p-group, metabelian, nilpotent (class 2), monomial
Aliases: C42.325D4, (C2×D4)⋊8C8, (C2×C8)⋊28D4, C2.15(C8×D4), C4⋊1(C22⋊C8), C23.8(C2×C8), C2.5(C8⋊6D4), C24.62(C2×C4), C4.48(C4⋊1D4), (C22×D4).31C4, C22.103(C4×D4), (C2×C4).47M4(2), C4.203(C4⋊D4), C4.88(C4.4D4), C22.34(C8○D4), C22.44(C22×C8), (C22×C8).54C22, (C23×C4).24C22, C23.273(C22×C4), C22.55(C2×M4(2)), (C2×C42).1070C22, (C22×C4).1637C23, C2.3(C24.3C22), (C2×C4×C8)⋊12C2, (C2×C4⋊C8)⋊15C2, (C2×C4×D4).22C2, (C2×C4⋊C4).60C4, (C2×C4).53(C2×C8), (C2×C22⋊C8)⋊16C2, C2.20(C2×C22⋊C8), (C2×C4).1543(C2×D4), (C2×C22⋊C4).31C4, (C2×C4).943(C4○D4), (C22×C4).127(C2×C4), (C2×C4).254(C22⋊C4), C2.4((C22×C8)⋊C2), C22.127(C2×C22⋊C4), SmallGroup(128,686)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.325D4
G = < a,b,c,d | a4=b4=1, c4=d2=b2, ab=ba, ac=ca, dad-1=a-1, bc=cb, bd=db, dcd-1=b-1c3 >
Subgroups: 364 in 198 conjugacy classes, 80 normal (22 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, C23, C23, C23, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C24, C4×C8, C22⋊C8, C4⋊C8, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C4×D4, C22×C8, C23×C4, C22×D4, C2×C4×C8, C2×C22⋊C8, C2×C4⋊C8, C2×C4×D4, C42.325D4
Quotients: C1, C2, C4, C22, C8, C2×C4, D4, C23, C22⋊C4, C2×C8, M4(2), C22×C4, C2×D4, C4○D4, C22⋊C8, C2×C22⋊C4, C4×D4, C4⋊D4, C4.4D4, C4⋊1D4, C22×C8, C2×M4(2), C8○D4, C24.3C22, C2×C22⋊C8, (C22×C8)⋊C2, C8×D4, C8⋊6D4, C42.325D4
►On 64 points
(1 43 33 20)(2 44 34 21)(3 45 35 22)(4 46 36 23)(5 47 37 24)(6 48 38 17)(7 41 39 18)(8 42 40 19)(9 61 29 51)(10 62 30 52)(11 63 31 53)(12 64 32 54)(13 57 25 55)(14 58 26 56)(15 59 27 49)(16 60 28 50)
(1 55 5 51)(2 56 6 52)(3 49 7 53)(4 50 8 54)(9 43 13 47)(10 44 14 48)(11 45 15 41)(12 46 16 42)(17 30 21 26)(18 31 22 27)(19 32 23 28)(20 25 24 29)(33 57 37 61)(34 58 38 62)(35 59 39 63)(36 60 40 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 18 5 22)(2 30 6 26)(3 20 7 24)(4 32 8 28)(9 59 13 63)(10 38 14 34)(11 61 15 57)(12 40 16 36)(17 56 21 52)(19 50 23 54)(25 53 29 49)(27 55 31 51)(33 41 37 45)(35 43 39 47)(42 60 46 64)(44 62 48 58)
G:=sub<Sym(64)| (1,43,33,20)(2,44,34,21)(3,45,35,22)(4,46,36,23)(5,47,37,24)(6,48,38,17)(7,41,39,18)(8,42,40,19)(9,61,29,51)(10,62,30,52)(11,63,31,53)(12,64,32,54)(13,57,25,55)(14,58,26,56)(15,59,27,49)(16,60,28,50), (1,55,5,51)(2,56,6,52)(3,49,7,53)(4,50,8,54)(9,43,13,47)(10,44,14,48)(11,45,15,41)(12,46,16,42)(17,30,21,26)(18,31,22,27)(19,32,23,28)(20,25,24,29)(33,57,37,61)(34,58,38,62)(35,59,39,63)(36,60,40,64), (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), (1,18,5,22)(2,30,6,26)(3,20,7,24)(4,32,8,28)(9,59,13,63)(10,38,14,34)(11,61,15,57)(12,40,16,36)(17,56,21,52)(19,50,23,54)(25,53,29,49)(27,55,31,51)(33,41,37,45)(35,43,39,47)(42,60,46,64)(44,62,48,58)>;
