p-group, metabelian, nilpotent (class 2), monomial
Aliases: C42.167D4, C23.442C24, C24.323C23, C22.1772- 1+4, C4⋊C4⋊27D4, (C2×D4)⋊16Q8, C2.22(D4×Q8), C4⋊3(C22⋊Q8), C42⋊9C4⋊28C2, C23.21(C2×Q8), C4.37(C4⋊1D4), C2.50(D4⋊6D4), C23.4Q8⋊21C2, C22.97(C22×Q8), (C22×C4).835C23, (C2×C42).547C22, (C23×C4).395C22, C22.293(C22×D4), (C22×D4).529C22, (C22×Q8).129C22, C24.3C22.44C2, C2.C42.548C22, C2.21(C23.38C23), (C4×C4⋊C4)⋊87C2, (C2×C4⋊Q8)⋊12C2, (C2×C4×D4).60C2, (C2×C4).73(C2×D4), C2.12(C2×C4⋊1D4), (C2×C22⋊Q8)⋊22C2, (C2×C4).309(C2×Q8), C2.30(C2×C22⋊Q8), (C2×C4).821(C4○D4), (C2×C4⋊C4).300C22, C22.319(C2×C4○D4), (C2×C22⋊C4).177C22, SmallGroup(128,1274)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.167D4
G = < a,b,c,d | a4=b4=c4=1, d2=b2, ab=ba, cac-1=ab2, dad-1=a-1, cbc-1=dbd-1=b-1, dcd-1=b2c-1 >
Subgroups: 612 in 338 conjugacy classes, 132 normal (18 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, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C24, C2.C42, C2×C42, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C4×D4, C22⋊Q8, C4⋊Q8, C23×C4, C22×D4, C22×Q8, C4×C4⋊C4, C42⋊9C4, C24.3C22, C23.4Q8, C2×C4×D4, C2×C22⋊Q8, C2×C4⋊Q8, C42.167D4
Quotients: C1, C2, C22, D4, Q8, C23, C2×D4, C2×Q8, C4○D4, C24, C22⋊Q8, C4⋊1D4, C22×D4, C22×Q8, C2×C4○D4, 2- 1+4, C2×C22⋊Q8, C2×C4⋊1D4, C23.38C23, D4⋊6D4, D4×Q8, C42.167D4
►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 20 13)(2 12 17 14)(3 9 18 15)(4 10 19 16)(5 28 63 29)(6 25 64 30)(7 26 61 31)(8 27 62 32)(21 44 52 45)(22 41 49 46)(23 42 50 47)(24 43 51 48)(33 57 39 56)(34 58 40 53)(35 59 37 54)(36 60 38 55)
(1 26 45 53)(2 32 46 59)(3 28 47 55)(4 30 48 57)(5 23 38 9)(6 51 39 16)(7 21 40 11)(8 49 37 14)(10 64 24 33)(12 62 22 35)(13 61 52 34)(15 63 50 36)(17 27 41 54)(18 29 42 60)(19 25 43 56)(20 31 44 58)
(1 37 20 35)(2 40 17 34)(3 39 18 33)(4 38 19 36)(5 43 63 48)(6 42 64 47)(7 41 61 46)(8 44 62 45)(9 57 15 56)(10 60 16 55)(11 59 13 54)(12 58 14 53)(21 32 52 27)(22 31 49 26)(23 30 50 25)(24 29 51 28)
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), (1,11,20,13)(2,12,17,14)(3,9,18,15)(4,10,19,16)(5,28,63,29)(6,25,64,30)(7,26,61,31)(8,27,62,32)(21,44,52,45)(22,41,49,46)(23,42,50,47)(24,43,51,48)(33,57,39,56)(34,58,40,53)(35,59,37,54)(36,60,38,55), (1,26,45,53)(2,32,46,59)(3,28,47,55)(4,30,48,57)(5,23,38,9)(6,51,39,16)(7,21,40,11)(8,49,37,14)(10,64,24,33)(12,62,22,35)(13,61,52,34)(15,63,50,36)(17,27,41,54)(18,29,42,60)(19,25,43,56)(20,31,44,58), (1,37,20,35)(2,40,17,34)(3,39,18,33)(4,38,19,36)(5,43,63,48)(6,42,64,47)(7,41,61,46)(8,44,62,45)(9,57,15,56)(10,60,16,55)(11,59,13,54)(12,58,14,53)(21,32,52,27)(22,31,49,26)(23,30,50,25)(24,29,51,28)>;
