p-group, metabelian, nilpotent (class 2), monomial
Aliases: C42.165D4, C24.319C23, C23.436C24, 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, C2.60(C22.46C24), C2.20(C23.38C23), (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
Subgroups: 548 in 302 conjugacy classes, 112 normal (20 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C4 [×4], C4 [×16], C22 [×3], C22 [×4], C22 [×20], C2×C4 [×14], C2×C4 [×40], D4 [×8], Q8 [×8], C23, C23 [×4], C23 [×12], C42 [×4], C42 [×4], C22⋊C4 [×20], C4⋊C4 [×20], C22×C4 [×3], C22×C4 [×10], C22×C4 [×12], C2×D4 [×6], C2×Q8 [×4], C2×Q8 [×4], C24 [×2], C2.C42 [×8], C2×C42, C2×C42 [×2], C2×C22⋊C4 [×10], C2×C4⋊C4 [×2], C2×C4⋊C4 [×8], C4×D4 [×4], C4×Q8 [×4], C22⋊Q8 [×8], C23×C4 [×2], C22×D4, C22×Q8, C23.7Q8 [×2], C42⋊8C4, C24.C22 [×4], C23.10D4 [×2], C23.81C23 [×2], C2×C4×D4, C2×C4×Q8, C2×C22⋊Q8 [×2], C42.165D4
Quotients:
C1, C2 [×15], C22 [×35], D4 [×8], C23 [×15], C2×D4 [×12], C4○D4 [×6], C24, C4⋊D4 [×4], C22×D4 [×2], C2×C4○D4 [×3], 2- (1+4) [×2], C2×C4⋊D4, C22.19C24, C23.38C23, Q8⋊5D4 [×2], C22.46C24 [×2], C42.165D4
Generators and relations
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 >
(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 27 14 10)(2 28 15 11)(3 25 16 12)(4 26 13 9)(5 37 24 32)(6 38 21 29)(7 39 22 30)(8 40 23 31)(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 33 13 44)(6 51 14 58)(7 35 15 42)(8 49 16 60)(9 45 37 18)(10 55 38 62)(11 47 39 20)(12 53 40 64)(17 27 48 29)(19 25 46 31)(26 56 32 63)(28 54 30 61)
(1 57 14 42)(2 43 15 58)(3 59 16 44)(4 41 13 60)(5 49 24 34)(6 35 21 50)(7 51 22 36)(8 33 23 52)(9 53 26 46)(10 47 27 54)(11 55 28 48)(12 45 25 56)(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), (1,27,14,10)(2,28,15,11)(3,25,16,12)(4,26,13,9)(5,37,24,32)(6,38,21,29)(7,39,22,30)(8,40,23,31)(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)(5,33,13,44)(6,51,14,58)(7,35,15,42)(8,49,16,60)(9,45,37,18)(10,55,38,62)(11,47,39,20)(12,53,40,64)(17,27,48,29)(19,25,46,31)(26,56,32,63)(28,54,30,61), (1,57,14,42)(2,43,15,58)(3,59,16,44)(4,41,13,60)(5,49,24,34)(6,35,21,50)(7,51,22,36)(8,33,23,52)(9,53,26,46)(10,47,27,54)(11,55,28,48)(12,45,25,56)(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), (1,27,14,10)(2,28,15,11)(3,25,16,12)(4,26,13,9)(5,37,24,32)(6,38,21,29)(7,39,22,30)(8,40,23,31)(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)(5,33,13,44)(6,51,14,58)(7,35,15,42)(8,49,16,60)(9,45,37,18)(10,55,38,62)(11,47,39,20)(12,53,40,64)(17,27,48,29)(19,25,46,31)(26,56,32,63)(28,54,30,61), (1,57,14,42)(2,43,15,58)(3,59,16,44)(4,41,13,60)(5,49,24,34)(6,35,21,50)(7,51,22,36)(8,33,23,52)(9,53,26,46)(10,47,27,54)(11,55,28,48)(12,45,25,56)(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)], [(1,27,14,10),(2,28,15,11),(3,25,16,12),(4,26,13,9),(5,37,24,32),(6,38,21,29),(7,39,22,30),(8,40,23,31),(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),(5,33,13,44),(6,51,14,58),(7,35,15,42),(8,49,16,60),(9,45,37,18),(10,55,38,62),(11,47,39,20),(12,53,40,64),(17,27,48,29),(19,25,46,31),(26,56,32,63),(28,54,30,61)], [(1,57,14,42),(2,43,15,58),(3,59,16,44),(4,41,13,60),(5,49,24,34),(6,35,21,50),(7,51,22,36),(8,33,23,52),(9,53,26,46),(10,47,27,54),(11,55,28,48),(12,45,25,56),(17,39,62,30),(18,31,63,40),(19,37,64,32),(20,29,61,38)])
Matrix representation ►G ⊆ 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] >;
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 |
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