p-group, metabelian, nilpotent (class 2), monomial
Aliases: C24.5C8, C22⋊2M5(2), (C2×C16)⋊9C22, C8.131(C2×D4), (C2×C8).387D4, C22⋊C16⋊13C2, (C2×M5(2))⋊6C2, (C22×C8).47C4, (C23×C8).23C2, C23.36(C2×C8), (C22×C4).13C8, (C23×C4).37C4, C2.6(C2×M5(2)), C4.24(C22⋊C8), C8.53(C22⋊C4), (C2×C8).624C23, (C2×C4).80M4(2), C4.62(C2×M4(2)), C22.46(C22×C8), C22.30(C22⋊C8), (C22×C8).498C22, (C2×C4).86(C2×C8), (C2×C8).249(C2×C4), C2.22(C2×C22⋊C8), C4.113(C2×C22⋊C4), (C22×C4).446(C2×C4), (C2×C4).609(C22×C4), (C2×C4).361(C22⋊C4), SmallGroup(128,844)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C24.5C8
G = < a,b,c,d,e | a2=b2=c2=d2=1, e8=d, ab=ba, eae-1=ac=ca, ad=da, bc=cb, ebe-1=bd=db, cd=dc, ce=ec, de=ed >
Subgroups: 196 in 130 conjugacy classes, 64 normal (18 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C8, C8, C2×C4, C2×C4, C2×C4, C23, C23, C23, C16, C2×C8, C2×C8, C2×C8, C22×C4, C22×C4, C22×C4, C24, C2×C16, M5(2), C22×C8, C22×C8, C22×C8, C23×C4, C22⋊C16, C2×M5(2), C23×C8, C24.5C8
Quotients: C1, C2, C4, C22, C8, C2×C4, D4, C23, C22⋊C4, C2×C8, M4(2), C22×C4, C2×D4, C22⋊C8, M5(2), C2×C22⋊C4, C22×C8, C2×M4(2), C2×C22⋊C8, C2×M5(2), C24.5C8
►On 32 points
(2 24)(4 26)(6 28)(8 30)(10 32)(12 18)(14 20)(16 22)
(1 31)(2 24)(3 17)(4 26)(5 19)(6 28)(7 21)(8 30)(9 23)(10 32)(11 25)(12 18)(13 27)(14 20)(15 29)(16 22)
(1 23)(2 24)(3 25)(4 26)(5 27)(6 28)(7 29)(8 30)(9 31)(10 32)(11 17)(12 18)(13 19)(14 20)(15 21)(16 22)
(1 9)(2 10)(3 11)(4 12)(5 13)(6 14)(7 15)(8 16)(17 25)(18 26)(19 27)(20 28)(21 29)(22 30)(23 31)(24 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)| (2,24)(4,26)(6,28)(8,30)(10,32)(12,18)(14,20)(16,22), (1,31)(2,24)(3,17)(4,26)(5,19)(6,28)(7,21)(8,30)(9,23)(10,32)(11,25)(12,18)(13,27)(14,20)(15,29)(16,22), (1,23)(2,24)(3,25)(4,26)(5,27)(6,28)(7,29)(8,30)(9,31)(10,32)(11,17)(12,18)(13,19)(14,20)(15,21)(16,22), (1,9)(2,10)(3,11)(4,12)(5,13)(6,14)(7,15)(8,16)(17,25)(18,26)(19,27)(20,28)(21,29)(22,30)(23,31)(24,32), (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( (2,24)(4,26)(6,28)(8,30)(10,32)(12,18)(14,20)(16,22), (1,31)(2,24)(3,17)(4,26)(5,19)(6,28)(7,21)(8,30)(9,23)(10,32)(11,25)(12,18)(13,27)(14,20)(15,29)(16,22), (1,23)(2,24)(3,25)(4,26)(5,27)(6,28)(7,29)(8,30)(9,31)(10,32)(11,17)(12,18)(13,19)(14,20)(15,21)(16,22), (1,9)(2,10)(3,11)(4,12)(5,13)(6,14)(7,15)(8,16)(17,25)(18,26)(19,27)(20,28)(21,29)(22,30)(23,31)(24,32), (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([[(2,24),(4,26),(6,28),(8,30),(10,32),(12,18),(14,20),(16,22)], [(1,31),(2,24),(3,17),(4,26),(5,19),(6,28),(7,21),(8,30),(9,23),(10,32),(11,25),(12,18),(13,27),(14,20),(15,29),(16,22)], [(1,23),(2,24),(3,25),(4,26),(5,27),(6,28),(7,29),(8,30),(9,31),(10,32),(11,17),(12,18),(13,19),(14,20),(15,21),(16,22)], [(1,9),(2,10),(3,11),(4,12),(5,13),(6,14),(7,15),(8,16),(17,25),(18,26),(19,27),(20,28),(21,29),(22,30),(23,31),(24,32)], [(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)]])
56 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | ··· | 2I | 4A | 4B | 4C | 4D | 4E | ··· | 4J | 8A | ··· | 8H | 8I | ··· | 8T | 16A | ··· | 16P |
| order | 1 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 8 | ··· | 8 | 8 | ··· | 8 | 16 | ··· | 16 |
| size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 |
| type | + | + | + | + | + | ||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C8 | C8 | D4 | M4(2) | M5(2) |
| kernel | C24.5C8 | C22⋊C16 | C2×M5(2) | C23×C8 | C22×C8 | C23×C4 | C22×C4 | C24 | C2×C8 | C2×C4 | C22 |
| # reps | 1 | 4 | 2 | 1 | 6 | 2 | 12 | 4 | 4 | 4 | 16 |
Matrix representation of C24.5C8 ►in GL4(𝔽17) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 8 | 16 |
| 16 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 |
| 16 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 |
| 9 | 0 | 0 | 0 |
| 0 | 0 | 9 | 2 |
| 0 | 0 | 11 | 8 |
G:=sub<GL(4,GF(17))| [1,0,0,0,0,1,0,0,0,0,1,8,0,0,0,16],[16,0,0,0,0,1,0,0,0,0,16,0,0,0,0,16],[1,0,0,0,0,1,0,0,0,0,16,0,0,0,0,16],[16,0,0,0,0,16,0,0,0,0,1,0,0,0,0,1],[0,9,0,0,1,0,0,0,0,0,9,11,0,0,2,8] >;
C24.5C8 in GAP, Magma, Sage, TeX
C_2^4._5C_8
% in TeX
G:=Group("C2^4.5C8"); // GroupNames label
G:=SmallGroup(128,844);
// by ID
G=gap.SmallGroup(128,844);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,-2,112,141,1430,102,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^2=1,e^8=d,a*b=b*a,e*a*e^-1=a*c=c*a,a*d=d*a,b*c=c*b,e*b*e^-1=b*d=d*b,c*d=d*c,c*e=e*c,d*e=e*d>;
// generators/relations