p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.16C23, C4:C8:7C22, (C2xQ8):22D4, Q8.7(C2xD4), C4:C4.337D4, C4:SD16:3C2, (C4xQ8):3C22, (C2xC8).18C23, C4.74(C22xD4), C4.35(C4:D4), C4:C4.384C23, (C2xC4).247C24, Q8.D4:14C2, C22:C4.138D4, (C2xQ16):17C22, (C2xD4).53C23, C23.444(C2xD4), D4:C4:76C22, C2.13(D4oSD16), Q8:C4:79C22, (C22xSD16):24C2, (C2xSD16):10C22, C4:1D4.58C22, C23.37D4:9C2, (C2xQ8).360C23, C23.24D4:24C2, C22.82(C4:D4), C42.6C22:5C2, (C22xC4).977C23, (C22xC8).341C22, C4.4D4.24C22, C22.507(C22xD4), C23.32C23:7C2, C22.29C24.10C2, (C22xD4).342C22, (C2xM4(2)).54C22, (C22xQ8).274C22, C42:C2.102C22, C4.157(C2xC4oD4), (C2xC4).467(C2xD4), C2.65(C2xC4:D4), (C2xC8.C22):16C2, (C2xC4).278(C4oD4), (C2xC4oD4).119C22, SmallGroup(128,1775)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.16C23
G = < a,b,c,d,e | a4=b4=c2=1, d2=b2, e2=a2b2, ab=ba, cac=a-1b2, ad=da, eae-1=ab2, cbc=dbd-1=b-1, be=eb, dcd-1=bc, ece-1=a2b2c, de=ed >
Subgroups: 508 in 245 conjugacy classes, 100 normal (26 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C8, C2xC4, C2xC4, C2xC4, D4, Q8, Q8, C23, C23, C42, C42, C22:C4, C22:C4, C4:C4, C4:C4, C2xC8, C2xC8, M4(2), SD16, Q16, C22xC4, C22xC4, C2xD4, C2xD4, C2xD4, C2xQ8, C2xQ8, C2xQ8, C4oD4, C24, D4:C4, Q8:C4, C4:C8, C42:C2, C42:C2, C4xQ8, C4xQ8, C22wrC2, C4:D4, C4.4D4, C4:1D4, C22xC8, C2xM4(2), C2xSD16, C2xSD16, C2xQ16, C8.C22, C22xD4, C22xQ8, C2xC4oD4, C23.24D4, C23.37D4, C42.6C22, C4:SD16, Q8.D4, C23.32C23, C22.29C24, C22xSD16, C2xC8.C22, C42.16C23
Quotients: C1, C2, C22, D4, C23, C2xD4, C4oD4, C24, C4:D4, C22xD4, C2xC4oD4, C2xC4:D4, D4oSD16, C42.16C23
(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 22 14 31)(2 23 15 32)(3 24 16 29)(4 21 13 30)(5 27 19 10)(6 28 20 11)(7 25 17 12)(8 26 18 9)
(2 13)(4 15)(5 10)(6 26)(7 12)(8 28)(9 20)(11 18)(17 25)(19 27)(21 23)(22 31)(24 29)(30 32)
(1 27 14 10)(2 28 15 11)(3 25 16 12)(4 26 13 9)(5 31 19 22)(6 32 20 23)(7 29 17 24)(8 30 18 21)
(1 4 16 15)(2 14 13 3)(5 8 17 20)(6 19 18 7)(9 25 28 10)(11 27 26 12)(21 29 32 22)(23 31 30 24)
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), (1,22,14,31)(2,23,15,32)(3,24,16,29)(4,21,13,30)(5,27,19,10)(6,28,20,11)(7,25,17,12)(8,26,18,9), (2,13)(4,15)(5,10)(6,26)(7,12)(8,28)(9,20)(11,18)(17,25)(19,27)(21,23)(22,31)(24,29)(30,32), (1,27,14,10)(2,28,15,11)(3,25,16,12)(4,26,13,9)(5,31,19,22)(6,32,20,23)(7,29,17,24)(8,30,18,21), (1,4,16,15)(2,14,13,3)(5,8,17,20)(6,19,18,7)(9,25,28,10)(11,27,26,12)(21,29,32,22)(23,31,30,24)>;
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), (1,22,14,31)(2,23,15,32)(3,24,16,29)(4,21,13,30)(5,27,19,10)(6,28,20,11)(7,25,17,12)(8,26,18,9), (2,13)(4,15)(5,10)(6,26)(7,12)(8,28)(9,20)(11,18)(17,25)(19,27)(21,23)(22,31)(24,29)(30,32), (1,27,14,10)(2,28,15,11)(3,25,16,12)(4,26,13,9)(5,31,19,22)(6,32,20,23)(7,29,17,24)(8,30,18,21), (1,4,16,15)(2,14,13,3)(5,8,17,20)(6,19,18,7)(9,25,28,10)(11,27,26,12)(21,29,32,22)(23,31,30,24) );
