p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.467C23, C4.472+ 1+4, (C8×D4)⋊20C2, C4⋊C4.408D4, D4⋊Q8⋊13C2, C4⋊2Q16⋊15C2, (C4×SD16)⋊41C2, (C2×D4).237D4, D4⋊6D4.6C2, C4.72(C4○D8), C2.46(Q8○D8), D4.19(C4○D4), D4.7D4⋊11C2, C8.18D4⋊14C2, C4⋊C8.297C22, C4⋊C4.402C23, (C2×C4).494C24, (C4×C8).274C22, (C2×C8).350C23, C22⋊C4.106D4, C4.SD16⋊30C2, C23.110(C2×D4), C4⋊Q8.143C22, C2.D8.56C22, (C2×D4).421C23, (C4×D4).335C22, C23.20D4⋊6C2, (C2×Q16).37C22, (C4×Q8).151C22, (C2×Q8).210C23, C2.130(D4⋊5D4), C4.Q8.166C22, C22⋊Q8.74C22, C23.24D4⋊12C2, C22⋊C8.203C22, (C22×C8).162C22, C22.754(C22×D4), D4⋊C4.167C22, C22.50C24⋊2C2, (C22×C4).1138C23, Q8⋊C4.114C22, (C2×SD16).157C22, C42⋊C2.182C22, C2.62(C2×C4○D8), C4.219(C2×C4○D4), (C2×C4).923(C2×D4), (C2×C4○D4).200C22, SmallGroup(128,2034)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.467C23
G = < a,b,c,d,e | a4=b4=d2=1, c2=a2, 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: 360 in 193 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, SD16, Q16, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C2×Q8, C4○D4, C4×C8, C22⋊C8, D4⋊C4, D4⋊C4, Q8⋊C4, Q8⋊C4, C4⋊C8, C4.Q8, C2.D8, C2×C4⋊C4, C42⋊C2, C4×D4, C4×Q8, C4×Q8, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C42⋊2C2, C4⋊Q8, C22×C8, C2×SD16, C2×Q16, C2×C4○D4, C23.24D4, C8×D4, C4×SD16, D4.7D4, C4⋊2Q16, C8.18D4, D4⋊Q8, C23.20D4, C4.SD16, D4⋊6D4, C22.50C24, C42.467C23
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, Q8○D8, C42.467C23
►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 46 17)(2 43 47 18)(3 44 48 19)(4 41 45 20)(5 62 33 39)(6 63 34 40)(7 64 35 37)(8 61 36 38)(9 15 23 49)(10 16 24 50)(11 13 21 51)(12 14 22 52)(25 56 60 31)(26 53 57 32)(27 54 58 29)(28 55 59 30)
(1 26 3 28)(2 25 4 27)(5 52 7 50)(6 51 8 49)(9 40 11 38)(10 39 12 37)(13 36 15 34)(14 35 16 33)(17 53 19 55)(18 56 20 54)(21 61 23 63)(22 64 24 62)(29 43 31 41)(30 42 32 44)(45 58 47 60)(46 57 48 59)
(1 3)(2 4)(5 64)(6 61)(7 62)(8 63)(9 11)(10 12)(13 49)(14 50)(15 51)(16 52)(17 44)(18 41)(19 42)(20 43)(21 23)(22 24)(25 29)(26 30)(27 31)(28 32)(33 37)(34 38)(35 39)(36 40)(45 47)(46 48)(53 59)(54 60)(55 57)(56 58)
(1 10 46 24)(2 9 47 23)(3 12 48 22)(4 11 45 21)(5 30 33 55)(6 29 34 54)(7 32 35 53)(8 31 36 56)(13 20 51 41)(14 19 52 44)(15 18 49 43)(16 17 50 42)(25 38 60 61)(26 37 57 64)(27 40 58 63)(28 39 59 62)
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), (1,42,46,17)(2,43,47,18)(3,44,48,19)(4,41,45,20)(5,62,33,39)(6,63,34,40)(7,64,35,37)(8,61,36,38)(9,15,23,49)(10,16,24,50)(11,13,21,51)(12,14,22,52)(25,56,60,31)(26,53,57,32)(27,54,58,29)(28,55,59,30), (1,26,3,28)(2,25,4,27)(5,52,7,50)(6,51,8,49)(9,40,11,38)(10,39,12,37)(13,36,15,34)(14,35,16,33)(17,53,19,55)(18,56,20,54)(21,61,23,63)(22,64,24,62)(29,43,31,41)(30,42,32,44)(45,58,47,60)(46,57,48,59), (1,3)(2,4)(5,64)(6,61)(7,62)(8,63)(9,11)(10,12)(13,49)(14,50)(15,51)(16,52)(17,44)(18,41)(19,42)(20,43)(21,23)(22,24)(25,29)(26,30)(27,31)(28,32)(33,37)(34,38)(35,39)(36,40)(45,47)(46,48)(53,59)(54,60)(55,57)(56,58), (1,10,46,24)(2,9,47,23)(3,12,48,22)(4,11,45,21)(5,30,33,55)(6,29,34,54)(7,32,35,53)(8,31,36,56)(13,20,51,41)(14,19,52,44)(15,18,49,43)(16,17,50,42)(25,38,60,61)(26,37,57,64)(27,40,58,63)(28,39,59,62)>;
