p-group, metabelian, nilpotent (class 3), monomial
Aliases: C23.5C42, C8○(C23⋊C4), (C22×C8)⋊8C4, C8○(C4.D4), (C2×C8).383D4, C4.D4⋊6C4, C4.107(C4×D4), C23⋊C4.3C4, (C2×C4).5C42, C8○(C4.10D4), C4.10D4⋊7C4, C22.25(C4×D4), C8.60(C22⋊C4), (C2×M4(2))⋊19C4, C23.2(C22×C4), C8○(M4(2)⋊4C4), C8○2M4(2)⋊21C2, M4(2).16(C2×C4), C22.12(C2×C42), M4(2)⋊4C4⋊22C2, C4.46(C42⋊C2), C8○(C23.C23), (C22×C4).652C23, (C22×C8).379C22, C42⋊C2.260C22, C8○(M4(2).8C22), (C2×M4(2)).305C22, C23.C23.11C2, M4(2).8C22.12C2, (C2×C8).10(C2×C4), (C2×C8○D4).11C2, C2.15(C4×C22⋊C4), (C2×D4).158(C2×C4), (C2×C4).44(C4○D4), (C2×C4).1301(C2×D4), C22⋊C4.28(C2×C4), C4.109(C2×C22⋊C4), (C22×C4).71(C2×C4), (C2×Q8).140(C2×C4), (C2×C4).521(C22×C4), (C2×C4○D4).249C22, SmallGroup(128,489)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C23.5C42
G = < a,b,c,d,e | a2=b2=c2=d4=1, e4=c, ab=ba, ac=ca, dad-1=abc, ae=ea, dbd-1=bc=cb, be=eb, cd=dc, ce=ec, ede-1=bcd >
Subgroups: 228 in 142 conjugacy classes, 74 normal (22 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C22, C8, C8, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, C2×C8, M4(2), M4(2), C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C4×C8, C8⋊C4, C23⋊C4, C4.D4, C4.10D4, C42⋊C2, C22×C8, C22×C8, C2×M4(2), C2×M4(2), C8○D4, C2×C4○D4, M4(2)⋊4C4, C8○2M4(2), C23.C23, M4(2).8C22, C2×C8○D4, C23.5C42
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C42, C22⋊C4, C22×C4, C2×D4, C4○D4, C2×C42, C2×C22⋊C4, C42⋊C2, C4×D4, C4×C22⋊C4, C23.5C42
►On 32 points
(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)
(9 13)(10 14)(11 15)(12 16)(25 29)(26 30)(27 31)(28 32)
(1 5)(2 6)(3 7)(4 8)(9 13)(10 14)(11 15)(12 16)(17 21)(18 22)(19 23)(20 24)(25 29)(26 30)(27 31)(28 32)
(1 30 23 16)(2 27 20 9)(3 32 17 10)(4 29 22 11)(5 26 19 12)(6 31 24 13)(7 28 21 14)(8 25 18 15)
(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)| (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), (9,13)(10,14)(11,15)(12,16)(25,29)(26,30)(27,31)(28,32), (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16)(17,21)(18,22)(19,23)(20,24)(25,29)(26,30)(27,31)(28,32), (1,30,23,16)(2,27,20,9)(3,32,17,10)(4,29,22,11)(5,26,19,12)(6,31,24,13)(7,28,21,14)(8,25,18,15), (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( (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), (9,13)(10,14)(11,15)(12,16)(25,29)(26,30)(27,31)(28,32), (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16)(17,21)(18,22)(19,23)(20,24)(25,29)(26,30)(27,31)(28,32), (1,30,23,16)(2,27,20,9)(3,32,17,10)(4,29,22,11)(5,26,19,12)(6,31,24,13)(7,28,21,14)(8,25,18,15), (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([[(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)], [(9,13),(10,14),(11,15),(12,16),(25,29),(26,30),(27,31),(28,32)], [(1,5),(2,6),(3,7),(4,8),(9,13),(10,14),(11,15),(12,16),(17,21),(18,22),(19,23),(20,24),(25,29),(26,30),(27,31),(28,32)], [(1,30,23,16),(2,27,20,9),(3,32,17,10),(4,29,22,11),(5,26,19,12),(6,31,24,13),(7,28,21,14),(8,25,18,15)], [(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 | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | 4F | ··· | 4O | 8A | 8B | 8C | 8D | 8E | ··· | 8J | 8K | ··· | 8V |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 8 | 8 | 8 | 8 | 8 | ··· | 8 | 8 | ··· | 8 |
| size | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 1 | 1 | 2 | 2 | 2 | 4 | ··· | 4 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 4 | ··· | 4 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C4 | C4 | D4 | C4○D4 | C23.5C42 |
| kernel | C23.5C42 | M4(2)⋊4C4 | C8○2M4(2) | C23.C23 | M4(2).8C22 | C2×C8○D4 | C23⋊C4 | C4.D4 | C4.10D4 | C22×C8 | C2×M4(2) | C2×C8 | C2×C4 | C1 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 8 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
Matrix representation of C23.5C42 ►in GL4(𝔽17) generated by
| 0 | 4 | 0 | 0 |
| 13 | 0 | 0 | 0 |
| 16 | 1 | 4 | 9 |
| 16 | 0 | 4 | 13 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 4 | 13 | 0 | 16 |
| 16 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 |
| 0 | 0 | 1 | 0 |
| 4 | 13 | 1 | 15 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 4 |
| 0 | 2 | 0 | 0 |
| 15 | 0 | 0 | 0 |
| 8 | 9 | 2 | 13 |
| 8 | 0 | 2 | 15 |
G:=sub<GL(4,GF(17))| [0,13,16,16,4,0,1,0,0,0,4,4,0,0,9,13],[1,0,0,4,0,1,0,13,0,0,16,0,0,0,0,16],[16,0,0,0,0,16,0,0,0,0,16,0,0,0,0,16],[0,4,0,0,0,13,1,0,1,1,0,0,0,15,0,4],[0,15,8,8,2,0,9,0,0,0,2,2,0,0,13,15] >;
C23.5C42 in GAP, Magma, Sage, TeX
C_2^3._5C_4^2
% in TeX
G:=Group("C2^3.5C4^2"); // GroupNames label
G:=SmallGroup(128,489);
// by ID
G=gap.SmallGroup(128,489);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,112,141,232,100,1018,172,4037]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^4=1,e^4=c,a*b=b*a,a*c=c*a,d*a*d^-1=a*b*c,a*e=e*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>;
// generators/relations