p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.23C23, C4○D4⋊2Q8, D4.7(C2×Q8), Q8.7(C2×Q8), C4⋊C4.344D4, Q8⋊Q8⋊4C2, D4⋊2Q8⋊4C2, D4.Q8⋊16C2, Q8.Q8⋊16C2, C4⋊C4.47C23, C4⋊C8.47C22, (C2×C8).31C23, C4.35(C22×Q8), (C2×C4).282C24, C22⋊C4.145D4, (C4×D4).71C22, C23.451(C2×D4), C4⋊Q8.104C22, C4.20(C22⋊Q8), (C4×Q8).68C22, C2.D8.82C22, C2.22(D4○SD16), (C2×D4).399C23, (C2×Q8).370C23, M4(2)⋊C4⋊21C2, C4.Q8.149C22, D4⋊C4.28C22, (C22×C8).345C22, Q8⋊C4.29C22, C23.36D4.4C2, C23.24D4.8C2, C22.542(C22×D4), C22.21(C22⋊Q8), C42.C2.11C22, C23.41C23⋊5C2, (C22×C4).1001C23, C42.6C22⋊11C2, (C2×M4(2)).71C22, C42⋊C2.121C22, C23.33C23.8C2, (C2×C4.Q8)⋊30C2, C4.92(C2×C4○D4), (C2×C4).484(C2×D4), (C2×C4).106(C2×Q8), C2.63(C2×C22⋊Q8), (C2×C4).484(C4○D4), (C2×C4⋊C4).608C22, (C2×C4○D4).135C22, SmallGroup(128,1816)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 332 in 188 conjugacy classes, 100 normal (38 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×2], C4 [×12], C22, C22 [×2], C22 [×6], C8 [×4], C2×C4 [×2], C2×C4 [×8], C2×C4 [×15], D4 [×2], D4 [×5], Q8 [×2], Q8 [×3], C23, C23, C42 [×2], C42 [×3], C22⋊C4 [×2], C22⋊C4 [×3], C4⋊C4 [×4], C4⋊C4 [×4], C4⋊C4 [×10], C2×C8 [×2], C2×C8 [×2], C2×C8 [×2], M4(2) [×2], C22×C4, C22×C4 [×5], C2×D4, C2×D4, C2×Q8, C2×Q8 [×2], C4○D4 [×4], C4○D4 [×2], D4⋊C4 [×2], D4⋊C4 [×2], Q8⋊C4 [×2], Q8⋊C4 [×2], C4⋊C8 [×4], C4.Q8 [×6], C2.D8 [×2], C2×C4⋊C4 [×2], C2×C4⋊C4, C42⋊C2 [×2], C42⋊C2, C4×D4 [×2], C4×D4 [×2], C4×Q8 [×2], C22⋊Q8 [×2], C42.C2 [×2], C42.C2, C4⋊Q8 [×2], C4⋊Q8, C22×C8, C2×M4(2), C2×C4○D4, C23.24D4, C23.36D4, C42.6C22, C2×C4.Q8, M4(2)⋊C4, Q8⋊Q8 [×2], D4⋊2Q8 [×2], D4.Q8 [×2], Q8.Q8 [×2], C23.33C23, C23.41C23, C42.23C23
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], Q8 [×4], C23 [×15], C2×D4 [×6], C2×Q8 [×6], C4○D4 [×2], C24, C22⋊Q8 [×4], C22×D4, C22×Q8, C2×C4○D4, C2×C22⋊Q8, D4○SD16 [×2], C42.23C23
Generators and relations
G = < a,b,c,d,e | a4=b4=d2=1, c2=a2b2, e2=a2, ab=ba, cac-1=a-1, ad=da, eae-1=ab2, cbc-1=dbd=b-1, be=eb, dcd=bc, ece-1=a2c, de=ed >
►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 42 49 12)(2 43 50 9)(3 44 51 10)(4 41 52 11)(5 31 40 56)(6 32 37 53)(7 29 38 54)(8 30 39 55)(13 20 46 21)(14 17 47 22)(15 18 48 23)(16 19 45 24)(25 59 33 62)(26 60 34 63)(27 57 35 64)(28 58 36 61)
(1 6 51 39)(2 5 52 38)(3 8 49 37)(4 7 50 40)(9 31 41 54)(10 30 42 53)(11 29 43 56)(12 32 44 55)(13 28 48 34)(14 27 45 33)(15 26 46 36)(16 25 47 35)(17 64 24 59)(18 63 21 58)(19 62 22 57)(20 61 23 60)