G:=Group( (1,43,33,20)(2,44,34,21)(3,45,35,22)(4,46,36,23)(5,47,37,24)(6,48,38,17)(7,41,39,18)(8,42,40,19)(9,61,29,51)(10,62,30,52)(11,63,31,53)(12,64,32,54)(13,57,25,55)(14,58,26,56)(15,59,27,49)(16,60,28,50), (1,55,5,51)(2,56,6,52)(3,49,7,53)(4,50,8,54)(9,43,13,47)(10,44,14,48)(11,45,15,41)(12,46,16,42)(17,30,21,26)(18,31,22,27)(19,32,23,28)(20,25,24,29)(33,57,37,61)(34,58,38,62)(35,59,39,63)(36,60,40,64), (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), (1,18,5,22)(2,30,6,26)(3,20,7,24)(4,32,8,28)(9,59,13,63)(10,38,14,34)(11,61,15,57)(12,40,16,36)(17,56,21,52)(19,50,23,54)(25,53,29,49)(27,55,31,51)(33,41,37,45)(35,43,39,47)(42,60,46,64)(44,62,48,58) );
G=PermutationGroup([[(1,43,33,20),(2,44,34,21),(3,45,35,22),(4,46,36,23),(5,47,37,24),(6,48,38,17),(7,41,39,18),(8,42,40,19),(9,61,29,51),(10,62,30,52),(11,63,31,53),(12,64,32,54),(13,57,25,55),(14,58,26,56),(15,59,27,49),(16,60,28,50)], [(1,55,5,51),(2,56,6,52),(3,49,7,53),(4,50,8,54),(9,43,13,47),(10,44,14,48),(11,45,15,41),(12,46,16,42),(17,30,21,26),(18,31,22,27),(19,32,23,28),(20,25,24,29),(33,57,37,61),(34,58,38,62),(35,59,39,63),(36,60,40,64)], [(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)], [(1,18,5,22),(2,30,6,26),(3,20,7,24),(4,32,8,28),(9,59,13,63),(10,38,14,34),(11,61,15,57),(12,40,16,36),(17,56,21,52),(19,50,23,54),(25,53,29,49),(27,55,31,51),(33,41,37,45),(35,43,39,47),(42,60,46,64),(44,62,48,58)]])
56 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 4A | ··· | 4H | 4I | ··· | 4P | 4Q | 4R | 4S | 4T | 8A | ··· | 8P | 8Q | ··· | 8X |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 8 | ··· | 8 |
| size | 1 | 1 | ··· | 1 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C8 | D4 | D4 | M4(2) | C4○D4 | C8○D4 |
| kernel | C42.325D4 | C2×C4×C8 | C2×C22⋊C8 | C2×C4⋊C8 | C2×C4×D4 | C2×C22⋊C4 | C2×C4⋊C4 | C22×D4 | C2×D4 | C42 | C2×C8 | C2×C4 | C2×C4 | C22 |
| # reps | 1 | 1 | 4 | 1 | 1 | 4 | 2 | 2 | 16 | 4 | 4 | 4 | 4 | 8 |
Matrix representation of C42.325D4 ►in GL6(𝔽17)
| 0 | 1 | 0 | 0 | 0 | 0 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 13 | 0 | 0 | 0 |
| 0 | 0 | 0 | 13 | 0 | 0 |
| 0 | 0 | 0 | 0 | 13 | 0 |
| 0 | 0 | 0 | 0 | 0 | 13 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 0 | 0 | 0 |
| 0 | 0 | 0 | 8 | 0 | 0 |
| 0 | 0 | 0 | 0 | 15 | 0 |
| 0 | 0 | 0 | 0 | 15 | 2 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 9 |
| 0 | 0 | 0 | 0 | 0 | 13 |
G:=sub<GL(6,GF(17))| [0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,13],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,8,0,0,0,0,0,0,8,0,0,0,0,0,0,15,15,0,0,0,0,0,2],[0,16,0,0,0,0,16,0,0,0,0,0,0,0,0,4,0,0,0,0,4,0,0,0,0,0,0,0,4,0,0,0,0,0,9,13] >;
C42.325D4 in GAP, Magma, Sage, TeX
C_4^2._{325}D_4 % in TeX
G:=Group("C4^2.325D4"); // GroupNames label
G:=SmallGroup(128,686);
// by ID
G=gap.SmallGroup(128,686);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,288,422,100,124]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^4=d^2=b^2,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,b*c=c*b,b*d=d*b,d*c*d^-1=b^-1*c^3>;
// generators/relations