p-group, metabelian, nilpotent (class 4), monomial
Aliases: C23.13D8, M5(2)⋊14C22, C4○D8⋊4C4, (C2×D8)⋊16C4, C4○(D8⋊2C4), D8⋊2C4⋊4C2, (C2×Q16)⋊16C4, D8.11(C2×C4), C8.117(C2×D4), (C2×C4).141D8, (C2×C8).122D4, C8.7(C22×C4), C8.9(C22⋊C4), Q16.11(C2×C4), (C2×C4).52SD16, C4.64(C2×SD16), C22.18(C2×D8), C4.Q8⋊41C22, (C2×M5(2))⋊16C2, (C2×C8).225C23, C4○D8.17C22, (C22×C4).335D4, C4.57(D4⋊C4), C23.25D4⋊19C2, (C22×C8).234C22, C22.32(D4⋊C4), (C2×C8).84(C2×C4), (C2×C4○D8).12C2, (C2×C4).270(C2×D4), C4.57(C2×C22⋊C4), C2.35(C2×D4⋊C4), (C2×C4).152(C22⋊C4), SmallGroup(128,877)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C23.13D8
G = < a,b,c,d,e | a2=b2=c2=1, d8=c, e2=b, ab=ba, eae-1=ac=ca, ad=da, dbd-1=bc=cb, be=eb, cd=dc, ce=ec, ede-1=bcd7 >
Subgroups: 260 in 112 conjugacy classes, 50 normal (34 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, D4, Q8, C23, C23, C16, C42, C22⋊C4, C4⋊C4, C2×C8, D8, D8, SD16, Q16, Q16, C22×C4, C22×C4, C2×D4, C2×Q8, C4○D4, C4.Q8, C2.D8, C2×C16, M5(2), M5(2), C42⋊C2, C22×C8, C2×D8, C2×SD16, C2×Q16, C4○D8, C4○D8, C2×C4○D4, D8⋊2C4, C23.25D4, C2×M5(2), C2×C4○D8, C23.13D8
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, D8, SD16, C22×C4, C2×D4, D4⋊C4, C2×C22⋊C4, C2×D8, C2×SD16, C2×D4⋊C4, C23.13D8
►On 32 points
(1 19)(2 20)(3 21)(4 22)(5 23)(6 24)(7 25)(8 26)(9 27)(10 28)(11 29)(12 30)(13 31)(14 32)(15 17)(16 18)
(1 19)(2 28)(3 21)(4 30)(5 23)(6 32)(7 25)(8 18)(9 27)(10 20)(11 29)(12 22)(13 31)(14 24)(15 17)(16 26)
(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)
(1 26 19 16)(2 7 28 25)(3 32 21 6)(4 13 30 31)(5 22 23 12)(8 9 18 27)(10 15 20 17)(11 24 29 14)
G:=sub<Sym(32)| (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,25)(8,26)(9,27)(10,28)(11,29)(12,30)(13,31)(14,32)(15,17)(16,18), (1,19)(2,28)(3,21)(4,30)(5,23)(6,32)(7,25)(8,18)(9,27)(10,20)(11,29)(12,22)(13,31)(14,24)(15,17)(16,26), (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), (1,26,19,16)(2,7,28,25)(3,32,21,6)(4,13,30,31)(5,22,23,12)(8,9,18,27)(10,15,20,17)(11,24,29,14)>;
G:=Group( (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,25)(8,26)(9,27)(10,28)(11,29)(12,30)(13,31)(14,32)(15,17)(16,18), (1,19)(2,28)(3,21)(4,30)(5,23)(6,32)(7,25)(8,18)(9,27)(10,20)(11,29)(12,22)(13,31)(14,24)(15,17)(16,26), (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), (1,26,19,16)(2,7,28,25)(3,32,21,6)(4,13,30,31)(5,22,23,12)(8,9,18,27)(10,15,20,17)(11,24,29,14) );
G=PermutationGroup([[(1,19),(2,20),(3,21),(4,22),(5,23),(6,24),(7,25),(8,26),(9,27),(10,28),(11,29),(12,30),(13,31),(14,32),(15,17),(16,18)], [(1,19),(2,28),(3,21),(4,30),(5,23),(6,32),(7,25),(8,18),(9,27),(10,20),(11,29),(12,22),(13,31),(14,24),(15,17),(16,26)], [(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)], [(1,26,19,16),(2,7,28,25),(3,32,21,6),(4,13,30,31),(5,22,23,12),(8,9,18,27),(10,15,20,17),(11,24,29,14)]])
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | 4F | ··· | 4K | 8A | 8B | 8C | 8D | 8E | 8F | 16A | ··· | 16H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 8 | 8 | 8 | 8 | 8 | 8 | 16 | ··· | 16 |
| size | 1 | 1 | 2 | 2 | 2 | 8 | 8 | 1 | 1 | 2 | 2 | 2 | 8 | ··· | 8 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | ··· | 4 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | D4 | D4 | D8 | SD16 | D8 | C23.13D8 |
| kernel | C23.13D8 | D8⋊2C4 | C23.25D4 | C2×M5(2) | C2×C4○D8 | C2×D8 | C2×Q16 | C4○D8 | C2×C8 | C22×C4 | C2×C4 | C2×C4 | C23 | C1 |
| # reps | 1 | 4 | 1 | 1 | 1 | 2 | 2 | 4 | 3 | 1 | 2 | 4 | 2 | 4 |
Matrix representation of C23.13D8 ►in GL4(𝔽17) generated by
| 0 | 4 | 0 | 0 |
| 13 | 0 | 0 | 0 |
| 0 | 0 | 0 | 13 |
| 0 | 0 | 4 | 0 |
| 0 | 4 | 0 | 0 |
| 13 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 |
| 0 | 0 | 13 | 0 |
| 16 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 |
| 0 | 0 | 0 | 16 |
| 0 | 0 | 16 | 0 |
| 5 | 12 | 0 | 0 |
| 12 | 12 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 4 | 0 | 0 |
| 13 | 0 | 0 | 0 |
G:=sub<GL(4,GF(17))| [0,13,0,0,4,0,0,0,0,0,0,4,0,0,13,0],[0,13,0,0,4,0,0,0,0,0,0,13,0,0,4,0],[16,0,0,0,0,16,0,0,0,0,16,0,0,0,0,16],[0,0,5,12,0,0,12,12,0,16,0,0,16,0,0,0],[0,0,0,13,0,0,4,0,1,0,0,0,0,1,0,0] >;
C23.13D8 in GAP, Magma, Sage, TeX
C_2^3._{13}D_8 % in TeX
G:=Group("C2^3.13D8"); // GroupNames label
G:=SmallGroup(128,877);
// by ID
G=gap.SmallGroup(128,877);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,-2,112,141,352,1123,1466,136,1411,4037,2028,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=1,d^8=c,e^2=b,a*b=b*a,e*a*e^-1=a*c=c*a,a*d=d*a,d*b*d^-1=b*c=c*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=b*c*d^7>;
// generators/relations