p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.17C23, Q8.8(C2×D4), C4⋊C4.338D4, C2.8(Q8○D8), C4⋊2Q16⋊20C2, C4⋊C8.40C22, (C2×C8).19C23, (C2×Q8).126D4, (C22×Q16)⋊9C2, C4.75(C22×D4), C4.36(C4⋊D4), C4⋊C4.385C23, (C2×C4).248C24, Q8.D4⋊15C2, C22⋊C4.139D4, (C2×D4).54C23, C23.445(C2×D4), C4⋊Q8.100C22, (C4×Q8).64C22, (C2×Q8).41C23, C23.38D4⋊9C2, (C2×Q16).53C22, (C2×SD16).6C22, C22.83(C4⋊D4), (C22×C8).140C22, C42.6C22⋊6C2, (C22×C4).978C23, C4.4D4.25C22, C23.24D4.3C2, C22.508(C22×D4), D4⋊C4.156C22, C23.32C23⋊8C2, Q8⋊C4.146C22, (C2×M4(2)).55C22, (C22×Q8).275C22, C42⋊C2.103C22, C23.38C23.10C2, C4.158(C2×C4○D4), (C2×C4).468(C2×D4), C2.66(C2×C4⋊D4), (C2×C4).279(C4○D4), (C2×C8.C22).11C2, (C2×C4○D4).120C22, SmallGroup(128,1776)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.17C23
G = < a,b,c,d,e | a4=b4=1, c2=d2=b2, e2=a2b2, ab=ba, cac-1=a-1b2, ad=da, eae-1=ab2, cbc-1=dbd-1=b-1, be=eb, dcd-1=bc, ece-1=a2b2c, de=ed >
Subgroups: 396 in 229 conjugacy classes, 100 normal (26 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, Q8, Q8, C23, C23, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), SD16, Q16, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C2×Q8, C4○D4, D4⋊C4, Q8⋊C4, C4⋊C8, C42⋊C2, C42⋊C2, C4×Q8, C4×Q8, C22⋊Q8, C22.D4, C4.4D4, C4⋊Q8, C22×C8, C2×M4(2), C2×SD16, C2×Q16, C2×Q16, C8.C22, C22×Q8, C2×C4○D4, C23.24D4, C23.38D4, C42.6C22, C4⋊2Q16, Q8.D4, C23.32C23, C23.38C23, C22×Q16, C2×C8.C22, C42.17C23
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C4⋊D4, C22×D4, C2×C4○D4, C2×C4⋊D4, Q8○D8, C42.17C23
►On 64 points
(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)(33 34 35 36)(37 38 39 40)(41 42 43 44)(45 46 47 48)(49 50 51 52)(53 54 55 56)(57 58 59 60)(61 62 63 64)
(1 60 34 23)(2 57 35 24)(3 58 36 21)(4 59 33 22)(5 12 47 26)(6 9 48 27)(7 10 45 28)(8 11 46 25)(13 18 50 53)(14 19 51 54)(15 20 52 55)(16 17 49 56)(29 61 39 44)(30 62 40 41)(31 63 37 42)(32 64 38 43)
(1 4 34 33)(2 36 35 3)(5 25 47 11)(6 10 48 28)(7 27 45 9)(8 12 46 26)(13 49 50 16)(14 15 51 52)(17 53 56 18)(19 55 54 20)(21 57 58 24)(22 23 59 60)(29 43 39 64)(30 63 40 42)(31 41 37 62)(32 61 38 44)
(1 28 34 10)(2 25 35 11)(3 26 36 12)(4 27 33 9)(5 58 47 21)(6 59 48 22)(7 60 45 23)(8 57 46 24)(13 39 50 29)(14 40 51 30)(15 37 52 31)(16 38 49 32)(17 64 56 43)(18 61 53 44)(19 62 54 41)(20 63 55 42)
(1 14 36 49)(2 52 33 13)(3 16 34 51)(4 50 35 15)(5 43 45 62)(6 61 46 42)(7 41 47 64)(8 63 48 44)(9 39 25 31)(10 30 26 38)(11 37 27 29)(12 32 28 40)(17 23 54 58)(18 57 55 22)(19 21 56 60)(20 59 53 24)
