p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.468C23, C4.482+ 1+4, (C8×D4)⋊21C2, (C4×D8)⋊13C2, D4⋊6D4⋊8C2, C8⋊8D4⋊40C2, C4⋊C4.409D4, D4⋊D4⋊11C2, D4⋊2Q8⋊40C2, (C2×D4).238D4, C4.46(C4○D8), (C4×C8).85C22, D4.20(C4○D4), D4.D4⋊44C2, C4⋊C4.403C23, C4⋊C8.344C22, (C2×C8).185C23, (C2×C4).495C24, C22⋊C4.107D4, C4.SD16⋊15C2, C23.111(C2×D4), C4⋊Q8.144C22, C2.70(D4○SD16), (C2×D8).139C22, (C4×D4).336C22, (C2×D4).225C23, C23.20D4⋊7C2, C4⋊D4.75C22, (C2×Q8).211C23, C2.131(D4⋊5D4), C4.Q8.102C22, C2.D8.191C22, C22⋊Q8.75C22, C23.24D4⋊28C2, C22⋊C8.204C22, (C22×C8).356C22, Q8⋊C4.13C22, (C2×SD16).97C22, C22.755(C22×D4), D4⋊C4.185C22, C22.49C24⋊3C2, (C22×C4).1139C23, C42⋊C2.183C22, C4⋊C4○(D4⋊C4), C2.63(C2×C4○D8), C4.220(C2×C4○D4), (C2×C4).924(C2×D4), (C2×C4○D4).201C22, SmallGroup(128,2035)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.468C23
G = < a,b,c,d,e | a4=b4=d2=1, c2=a2b2, e2=b2, ab=ba, cac-1=eae-1=a-1, ad=da, cbc-1=dbd=b-1, be=eb, dcd=bc, ece-1=a2c, de=ed >
Subgroups: 400 in 199 conjugacy classes, 88 normal (38 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, D4, Q8, C23, C23, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C2×C8, D8, SD16, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C4○D4, C4×C8, C22⋊C8, D4⋊C4, D4⋊C4, Q8⋊C4, C4⋊C8, C4.Q8, C2.D8, C2×C4⋊C4, C42⋊C2, C42⋊C2, C4×D4, C4⋊D4, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C4⋊Q8, C22×C8, C2×D8, C2×SD16, C2×C4○D4, C23.24D4, C8×D4, C4×D8, D4⋊D4, D4.D4, C8⋊8D4, D4⋊2Q8, C23.20D4, C4.SD16, D4⋊6D4, C22.49C24, C42.468C23
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C4○D8, C22×D4, C2×C4○D4, 2+ 1+4, D4⋊5D4, C2×C4○D8, D4○SD16, C42.468C23
►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 23 57 12)(2 24 58 9)(3 21 59 10)(4 22 60 11)(5 47 40 27)(6 48 37 28)(7 45 38 25)(8 46 39 26)(13 36 19 53)(14 33 20 54)(15 34 17 55)(16 35 18 56)(29 61 44 52)(30 62 41 49)(31 63 42 50)(32 64 43 51)
(1 35 59 54)(2 34 60 53)(3 33 57 56)(4 36 58 55)(5 44 38 31)(6 43 39 30)(7 42 40 29)(8 41 37 32)(9 17 22 13)(10 20 23 16)(11 19 24 15)(12 18 21 14)(25 50 47 61)(26 49 48 64)(27 52 45 63)(28 51 46 62)
(1 37)(2 38)(3 39)(4 40)(5 60)(6 57)(7 58)(8 59)(9 25)(10 26)(11 27)(12 28)(13 29)(14 30)(15 31)(16 32)(17 42)(18 43)(19 44)(20 41)(21 46)(22 47)(23 48)(24 45)(33 49)(34 50)(35 51)(36 52)(53 61)(54 62)(55 63)(56 64)
(1 56 57 35)(2 55 58 34)(3 54 59 33)(4 53 60 36)(5 52 40 61)(6 51 37 64)(7 50 38 63)(8 49 39 62)(9 17 24 15)(10 20 21 14)(11 19 22 13)(12 18 23 16)(25 42 45 31)(26 41 46 30)(27 44 47 29)(28 43 48 32)
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), (1,23,57,12)(2,24,58,9)(3,21,59,10)(4,22,60,11)(5,47,40,27)(6,48,37,28)(7,45,38,25)(8,46,39,26)(13,36,19,53)(14,33,20,54)(15,34,17,55)(16,35,18,56)(29,61,44,52)(30,62,41,49)(31,63,42,50)(32,64,43,51), (1,35,59,54)(2,34,60,53)(3,33,57,56)(4,36,58,55)(5,44,38,31)(6,43,39,30)(7,42,40,29)(8,41,37,32)(9,17,22,13)(10,20,23,16)(11,19,24,15)(12,18,21,14)(25,50,47,61)(26,49,48,64)(27,52,45,63)(28,51,46,62), (1,37)(2,38)(3,39)(4,40)(5,60)(6,57)(7,58)(8,59)(9,25)(10,26)(11,27)(12,28)(13,29)(14,30)(15,31)(16,32)(17,42)(18,43)(19,44)(20,41)(21,46)(22,47)(23,48)(24,45)(33,49)(34,50)(35,51)(36,52)(53,61)(54,62)(55,63)(56,64), (1,56,57,35)(2,55,58,34)(3,54,59,33)(4,53,60,36)(5,52,40,61)(6,51,37,64)(7,50,38,63)(8,49,39,62)(9,17,24,15)(10,20,21,14)(11,19,22,13)(12,18,23,16)(25,42,45,31)(26,41,46,30)(27,44,47,29)(28,43,48,32)>;
