0 | 0 | 0 | 0 | 0 |
0 | -1 | 0 | 0 | 0 | 0 |
0 | 0 | -1 | 0 | 2 | 0 |
0 | 0 | 0 | 0 | 1 | 1 |
0 | 0 | 0 | 0 | 1 | 0 |
0 | 0 | 0 | 1 | -1 | 0 |
G:=sub<GL(6,Integers())| [0,-1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,-1,-1,0,1,0,0,2,1,1,-1,0,0,0,0,0,-1,0,0,0,0,1,0],[-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,2,1,1,-1,0,0,0,0,0,1,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,-1,1,0,0,0,0,0,-1,0,0,2,1,1,-1,0,0,0,1,0,0],[-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,2,1,1,-1,0,0,0,1,0,0] >;
C4:2+ 1+4 in GAP, Magma, Sage, TeX
C_4\rtimes 2_+^{1+4}
% in TeX
G:=Group("C4:ES+(2,2)");
// GroupNames label
G:=SmallGroup(128,2228);
// by ID
G=gap.SmallGroup(128,2228);
# by ID
G:=PCGroup([7,-2,2,2,2,2,-2,2,477,1430,352,570,1684]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^4=c^2=e^2=1,d^2=b^2,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=b^2*d>;
// generators/relations