G:=sub<Sym(64)| (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)(33,34,35,36)(37,38,39,40)(41,42,43,44)(45,46,47,48)(49,50,51,52)(53,54,55,56)(57,58,59,60)(61,62,63,64), (1,60,34,23)(2,57,35,24)(3,58,36,21)(4,59,33,22)(5,12,47,26)(6,9,48,27)(7,10,45,28)(8,11,46,25)(13,18,50,53)(14,19,51,54)(15,20,52,55)(16,17,49,56)(29,61,39,44)(30,62,40,41)(31,63,37,42)(32,64,38,43), (1,4,34,33)(2,36,35,3)(5,25,47,11)(6,10,48,28)(7,27,45,9)(8,12,46,26)(13,49,50,16)(14,15,51,52)(17,53,56,18)(19,55,54,20)(21,57,58,24)(22,23,59,60)(29,43,39,64)(30,63,40,42)(31,41,37,62)(32,61,38,44), (1,28,34,10)(2,25,35,11)(3,26,36,12)(4,27,33,9)(5,58,47,21)(6,59,48,22)(7,60,45,23)(8,57,46,24)(13,39,50,29)(14,40,51,30)(15,37,52,31)(16,38,49,32)(17,64,56,43)(18,61,53,44)(19,62,54,41)(20,63,55,42), (1,14,36,49)(2,52,33,13)(3,16,34,51)(4,50,35,15)(5,43,45,62)(6,61,46,42)(7,41,47,64)(8,63,48,44)(9,39,25,31)(10,30,26,38)(11,37,27,29)(12,32,28,40)(17,23,54,58)(18,57,55,22)(19,21,56,60)(20,59,53,24)>;
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)(33,34,35,36)(37,38,39,40)(41,42,43,44)(45,46,47,48)(49,50,51,52)(53,54,55,56)(57,58,59,60)(61,62,63,64), (1,60,34,23)(2,57,35,24)(3,58,36,21)(4,59,33,22)(5,12,47,26)(6,9,48,27)(7,10,45,28)(8,11,46,25)(13,18,50,53)(14,19,51,54)(15,20,52,55)(16,17,49,56)(29,61,39,44)(30,62,40,41)(31,63,37,42)(32,64,38,43), (1,4,34,33)(2,36,35,3)(5,25,47,11)(6,10,48,28)(7,27,45,9)(8,12,46,26)(13,49,50,16)(14,15,51,52)(17,53,56,18)(19,55,54,20)(21,57,58,24)(22,23,59,60)(29,43,39,64)(30,63,40,42)(31,41,37,62)(32,61,38,44), (1,28,34,10)(2,25,35,11)(3,26,36,12)(4,27,33,9)(5,58,47,21)(6,59,48,22)(7,60,45,23)(8,57,46,24)(13,39,50,29)(14,40,51,30)(15,37,52,31)(16,38,49,32)(17,64,56,43)(18,61,53,44)(19,62,54,41)(20,63,55,42), (1,14,36,49)(2,52,33,13)(3,16,34,51)(4,50,35,15)(5,43,45,62)(6,61,46,42)(7,41,47,64)(8,63,48,44)(9,39,25,31)(10,30,26,38)(11,37,27,29)(12,32,28,40)(17,23,54,58)(18,57,55,22)(19,21,56,60)(20,59,53,24) );
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),(33,34,35,36),(37,38,39,40),(41,42,43,44),(45,46,47,48),(49,50,51,52),(53,54,55,56),(57,58,59,60),(61,62,63,64)], [(1,60,34,23),(2,57,35,24),(3,58,36,21),(4,59,33,22),(5,12,47,26),(6,9,48,27),(7,10,45,28),(8,11,46,25),(13,18,50,53),(14,19,51,54),(15,20,52,55),(16,17,49,56),(29,61,39,44),(30,62,40,41),(31,63,37,42),(32,64,38,43)], [(1,4,34,33),(2,36,35,3),(5,25,47,11),(6,10,48,28),(7,27,45,9),(8,12,46,26),(13,49,50,16),(14,15,51,52),(17,53,56,18),(19,55,54,20),(21,57,58,24),(22,23,59,60),(29,43,39,64),(30,63,40,42),(31,41,37,62),(32,61,38,44)], [(1,28,34,10),(2,25,35,11),(3,26,36,12),(4,27,33,9),(5,58,47,21),(6,59,48,22),(7,60,45,23),(8,57,46,24),(13,39,50,29),(14,40,51,30),(15,37,52,31),(16,38,49,32),(17,64,56,43),(18,61,53,44),(19,62,54,41),(20,63,55,42)], [(1,14,36,49),(2,52,33,13),(3,16,34,51),(4,50,35,15),(5,43,45,62),(6,61,46,42),(7,41,47,64),(8,63,48,44),(9,39,25,31),(10,30,26,38),(11,37,27,29),(12,32,28,40),(17,23,54,58),(18,57,55,22),(19,21,56,60),(20,59,53,24)]])