(5 56)(6 53)(7 54)(8 55)(9 43)(10 44)(11 41)(12 42)(13 46)(14 47)(15 48)(16 45)(25 59)(26 60)(27 57)(28 58)(29 38)(30 39)(31 40)(32 37)(33 62)(34 63)(35 64)(36 61)
(1 17 3 19)(2 23 4 21)(5 58 7 60)(6 62 8 64)(9 48 11 46)(10 16 12 14)(13 43 15 41)(18 52 20 50)(22 51 24 49)(25 30 27 32)(26 56 28 54)(29 34 31 36)(33 55 35 53)(37 59 39 57)(38 63 40 61)(42 47 44 45)
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), (1,42,49,12)(2,43,50,9)(3,44,51,10)(4,41,52,11)(5,31,40,56)(6,32,37,53)(7,29,38,54)(8,30,39,55)(13,20,46,21)(14,17,47,22)(15,18,48,23)(16,19,45,24)(25,59,33,62)(26,60,34,63)(27,57,35,64)(28,58,36,61), (1,6,51,39)(2,5,52,38)(3,8,49,37)(4,7,50,40)(9,31,41,54)(10,30,42,53)(11,29,43,56)(12,32,44,55)(13,28,48,34)(14,27,45,33)(15,26,46,36)(16,25,47,35)(17,64,24,59)(18,63,21,58)(19,62,22,57)(20,61,23,60), (5,56)(6,53)(7,54)(8,55)(9,43)(10,44)(11,41)(12,42)(13,46)(14,47)(15,48)(16,45)(25,59)(26,60)(27,57)(28,58)(29,38)(30,39)(31,40)(32,37)(33,62)(34,63)(35,64)(36,61), (1,17,3,19)(2,23,4,21)(5,58,7,60)(6,62,8,64)(9,48,11,46)(10,16,12,14)(13,43,15,41)(18,52,20,50)(22,51,24,49)(25,30,27,32)(26,56,28,54)(29,34,31,36)(33,55,35,53)(37,59,39,57)(38,63,40,61)(42,47,44,45)>;
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), (1,42,49,12)(2,43,50,9)(3,44,51,10)(4,41,52,11)(5,31,40,56)(6,32,37,53)(7,29,38,54)(8,30,39,55)(13,20,46,21)(14,17,47,22)(15,18,48,23)(16,19,45,24)(25,59,33,62)(26,60,34,63)(27,57,35,64)(28,58,36,61), (1,6,51,39)(2,5,52,38)(3,8,49,37)(4,7,50,40)(9,31,41,54)(10,30,42,53)(11,29,43,56)(12,32,44,55)(13,28,48,34)(14,27,45,33)(15,26,46,36)(16,25,47,35)(17,64,24,59)(18,63,21,58)(19,62,22,57)(20,61,23,60), (5,56)(6,53)(7,54)(8,55)(9,43)(10,44)(11,41)(12,42)(13,46)(14,47)(15,48)(16,45)(25,59)(26,60)(27,57)(28,58)(29,38)(30,39)(31,40)(32,37)(33,62)(34,63)(35,64)(36,61), (1,17,3,19)(2,23,4,21)(5,58,7,60)(6,62,8,64)(9,48,11,46)(10,16,12,14)(13,43,15,41)(18,52,20,50)(22,51,24,49)(25,30,27,32)(26,56,28,54)(29,34,31,36)(33,55,35,53)(37,59,39,57)(38,63,40,61)(42,47,44,45) );
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)], [(1,42,49,12),(2,43,50,9),(3,44,51,10),(4,41,52,11),(5,31,40,56),(6,32,37,53),(7,29,38,54),(8,30,39,55),(13,20,46,21),(14,17,47,22),(15,18,48,23),(16,19,45,24),(25,59,33,62),(26,60,34,63),(27,57,35,64),(28,58,36,61)], [(1,6,51,39),(2,5,52,38),(3,8,49,37),(4,7,50,40),(9,31,41,54),(10,30,42,53),(11,29,43,56),(12,32,44,55),(13,28,48,34),(14,27,45,33),(15,26,46,36),(16,25,47,35),(17,64,24,59),(18,63,21,58),(19,62,22,57),(20,61,23,60)], [(5,56),(6,53),(7,54),(8,55),(9,43),(10,44),(11,41),(12,42),(13,46),(14,47),(15,48),(16,45),(25,59),(26,60),(27,57),(28,58),(29,38),(30,39),(31,40),(32,37),(33,62),(34,63),(35,64),(36,61)], [(1,17,3,19),(2,23,4,21),(5,58,7,60),(6,62,8,64),(9,48,11,46),(10,16,12,14),(13,43,15,41),(18,52,20,50),(22,51,24,49),(25,30,27,32),(26,56,28,54),(29,34,31,36),(33,55,35,53),(37,59,39,57),(38,63,40,61),(42,47,44,45)])
