p-group, metabelian, nilpotent (class 3), monomial
Aliases: C23.C16, C16.25D4, M6(2)⋊3C2, C4.7M5(2), C8.21M4(2), (C2×C4).C16, (C2×C8).4C8, (C2×C16).6C4, (C22×C4).5C8, C22.4(C2×C16), (C22×C8).16C4, C8.58(C22⋊C4), C2.7(C22⋊C16), C4.34(C22⋊C8), (C2×C16).49C22, (C2×M5(2)).18C2, (C2×C4).77(C2×C8), (C2×C8).241(C2×C4), SmallGroup(128,132)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C23.C16
G = < a,b,c,d | a2=b2=c2=1, d16=c, ab=ba, ac=ca, dad-1=abc, dbd-1=bc=cb, cd=dc >
Smallest permutation representation of C23.C16
►On 32 points
Generators in S32
(2 18)(3 19)(6 22)(7 23)(10 26)(11 27)(14 30)(15 31)
(2 18)(4 20)(6 22)(8 24)(10 26)(12 28)(14 30)(16 32)
(1 17)(2 18)(3 19)(4 20)(5 21)(6 22)(7 23)(8 24)(9 25)(10 26)(11 27)(12 28)(13 29)(14 30)(15 31)(16 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)
G:=sub<Sym(32)| (2,18)(3,19)(6,22)(7,23)(10,26)(11,27)(14,30)(15,31), (2,18)(4,20)(6,22)(8,24)(10,26)(12,28)(14,30)(16,32), (1,17)(2,18)(3,19)(4,20)(5,21)(6,22)(7,23)(8,24)(9,25)(10,26)(11,27)(12,28)(13,29)(14,30)(15,31)(16,32), (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)>;
G:=Group( (2,18)(3,19)(6,22)(7,23)(10,26)(11,27)(14,30)(15,31), (2,18)(4,20)(6,22)(8,24)(10,26)(12,28)(14,30)(16,32), (1,17)(2,18)(3,19)(4,20)(5,21)(6,22)(7,23)(8,24)(9,25)(10,26)(11,27)(12,28)(13,29)(14,30)(15,31)(16,32), (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) );
G=PermutationGroup([[(2,18),(3,19),(6,22),(7,23),(10,26),(11,27),(14,30),(15,31)], [(2,18),(4,20),(6,22),(8,24),(10,26),(12,28),(14,30),(16,32)], [(1,17),(2,18),(3,19),(4,20),(5,21),(6,22),(7,23),(8,24),(9,25),(10,26),(11,27),(12,28),(13,29),(14,30),(15,31),(16,32)], [(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)]])
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 16A | ··· | 16H | 16I | 16J | 16K | 16L | 32A | ··· | 32P |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 16 | ··· | 16 | 16 | 16 | 16 | 16 | 32 | ··· | 32 |
| size | 1 | 1 | 2 | 4 | 1 | 1 | 2 | 4 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | |||||||||
| image | C1 | C2 | C2 | C4 | C4 | C8 | C8 | C16 | C16 | D4 | M4(2) | M5(2) | C23.C16 |
| kernel | C23.C16 | M6(2) | C2×M5(2) | C2×C16 | C22×C8 | C2×C8 | C22×C4 | C2×C4 | C23 | C16 | C8 | C4 | C1 |
| # reps | 1 | 2 | 1 | 2 | 2 | 4 | 4 | 8 | 8 | 2 | 2 | 4 | 4 |
Matrix representation of C23.C16 ►in GL4(𝔽97) generated by
| 1 | 0 | 0 | 0 |
| 61 | 96 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 27 | 0 | 0 | 96 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 68 | 0 | 96 | 0 |
| 27 | 0 | 0 | 96 |
| 96 | 0 | 0 | 0 |
| 0 | 96 | 0 | 0 |
| 0 | 0 | 96 | 0 |
| 0 | 0 | 0 | 96 |
| 68 | 0 | 95 | 0 |
| 72 | 0 | 36 | 1 |
| 2 | 1 | 29 | 0 |
| 13 | 0 | 70 | 0 |
G:=sub<GL(4,GF(97))| [1,61,0,27,0,96,0,0,0,0,1,0,0,0,0,96],[1,0,68,27,0,1,0,0,0,0,96,0,0,0,0,96],[96,0,0,0,0,96,0,0,0,0,96,0,0,0,0,96],[68,72,2,13,0,0,1,0,95,36,29,70,0,1,0,0] >;
C23.C16 in GAP, Magma, Sage, TeX
C_2^3.C_{16} % in TeX
G:=Group("C2^3.C16"); // GroupNames label
G:=SmallGroup(128,132);
// by ID
G=gap.SmallGroup(128,132);
# by ID
G:=PCGroup([7,-2,2,-2,2,-2,-2,-2,56,85,1430,1018,80,102,124]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^2=c^2=1,d^16=c,a*b=b*a,a*c=c*a,d*a*d^-1=a*b*c,d*b*d^-1=b*c=c*b,c*d=d*c>;
// generators/relations
Export