p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.491C23, C4.772- 1+4, C4⋊C4.414D4, (C4×Q16)⋊15C2, C8⋊2Q8⋊20C2, (C8×D4).12C2, Q8⋊Q8⋊44C2, (C2×D4).246D4, C8.79(C4○D4), C4.91(C4○D8), (C4×C8).90C22, C2.56(Q8○D8), C8.18D4⋊24C2, C4⋊C8.348C22, C4⋊C4.247C23, (C2×C4).534C24, (C2×C8).363C23, C22⋊C4.118D4, C23.119(C2×D4), C4⋊Q8.166C22, C2.87(D4⋊6D4), C2.D8.65C22, (C4×D4).347C22, (C2×Q8).238C23, (C4×Q8).175C22, C4.Q8.171C22, C23.20D4⋊11C2, C23.25D4⋊14C2, C22⋊C8.212C22, (C22×C8).201C22, (C2×Q16).140C22, C22.794(C22×D4), C22⋊Q8.101C22, (C22×C4).1166C23, Q8⋊C4.119C22, C42⋊C2.205C22, C22.50C24.5C2, C2.71(C2×C4○D8), C4.116(C2×C4○D4), (C2×C4).936(C2×D4), SmallGroup(128,2074)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.491C23
G = < a,b,c,d,e | a4=b4=1, c2=a2, d2=a2b2, e2=b2, ab=ba, cac-1=eae-1=a-1, ad=da, cbc-1=dbd-1=b-1, be=eb, dcd-1=bc, ece-1=a2c, de=ed >
Subgroups: 280 in 170 conjugacy classes, 88 normal (24 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C8, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C2×C8, Q16, C22×C4, C2×D4, C2×Q8, C2×Q8, C4×C8, C22⋊C8, Q8⋊C4, C4⋊C8, C4.Q8, C2.D8, C2.D8, C42⋊C2, C4×D4, C4×Q8, C4×Q8, C22⋊Q8, C4.4D4, C42⋊2C2, C4⋊Q8, C22×C8, C2×Q16, C23.25D4, C8×D4, C4×Q16, C8.18D4, Q8⋊Q8, C23.20D4, C8⋊2Q8, C22.50C24, C42.491C23
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C4○D8, C22×D4, C2×C4○D4, 2- 1+4, D4⋊6D4, C2×C4○D8, Q8○D8, C42.491C23
►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 22 26 18)(2 23 27 19)(3 24 28 20)(4 21 25 17)(5 64 15 9)(6 61 16 10)(7 62 13 11)(8 63 14 12)(29 44 40 34)(30 41 37 35)(31 42 38 36)(32 43 39 33)(45 57 55 49)(46 58 56 50)(47 59 53 51)(48 60 54 52)
(1 54 3 56)(2 53 4 55)(5 44 7 42)(6 43 8 41)(9 40 11 38)(10 39 12 37)(13 36 15 34)(14 35 16 33)(17 49 19 51)(18 52 20 50)(21 57 23 59)(22 60 24 58)(25 45 27 47)(26 48 28 46)(29 62 31 64)(30 61 32 63)
(1 29 28 38)(2 30 25 39)(3 31 26 40)(4 32 27 37)(5 48 13 56)(6 45 14 53)(7 46 15 54)(8 47 16 55)(9 60 62 50)(10 57 63 51)(11 58 64 52)(12 59 61 49)(17 43 23 35)(18 44 24 36)(19 41 21 33)(20 42 22 34)
(1 18 26 22)(2 17 27 21)(3 20 28 24)(4 19 25 23)(5 11 15 62)(6 10 16 61)(7 9 13 64)(8 12 14 63)(29 44 40 34)(30 43 37 33)(31 42 38 36)(32 41 39 35)(45 57 55 49)(46 60 56 52)(47 59 53 51)(48 58 54 50)
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), (1,22,26,18)(2,23,27,19)(3,24,28,20)(4,21,25,17)(5,64,15,9)(6,61,16,10)(7,62,13,11)(8,63,14,12)(29,44,40,34)(30,41,37,35)(31,42,38,36)(32,43,39,33)(45,57,55,49)(46,58,56,50)(47,59,53,51)(48,60,54,52), (1,54,3,56)(2,53,4,55)(5,44,7,42)(6,43,8,41)(9,40,11,38)(10,39,12,37)(13,36,15,34)(14,35,16,33)(17,49,19,51)(18,52,20,50)(21,57,23,59)(22,60,24,58)(25,45,27,47)(26,48,28,46)(29,62,31,64)(30,61,32,63), (1,29,28,38)(2,30,25,39)(3,31,26,40)(4,32,27,37)(5,48,13,56)(6,45,14,53)(7,46,15,54)(8,47,16,55)(9,60,62,50)(10,57,63,51)(11,58,64,52)(12,59,61,49)(17,43,23,35)(18,44,24,36)(19,41,21,33)(20,42,22,34), (1,18,26,22)(2,17,27,21)(3,20,28,24)(4,19,25,23)(5,11,15,62)(6,10,16,61)(7,9,13,64)(8,12,14,63)(29,44,40,34)(30,43,37,33)(31,42,38,36)(32,41,39,35)(45,57,55,49)(46,60,56,52)(47,59,53,51)(48,58,54,50)>;