Matrix representation ►G ⊆ GL6(𝔽17)
| 13 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 16 | 0 | 1 | 0 |
| 0 | 0 | 16 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 15 | 0 | 0 |
| 0 | 0 | 1 | 16 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 1 |
| 0 | 0 | 1 | 16 | 16 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 7 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 12 | 5 |
| 0 | 0 | 12 | 12 | 5 | 5 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 1 | 0 | 0 | 16 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 2 | 0 |
| 0 | 0 | 0 | 0 | 1 | 16 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 16 | 1 | 0 |
G:=sub<GL(6,GF(17))| [13,0,0,0,0,0,0,4,0,0,0,0,0,0,16,0,16,16,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,1,0,1,0,0,15,16,16,16,0,0,0,0,0,16,0,0,0,0,1,0],[0,1,0,0,0,0,16,0,0,0,0,0,0,0,0,12,0,12,0,0,7,0,12,12,0,0,0,0,12,5,0,0,0,0,5,5],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,1,0,1,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16],[4,0,0,0,0,0,0,13,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,2,1,1,1,0,0,0,16,0,0] >;
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | ··· | 4N | 4O | 4P | 4Q | 4R | 8A | 8B | 8C | 8D | 8E | 8F |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 8 | 4 | 4 | 4 | 4 | 8 | 8 |
32 irreducible representations
| dim | 1 | 1 | 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 | C2 | C2 | D4 | D4 | Q8 | C4○D4 | D4○SD16 |
| kernel | C42.23C23 | C23.24D4 | C23.36D4 | C42.6C22 | C2×C4.Q8 | M4(2)⋊C4 | Q8⋊Q8 | D4⋊2Q8 | D4.Q8 | Q8.Q8 | C23.33C23 | C23.41C23 | C22⋊C4 | C4⋊C4 | C4○D4 | C2×C4 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 2 | 2 | 4 | 4 | 4 |
In GAP, Magma, Sage, TeX
C_4^2._{23}C_2^3 % in TeX
G:=Group("C4^2.23C2^3"); // GroupNames label
G:=SmallGroup(128,1816);
// by ID
G=gap.SmallGroup(128,1816);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,560,253,120,758,1018,248,4037,1027,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^4=d^2=1,c^2=a^2*b^2,e^2=a^2,a*b=b*a,c*a*c^-1=a^-1,a*d=d*a,e*a*e^-1=a*b^2,c*b*c^-1=d*b*d=b^-1,b*e=e*b,d*c*d=b*c,e*c*e^-1=a^2*c,d*e=e*d>;
// generators/relations