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), (1,23,57,12)(2,24,58,9)(3,21,59,10)(4,22,60,11)(5,47,40,27)(6,48,37,28)(7,45,38,25)(8,46,39,26)(13,36,19,53)(14,33,20,54)(15,34,17,55)(16,35,18,56)(29,61,44,52)(30,62,41,49)(31,63,42,50)(32,64,43,51), (1,35,59,54)(2,34,60,53)(3,33,57,56)(4,36,58,55)(5,44,38,31)(6,43,39,30)(7,42,40,29)(8,41,37,32)(9,17,22,13)(10,20,23,16)(11,19,24,15)(12,18,21,14)(25,50,47,61)(26,49,48,64)(27,52,45,63)(28,51,46,62), (1,37)(2,38)(3,39)(4,40)(5,60)(6,57)(7,58)(8,59)(9,25)(10,26)(11,27)(12,28)(13,29)(14,30)(15,31)(16,32)(17,42)(18,43)(19,44)(20,41)(21,46)(22,47)(23,48)(24,45)(33,49)(34,50)(35,51)(36,52)(53,61)(54,62)(55,63)(56,64), (1,56,57,35)(2,55,58,34)(3,54,59,33)(4,53,60,36)(5,52,40,61)(6,51,37,64)(7,50,38,63)(8,49,39,62)(9,17,24,15)(10,20,21,14)(11,19,22,13)(12,18,23,16)(25,42,45,31)(26,41,46,30)(27,44,47,29)(28,43,48,32) );
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)], [(1,23,57,12),(2,24,58,9),(3,21,59,10),(4,22,60,11),(5,47,40,27),(6,48,37,28),(7,45,38,25),(8,46,39,26),(13,36,19,53),(14,33,20,54),(15,34,17,55),(16,35,18,56),(29,61,44,52),(30,62,41,49),(31,63,42,50),(32,64,43,51)], [(1,35,59,54),(2,34,60,53),(3,33,57,56),(4,36,58,55),(5,44,38,31),(6,43,39,30),(7,42,40,29),(8,41,37,32),(9,17,22,13),(10,20,23,16),(11,19,24,15),(12,18,21,14),(25,50,47,61),(26,49,48,64),(27,52,45,63),(28,51,46,62)], [(1,37),(2,38),(3,39),(4,40),(5,60),(6,57),(7,58),(8,59),(9,25),(10,26),(11,27),(12,28),(13,29),(14,30),(15,31),(16,32),(17,42),(18,43),(19,44),(20,41),(21,46),(22,47),(23,48),(24,45),(33,49),(34,50),(35,51),(36,52),(53,61),(54,62),(55,63),(56,64)], [(1,56,57,35),(2,55,58,34),(3,54,59,33),(4,53,60,36),(5,52,40,61),(6,51,37,64),(7,50,38,63),(8,49,39,62),(9,17,24,15),(10,20,21,14),(11,19,22,13),(12,18,23,16),(25,42,45,31),(26,41,46,30),(27,44,47,29),(28,43,48,32)]])
35 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 4A | ··· | 4H | 4I | 4J | 4K | 4L | ··· | 4P | 8A | 8B | 8C | 8D | 8E | ··· | 8J |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 8 | 8 | 8 | 8 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 8 | 2 | ··· | 2 | 4 | 4 | 4 | 8 | ··· | 8 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
35 irreducible representations
| dim | 1 | 1 | 1 | 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 | C2 | C2 | C2 | D4 | D4 | D4 | C4○D4 | C4○D8 | 2+ 1+4 | D4○SD16 |
| kernel | C42.468C23 | C23.24D4 | C8×D4 | C4×D8 | D4⋊D4 | D4.D4 | C8⋊8D4 | D4⋊2Q8 | C23.20D4 | C4.SD16 | D4⋊6D4 | C22.49C24 | C22⋊C4 | C4⋊C4 | C2×D4 | D4 | C4 | C4 | C2 |
| # reps | 1 | 2 | 1 | 1 | 2 | 1 | 2 | 1 | 2 | 1 | 1 | 1 | 2 | 1 | 1 | 4 | 8 | 1 | 2 |
Matrix representation of C42.468C23 ►in GL4(𝔽17) generated by
| 4 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 15 |
| 0 | 0 | 1 | 16 |
| 0 | 4 | 0 | 0 |
| 4 | 0 | 0 | 0 |
| 0 | 0 | 13 | 0 |
| 0 | 0 | 13 | 4 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 11 | 6 |
| 0 | 0 | 14 | 6 |
| 0 | 13 | 0 | 0 |
| 4 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 4 |
G:=sub<GL(4,GF(17))| [4,0,0,0,0,13,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,1,1,0,0,15,16],[0,4,0,0,4,0,0,0,0,0,13,13,0,0,0,4],[1,0,0,0,0,1,0,0,0,0,11,14,0,0,6,6],[0,4,0,0,13,0,0,0,0,0,4,0,0,0,0,4] >;
C42.468C23 in GAP, Magma, Sage, TeX
C_4^2._{468}C_2^3 % in TeX
G:=Group("C4^2.468C2^3"); // GroupNames label
G:=SmallGroup(128,2035);
// by ID
G=gap.SmallGroup(128,2035);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,560,253,456,758,346,248,4037,1027,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^4=d^2=1,c^2=a^2*b^2,e^2=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=b^-1,b*e=e*b,d*c*d=b*c,e*c*e^-1=a^2*c,d*e=e*d>;
// generators/relations