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), (1,11,20,13)(2,12,17,14)(3,9,18,15)(4,10,19,16)(5,28,63,29)(6,25,64,30)(7,26,61,31)(8,27,62,32)(21,44,52,45)(22,41,49,46)(23,42,50,47)(24,43,51,48)(33,57,39,56)(34,58,40,53)(35,59,37,54)(36,60,38,55), (1,26,45,53)(2,32,46,59)(3,28,47,55)(4,30,48,57)(5,23,38,9)(6,51,39,16)(7,21,40,11)(8,49,37,14)(10,64,24,33)(12,62,22,35)(13,61,52,34)(15,63,50,36)(17,27,41,54)(18,29,42,60)(19,25,43,56)(20,31,44,58), (1,37,20,35)(2,40,17,34)(3,39,18,33)(4,38,19,36)(5,43,63,48)(6,42,64,47)(7,41,61,46)(8,44,62,45)(9,57,15,56)(10,60,16,55)(11,59,13,54)(12,58,14,53)(21,32,52,27)(22,31,49,26)(23,30,50,25)(24,29,51,28) );
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)], [(1,11,20,13),(2,12,17,14),(3,9,18,15),(4,10,19,16),(5,28,63,29),(6,25,64,30),(7,26,61,31),(8,27,62,32),(21,44,52,45),(22,41,49,46),(23,42,50,47),(24,43,51,48),(33,57,39,56),(34,58,40,53),(35,59,37,54),(36,60,38,55)], [(1,26,45,53),(2,32,46,59),(3,28,47,55),(4,30,48,57),(5,23,38,9),(6,51,39,16),(7,21,40,11),(8,49,37,14),(10,64,24,33),(12,62,22,35),(13,61,52,34),(15,63,50,36),(17,27,41,54),(18,29,42,60),(19,25,43,56),(20,31,44,58)], [(1,37,20,35),(2,40,17,34),(3,39,18,33),(4,38,19,36),(5,43,63,48),(6,42,64,47),(7,41,61,46),(8,44,62,45),(9,57,15,56),(10,60,16,55),(11,59,13,54),(12,58,14,53),(21,32,52,27),(22,31,49,26),(23,30,50,25),(24,29,51,28)]])
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 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | - | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | Q8 | C4○D4 | 2- 1+4 |
| kernel | C42.167D4 | C4×C4⋊C4 | C42⋊9C4 | C24.3C22 | C23.4Q8 | C2×C4×D4 | C2×C22⋊Q8 | C2×C4⋊Q8 | C42 | C4⋊C4 | C2×D4 | C2×C4 | C22 |
| # reps | 1 | 1 | 2 | 2 | 4 | 1 | 4 | 1 | 4 | 8 | 4 | 4 | 2 |
Matrix representation of C42.167D4 ►in GL6(𝔽5)
| 4 | 2 | 0 | 0 | 0 | 0 |
| 4 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 0 | 1 | 2 |
| 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 | 4 | 3 |
| 1 | 3 | 0 | 0 | 0 | 0 |
| 1 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 2 | 0 | 0 |
| 0 | 0 | 4 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 2 |
| 0 | 0 | 0 | 0 | 1 | 2 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 4 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 3 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 2 |
| 0 | 0 | 0 | 0 | 0 | 2 |
G:=sub<GL(6,GF(5))| [4,4,0,0,0,0,2,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,3,1,0,0,0,0,0,2],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,2,4,0,0,0,0,0,3],[1,1,0,0,0,0,3,4,0,0,0,0,0,0,4,4,0,0,0,0,2,1,0,0,0,0,0,0,3,1,0,0,0,0,2,2],[4,4,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,3,4,0,0,0,0,0,0,3,0,0,0,0,0,2,2] >;
C42.167D4 in GAP, Magma, Sage, TeX
C_4^2._{167}D_4 % in TeX
G:=Group("C4^2.167D4"); // GroupNames label
G:=SmallGroup(128,1274);
// by ID
G=gap.SmallGroup(128,1274);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,2,336,253,568,758,723,675,80]);
// 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*b^2,d*a*d^-1=a^-1,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=b^2*c^-1>;
// generators/relations