p-group, metabelian, nilpotent (class 2), monomial
Aliases: C42.165D4, C23.436C24, C24.319C23, C22.1732- 1+4, (C2×Q8)⋊24D4, C42⋊8C4⋊41C2, C4.166(C4⋊D4), C2.36(Q8⋊5D4), C23.50(C4○D4), (C22×C4).93C23, C23.7Q8⋊64C2, (C23×C4).389C22, (C2×C42).542C22, C22.287(C22×D4), C24.C22⋊77C2, C23.10D4.18C2, (C22×D4).527C22, (C22×Q8).431C22, C23.81C23⋊35C2, C2.59(C22.19C24), C2.C42.179C22, C2.20(C23.38C23), C2.60(C22.46C24), (C2×C4×Q8)⋊21C2, (C2×C4×D4).58C2, C2.31(C2×C4⋊D4), (C2×C22⋊Q8)⋊19C2, (C2×C4).1194(C2×D4), (C2×C4).818(C4○D4), (C2×C4⋊C4).296C22, C22.313(C2×C4○D4), (C2×C22⋊C4).172C22, SmallGroup(128,1268)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.165D4
G = < a,b,c,d | a4=b4=c4=1, d2=b2, ab=ba, cac-1=a-1b2, dad-1=ab2, cbc-1=dbd-1=b-1, dcd-1=b2c-1 >
Subgroups: 548 in 302 conjugacy classes, 112 normal (20 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C2×C4, C2×C4, D4, Q8, C23, C23, C23, C42, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×D4, C2×Q8, C2×Q8, C24, C2.C42, C2×C42, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C4×D4, C4×Q8, C22⋊Q8, C23×C4, C22×D4, C22×Q8, C23.7Q8, C42⋊8C4, C24.C22, C23.10D4, C23.81C23, C2×C4×D4, C2×C4×Q8, C2×C22⋊Q8, C42.165D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C4⋊D4, C22×D4, C2×C4○D4, 2- 1+4, C2×C4⋊D4, C22.19C24, C23.38C23, Q8⋊5D4, C22.46C24, C42.165D4
►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 11 14 25)(2 12 15 26)(3 9 16 27)(4 10 13 28)(5 39 22 30)(6 40 23 31)(7 37 24 32)(8 38 21 29)(17 36 62 51)(18 33 63 52)(19 34 64 49)(20 35 61 50)(41 53 60 46)(42 54 57 47)(43 55 58 48)(44 56 59 45)
(1 43 21 36)(2 57 22 50)(3 41 23 34)(4 59 24 52)(5 35 15 42)(6 49 16 60)(7 33 13 44)(8 51 14 58)(9 46 31 19)(10 56 32 63)(11 48 29 17)(12 54 30 61)(18 28 45 37)(20 26 47 39)(25 55 38 62)(27 53 40 64)
(1 57 14 42)(2 43 15 58)(3 59 16 44)(4 41 13 60)(5 51 22 36)(6 33 23 52)(7 49 24 34)(8 35 21 50)(9 56 27 45)(10 46 28 53)(11 54 25 47)(12 48 26 55)(17 39 62 30)(18 31 63 40)(19 37 64 32)(20 29 61 38)
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), (1,11,14,25)(2,12,15,26)(3,9,16,27)(4,10,13,28)(5,39,22,30)(6,40,23,31)(7,37,24,32)(8,38,21,29)(17,36,62,51)(18,33,63,52)(19,34,64,49)(20,35,61,50)(41,53,60,46)(42,54,57,47)(43,55,58,48)(44,56,59,45), (1,43,21,36)(2,57,22,50)(3,41,23,34)(4,59,24,52)(5,35,15,42)(6,49,16,60)(7,33,13,44)(8,51,14,58)(9,46,31,19)(10,56,32,63)(11,48,29,17)(12,54,30,61)(18,28,45,37)(20,26,47,39)(25,55,38,62)(27,53,40,64), (1,57,14,42)(2,43,15,58)(3,59,16,44)(4,41,13,60)(5,51,22,36)(6,33,23,52)(7,49,24,34)(8,35,21,50)(9,56,27,45)(10,46,28,53)(11,54,25,47)(12,48,26,55)(17,39,62,30)(18,31,63,40)(19,37,64,32)(20,29,61,38)>;