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), (1,22,26,18)(2,23,27,19)(3,24,28,20)(4,21,25,17)(5,64,15,9)(6,61,16,10)(7,62,13,11)(8,63,14,12)(29,44,40,34)(30,41,37,35)(31,42,38,36)(32,43,39,33)(45,57,55,49)(46,58,56,50)(47,59,53,51)(48,60,54,52), (1,54,3,56)(2,53,4,55)(5,44,7,42)(6,43,8,41)(9,40,11,38)(10,39,12,37)(13,36,15,34)(14,35,16,33)(17,49,19,51)(18,52,20,50)(21,57,23,59)(22,60,24,58)(25,45,27,47)(26,48,28,46)(29,62,31,64)(30,61,32,63), (1,29,28,38)(2,30,25,39)(3,31,26,40)(4,32,27,37)(5,48,13,56)(6,45,14,53)(7,46,15,54)(8,47,16,55)(9,60,62,50)(10,57,63,51)(11,58,64,52)(12,59,61,49)(17,43,23,35)(18,44,24,36)(19,41,21,33)(20,42,22,34), (1,18,26,22)(2,17,27,21)(3,20,28,24)(4,19,25,23)(5,11,15,62)(6,10,16,61)(7,9,13,64)(8,12,14,63)(29,44,40,34)(30,43,37,33)(31,42,38,36)(32,41,39,35)(45,57,55,49)(46,60,56,52)(47,59,53,51)(48,58,54,50) );
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)], [(1,22,26,18),(2,23,27,19),(3,24,28,20),(4,21,25,17),(5,64,15,9),(6,61,16,10),(7,62,13,11),(8,63,14,12),(29,44,40,34),(30,41,37,35),(31,42,38,36),(32,43,39,33),(45,57,55,49),(46,58,56,50),(47,59,53,51),(48,60,54,52)], [(1,54,3,56),(2,53,4,55),(5,44,7,42),(6,43,8,41),(9,40,11,38),(10,39,12,37),(13,36,15,34),(14,35,16,33),(17,49,19,51),(18,52,20,50),(21,57,23,59),(22,60,24,58),(25,45,27,47),(26,48,28,46),(29,62,31,64),(30,61,32,63)], [(1,29,28,38),(2,30,25,39),(3,31,26,40),(4,32,27,37),(5,48,13,56),(6,45,14,53),(7,46,15,54),(8,47,16,55),(9,60,62,50),(10,57,63,51),(11,58,64,52),(12,59,61,49),(17,43,23,35),(18,44,24,36),(19,41,21,33),(20,42,22,34)], [(1,18,26,22),(2,17,27,21),(3,20,28,24),(4,19,25,23),(5,11,15,62),(6,10,16,61),(7,9,13,64),(8,12,14,63),(29,44,40,34),(30,43,37,33),(31,42,38,36),(32,41,39,35),(45,57,55,49),(46,60,56,52),(47,59,53,51),(48,58,54,50)]])
35 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | ··· | 4H | 4I | ··· | 4M | 4N | ··· | 4S | 8A | 8B | 8C | 8D | 8E | ··· | 8J |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | 8 | 8 | 8 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
35 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | - | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | C4○D4 | C4○D8 | 2- 1+4 | Q8○D8 |
| kernel | C42.491C23 | C23.25D4 | C8×D4 | C4×Q16 | C8.18D4 | Q8⋊Q8 | C23.20D4 | C8⋊2Q8 | C22.50C24 | C22⋊C4 | C4⋊C4 | C2×D4 | C8 | C4 | C4 | C2 |
| # reps | 1 | 2 | 1 | 1 | 2 | 2 | 4 | 1 | 2 | 2 | 1 | 1 | 4 | 8 | 1 | 2 |
Matrix representation of C42.491C23 ►in GL4(𝔽17) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 16 |
| 0 | 0 | 1 | 0 |
| 4 | 0 | 0 | 0 |
| 4 | 13 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 8 | 1 | 0 | 0 |
| 5 | 9 | 0 | 0 |
| 0 | 0 | 0 | 13 |
| 0 | 0 | 13 | 0 |
| 1 | 15 | 0 | 0 |
| 1 | 16 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 4 |
| 13 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 16 |
G:=sub<GL(4,GF(17))| [1,0,0,0,0,1,0,0,0,0,0,1,0,0,16,0],[4,4,0,0,0,13,0,0,0,0,1,0,0,0,0,1],[8,5,0,0,1,9,0,0,0,0,0,13,0,0,13,0],[1,1,0,0,15,16,0,0,0,0,4,0,0,0,0,4],[13,0,0,0,0,13,0,0,0,0,1,0,0,0,0,16] >;
C42.491C23 in GAP, Magma, Sage, TeX
C_4^2._{491}C_2^3 % in TeX
G:=Group("C4^2.491C2^3"); // GroupNames label
G:=SmallGroup(128,2074);
// by ID
G=gap.SmallGroup(128,2074);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,112,253,456,758,436,346,248,4037,1027,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^4=1,c^2=a^2,d^2=a^2*b^2,e^2=b^2,a*b=b*a,c*a*c^-1=e*a*e^-1=a^-1,a*d=d*a,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*c,d*e=e*d>;
// generators/relations