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | ··· | 4P | 4Q | 4R | 4S | 8A | 8B | 8C | 8D | 8E | 8F |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 8 | 8 | 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 | C4○D4 | Q8○D8 |
| kernel | C42.17C23 | C23.24D4 | C23.38D4 | C42.6C22 | C4⋊2Q16 | Q8.D4 | C23.32C23 | C23.38C23 | C22×Q16 | C2×C8.C22 | C22⋊C4 | C4⋊C4 | C2×Q8 | C2×C4 | C2 |
| # reps | 1 | 1 | 1 | 1 | 4 | 4 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 |
Matrix representation of C42.17C23 ►in GL6(𝔽17)
| 14 | 15 | 0 | 0 | 0 | 0 |
| 5 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 4 | 13 | 4 | 9 |
| 0 | 0 | 13 | 0 | 0 | 0 |
| 0 | 0 | 13 | 4 | 0 | 4 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 1 | 16 | 1 | 15 |
| 0 | 0 | 1 | 0 | 1 | 16 |
| 14 | 15 | 0 | 0 | 0 | 0 |
| 4 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 13 | 0 |
| 0 | 0 | 4 | 13 | 4 | 9 |
| 0 | 0 | 13 | 0 | 0 | 0 |
| 0 | 0 | 13 | 0 | 13 | 4 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 5 | 0 | 0 |
| 0 | 0 | 5 | 12 | 0 | 0 |
| 0 | 0 | 5 | 12 | 10 | 7 |
| 0 | 0 | 5 | 0 | 5 | 7 |
| 14 | 15 | 0 | 0 | 0 | 0 |
| 5 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 13 | 0 | 0 | 0 |
| 0 | 0 | 0 | 13 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 13 | 4 | 0 | 4 |
G:=sub<GL(6,GF(17))| [14,5,0,0,0,0,15,3,0,0,0,0,0,0,0,4,13,13,0,0,0,13,0,4,0,0,4,4,0,0,0,0,0,9,0,4],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,16,1,1,0,0,1,0,16,0,0,0,0,0,1,1,0,0,0,0,15,16],[14,4,0,0,0,0,15,3,0,0,0,0,0,0,0,4,13,13,0,0,0,13,0,0,0,0,13,4,0,13,0,0,0,9,0,4],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,5,5,5,5,0,0,5,12,12,0,0,0,0,0,10,5,0,0,0,0,7,7],[14,5,0,0,0,0,15,3,0,0,0,0,0,0,13,0,0,13,0,0,0,13,0,4,0,0,0,0,4,0,0,0,0,0,0,4] >;
C42.17C23 in GAP, Magma, Sage, TeX
C_4^2._{17}C_2^3 % in TeX
G:=Group("C4^2.17C2^3"); // GroupNames label
G:=SmallGroup(128,1776);
// by ID
G=gap.SmallGroup(128,1776);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,120,758,352,1018,4037,1027,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^4=1,c^2=d^2=b^2,e^2=a^2*b^2,a*b=b*a,c*a*c^-1=a^-1*b^2,a*d=d*a,e*a*e^-1=a*b^2,c*b*c^-1=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