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), (1,11,14,25)(2,12,15,26)(3,9,16,27)(4,10,13,28)(5,39,22,30)(6,40,23,31)(7,37,24,32)(8,38,21,29)(17,36,62,51)(18,33,63,52)(19,34,64,49)(20,35,61,50)(41,53,60,46)(42,54,57,47)(43,55,58,48)(44,56,59,45), (1,43,21,36)(2,57,22,50)(3,41,23,34)(4,59,24,52)(5,35,15,42)(6,49,16,60)(7,33,13,44)(8,51,14,58)(9,46,31,19)(10,56,32,63)(11,48,29,17)(12,54,30,61)(18,28,45,37)(20,26,47,39)(25,55,38,62)(27,53,40,64), (1,57,14,42)(2,43,15,58)(3,59,16,44)(4,41,13,60)(5,51,22,36)(6,33,23,52)(7,49,24,34)(8,35,21,50)(9,56,27,45)(10,46,28,53)(11,54,25,47)(12,48,26,55)(17,39,62,30)(18,31,63,40)(19,37,64,32)(20,29,61,38) );
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)], [(1,11,14,25),(2,12,15,26),(3,9,16,27),(4,10,13,28),(5,39,22,30),(6,40,23,31),(7,37,24,32),(8,38,21,29),(17,36,62,51),(18,33,63,52),(19,34,64,49),(20,35,61,50),(41,53,60,46),(42,54,57,47),(43,55,58,48),(44,56,59,45)], [(1,43,21,36),(2,57,22,50),(3,41,23,34),(4,59,24,52),(5,35,15,42),(6,49,16,60),(7,33,13,44),(8,51,14,58),(9,46,31,19),(10,56,32,63),(11,48,29,17),(12,54,30,61),(18,28,45,37),(20,26,47,39),(25,55,38,62),(27,53,40,64)], [(1,57,14,42),(2,43,15,58),(3,59,16,44),(4,41,13,60),(5,51,22,36),(6,33,23,52),(7,49,24,34),(8,35,21,50),(9,56,27,45),(10,46,28,53),(11,54,25,47),(12,48,26,55),(17,39,62,30),(18,31,63,40),(19,37,64,32),(20,29,61,38)]])
38 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 4A | ··· | 4H | 4I | ··· | 4V | 4W | 4X | 4Y | 4Z |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 |
| size | 1 | 1 | ··· | 1 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
38 irreducible representations
| dim | 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 | D4 | D4 | C4○D4 | C4○D4 | 2- 1+4 |
| kernel | C42.165D4 | C23.7Q8 | C42⋊8C4 | C24.C22 | C23.10D4 | C23.81C23 | C2×C4×D4 | C2×C4×Q8 | C2×C22⋊Q8 | C42 | C2×Q8 | C2×C4 | C23 | C22 |
| # reps | 1 | 2 | 1 | 4 | 2 | 2 | 1 | 1 | 2 | 4 | 4 | 4 | 8 | 2 |
Matrix representation of C42.165D4 ►in GL6(𝔽5)
| 1 | 3 | 0 | 0 | 0 | 0 |
| 1 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 4 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 0 | 2 | 3 |
| 2 | 0 | 0 | 0 | 0 | 0 |
| 2 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 2 |
| 0 | 0 | 0 | 0 | 4 | 1 |
| 3 | 4 | 0 | 0 | 0 | 0 |
| 3 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 3 |
| 0 | 0 | 0 | 0 | 1 | 4 |
G:=sub<GL(6,GF(5))| [1,1,0,0,0,0,3,4,0,0,0,0,0,0,0,1,0,0,0,0,4,0,0,0,0,0,0,0,4,4,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,2,2,0,0,0,0,0,3],[2,2,0,0,0,0,0,3,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,4,4,0,0,0,0,2,1],[3,3,0,0,0,0,4,2,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,3,4] >;
C42.165D4 in GAP, Magma, Sage, TeX
C_4^2._{165}D_4 % in TeX
G:=Group("C4^2.165D4"); // GroupNames label
G:=SmallGroup(128,1268);
// by ID
G=gap.SmallGroup(128,1268);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,2,112,253,456,758,723,675,80]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^4=1,d^2=b^2,a*b=b*a,c*a*c^-1=a^-1*b^2,d*a*d^-1=a*b^2,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=b^2*c^-1>;
// generators/relations