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), (1,42,46,17)(2,43,47,18)(3,44,48,19)(4,41,45,20)(5,62,33,39)(6,63,34,40)(7,64,35,37)(8,61,36,38)(9,15,23,49)(10,16,24,50)(11,13,21,51)(12,14,22,52)(25,56,60,31)(26,53,57,32)(27,54,58,29)(28,55,59,30), (1,26,3,28)(2,25,4,27)(5,52,7,50)(6,51,8,49)(9,40,11,38)(10,39,12,37)(13,36,15,34)(14,35,16,33)(17,53,19,55)(18,56,20,54)(21,61,23,63)(22,64,24,62)(29,43,31,41)(30,42,32,44)(45,58,47,60)(46,57,48,59), (1,3)(2,4)(5,64)(6,61)(7,62)(8,63)(9,11)(10,12)(13,49)(14,50)(15,51)(16,52)(17,44)(18,41)(19,42)(20,43)(21,23)(22,24)(25,29)(26,30)(27,31)(28,32)(33,37)(34,38)(35,39)(36,40)(45,47)(46,48)(53,59)(54,60)(55,57)(56,58), (1,10,46,24)(2,9,47,23)(3,12,48,22)(4,11,45,21)(5,30,33,55)(6,29,34,54)(7,32,35,53)(8,31,36,56)(13,20,51,41)(14,19,52,44)(15,18,49,43)(16,17,50,42)(25,38,60,61)(26,37,57,64)(27,40,58,63)(28,39,59,62) );
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)], [(1,42,46,17),(2,43,47,18),(3,44,48,19),(4,41,45,20),(5,62,33,39),(6,63,34,40),(7,64,35,37),(8,61,36,38),(9,15,23,49),(10,16,24,50),(11,13,21,51),(12,14,22,52),(25,56,60,31),(26,53,57,32),(27,54,58,29),(28,55,59,30)], [(1,26,3,28),(2,25,4,27),(5,52,7,50),(6,51,8,49),(9,40,11,38),(10,39,12,37),(13,36,15,34),(14,35,16,33),(17,53,19,55),(18,56,20,54),(21,61,23,63),(22,64,24,62),(29,43,31,41),(30,42,32,44),(45,58,47,60),(46,57,48,59)], [(1,3),(2,4),(5,64),(6,61),(7,62),(8,63),(9,11),(10,12),(13,49),(14,50),(15,51),(16,52),(17,44),(18,41),(19,42),(20,43),(21,23),(22,24),(25,29),(26,30),(27,31),(28,32),(33,37),(34,38),(35,39),(36,40),(45,47),(46,48),(53,59),(54,60),(55,57),(56,58)], [(1,10,46,24),(2,9,47,23),(3,12,48,22),(4,11,45,21),(5,30,33,55),(6,29,34,54),(7,32,35,53),(8,31,36,56),(13,20,51,41),(14,19,52,44),(15,18,49,43),(16,17,50,42),(25,38,60,61),(26,37,57,64),(27,40,58,63),(28,39,59,62)]])
35 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | ··· | 4H | 4I | 4J | 4K | 4L | ··· | 4Q | 8A | 8B | 8C | 8D | 8E | ··· | 8J |
| order | 1 | 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 | 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 | Q8○D8 |
| kernel | C42.467C23 | C23.24D4 | C8×D4 | C4×SD16 | D4.7D4 | C4⋊2Q16 | C8.18D4 | D4⋊Q8 | C23.20D4 | C4.SD16 | D4⋊6D4 | C22.50C24 | 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 C42.467C23 ►in GL4(𝔽17) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 7 | 4 |
| 0 | 0 | 13 | 10 |
| 0 | 1 | 0 | 0 |
| 16 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 3 | 14 | 0 | 0 |
| 14 | 14 | 0 | 0 |
| 0 | 0 | 0 | 4 |
| 0 | 0 | 4 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 |
| 13 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 |
| 0 | 0 | 13 | 10 |
| 0 | 0 | 7 | 4 |
G:=sub<GL(4,GF(17))| [1,0,0,0,0,1,0,0,0,0,7,13,0,0,4,10],[0,16,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[3,14,0,0,14,14,0,0,0,0,0,4,0,0,4,0],[1,0,0,0,0,16,0,0,0,0,16,0,0,0,0,16],[13,0,0,0,0,13,0,0,0,0,13,7,0,0,10,4] >;
C42.467C23 in GAP, Magma, Sage, TeX
C_4^2._{467}C_2^3 % in TeX
G:=Group("C4^2.467C2^3"); // GroupNames label
G:=SmallGroup(128,2034);
// by ID
G=gap.SmallGroup(128,2034);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,112,253,456,758,346,248,4037,1027,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^4=d^2=1,c^2=a^2,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