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)], [(1,22,14,31),(2,23,15,32),(3,24,16,29),(4,21,13,30),(5,27,19,10),(6,28,20,11),(7,25,17,12),(8,26,18,9)], [(2,13),(4,15),(5,10),(6,26),(7,12),(8,28),(9,20),(11,18),(17,25),(19,27),(21,23),(22,31),(24,29),(30,32)], [(1,27,14,10),(2,28,15,11),(3,25,16,12),(4,26,13,9),(5,31,19,22),(6,32,20,23),(7,29,17,24),(8,30,18,21)], [(1,4,16,15),(2,14,13,3),(5,8,17,20),(6,19,18,7),(9,25,28,10),(11,27,26,12),(21,29,32,22),(23,31,30,24)]])
32 conjugacy classes
class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 4A | 4B | 4C | 4D | 4E | ··· | 4P | 4Q | 8A | 8B | 8C | 8D | 8E | 8F |
order | 1 | 2 | 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 | 2 | 2 | 8 | 8 | 8 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 8 | 4 | 4 | 4 | 4 | 8 | 8 |
32 irreducible representations
dim | 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 | D4 | D4 | D4 | C4oD4 | D4oSD16 |
kernel | C42.16C23 | C23.24D4 | C23.37D4 | C42.6C22 | C4:SD16 | Q8.D4 | C23.32C23 | C22.29C24 | C22xSD16 | C2xC8.C22 | C22:C4 | C4:C4 | C2xQ8 | C2xC4 | C2 |
# reps | 1 | 1 | 1 | 1 | 4 | 4 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 |
Matrix representation of C42.16C23 ►in GL6(F17)
2 | 15 | 0 | 0 | 0 | 0 |
11 | 15 | 0 | 0 | 0 | 0 |
0 | 0 | 16 | 0 | 15 | 0 |
0 | 0 | 0 | 0 | 1 | 1 |
0 | 0 | 1 | 0 | 1 | 0 |
0 | 0 | 16 | 16 | 16 | 0 |
1 | 0 | 0 | 0 | 0 | 0 |
0 | 1 | 0 | 0 | 0 | 0 |
0 | 0 | 16 | 15 | 0 | 0 |
0 | 0 | 1 | 1 | 0 | 0 |
0 | 0 | 0 | 1 | 0 | 1 |
0 | 0 | 16 | 16 | 16 | 0 |
1 | 0 | 0 | 0 | 0 | 0 |
2 | 16 | 0 | 0 | 0 | 0 |
0 | 0 | 1 | 0 | 0 | 0 |
0 | 0 | 16 | 16 | 0 | 0 |
0 | 0 | 0 | 0 | 1 | 0 |
0 | 0 | 1 | 0 | 0 | 16 |
16 | 0 | 0 | 0 | 0 | 0 |
0 | 16 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 10 | 0 | 0 |
0 | 0 | 5 | 0 | 0 | 0 |
0 | 0 | 0 | 12 | 12 | 12 |
0 | 0 | 12 | 5 | 12 | 5 |
2 | 15 | 0 | 0 | 0 | 0 |
11 | 15 | 0 | 0 | 0 | 0 |
0 | 0 | 16 | 0 | 15 | 0 |
0 | 0 | 0 | 0 | 1 | 1 |
0 | 0 | 0 | 0 | 1 | 0 |
0 | 0 | 0 | 1 | 16 | 0 |
G:=sub<GL(6,GF(17))| [2,11,0,0,0,0,15,15,0,0,0,0,0,0,16,0,1,16,0,0,0,0,0,16,0,0,15,1,1,16,0,0,0,1,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,1,0,16,0,0,15,1,1,16,0,0,0,0,0,16,0,0,0,0,1,0],[1,2,0,0,0,0,0,16,0,0,0,0,0,0,1,16,0,1,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,5,0,12,0,0,10,0,12,5,0,0,0,0,12,12,0,0,0,0,12,5],[2,11,0,0,0,0,15,15,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,15,1,1,16,0,0,0,1,0,0] >;
C42.16C23 in GAP, Magma, Sage, TeX
C_4^2._{16}C_2^3
% in TeX
G:=Group("C4^2.16C2^3");
// GroupNames label
G:=SmallGroup(128,1775);
// by ID
G=gap.SmallGroup(128,1775);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,120,758,352,1018,4037,1027,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^4=c^2=1,d^2=b^2,e^2=a^2*b^2,a*b=b*a,c*a*c=a^-1*b^2,a*d=d*a,e*a*e^-1=a*b^2,c*b*c=d*b*d^-1=b^-1,b*e=e*b,d*c*d^-1=b*c,e*c*e^-1=a^2*b^2*c,d*e=e*d>;
// generators/relations