p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.100D4, (C4×D4)⋊14C4, D4.1(C4⋊C4), (C2×D4).23Q8, C42⋊8C4⋊3C2, (C2×D4).199D4, C2.1(D4.Q8), C22.4Q16⋊2C2, C42.138(C2×C4), C23.742(C2×D4), (C22×C4).268D4, C4.95(C22⋊Q8), C4.120(C4⋊D4), C2.1(D4.2D4), C22.41(C4○D8), (C22×C8).12C22, C4.33(C42⋊C2), C22.61(C8⋊C22), (C2×C42).249C22, C22.69(C22⋊Q8), C22.107(C4⋊D4), (C22×C4).1326C23, (C22×D4).457C22, C2.13(C23.7Q8), C2.18(C23.37D4), C2.21(C23.24D4), C4.3(C2×C4⋊C4), (C2×C4⋊C8)⋊12C2, (C2×C4×D4).16C2, C4⋊C4.191(C2×C4), (C2×C4).261(C2×Q8), (C2×D4⋊C4).2C2, (C2×D4).206(C2×C4), (C2×C4).1316(C2×D4), (C2×C4⋊C4).34C22, (C2×C4).862(C4○D4), (C2×C4).364(C22×C4), (C2×C4).331(C22⋊C4), C22.249(C2×C22⋊C4), SmallGroup(128,536)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.100D4
G = < a,b,c,d | a4=b4=c4=1, d2=cbc-1=b-1, ab=ba, cac-1=dad-1=a-1b2, bd=db, dcd-1=b-1c-1 >
Subgroups: 388 in 178 conjugacy classes, 68 normal (28 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, D4, C23, C23, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C22×C4, C22×C4, C2×D4, C2×D4, C24, C2.C42, D4⋊C4, C4⋊C8, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C4×D4, C4×D4, C22×C8, C23×C4, C22×D4, C22.4Q16, C42⋊8C4, C2×D4⋊C4, C2×C4⋊C8, C2×C4×D4, C42.100D4
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×Q8, C4○D4, C2×C22⋊C4, C2×C4⋊C4, C42⋊C2, C4⋊D4, C22⋊Q8, C4○D8, C8⋊C22, C23.7Q8, C23.24D4, C23.37D4, D4.2D4, D4.Q8, C42.100D4
►On 64 points
(1 59 51 47)(2 44 52 64)(3 61 53 41)(4 46 54 58)(5 63 55 43)(6 48 56 60)(7 57 49 45)(8 42 50 62)(9 33 26 22)(10 19 27 38)(11 35 28 24)(12 21 29 40)(13 37 30 18)(14 23 31 34)(15 39 32 20)(16 17 25 36)
(1 7 5 3)(2 8 6 4)(9 15 13 11)(10 16 14 12)(17 23 21 19)(18 24 22 20)(25 31 29 27)(26 32 30 28)(33 39 37 35)(34 40 38 36)(41 47 45 43)(42 48 46 44)(49 55 53 51)(50 56 54 52)(57 63 61 59)(58 64 62 60)
(1 34 20 54)(2 53 21 33)(3 40 22 52)(4 51 23 39)(5 38 24 50)(6 49 17 37)(7 36 18 56)(8 55 19 35)(9 48 61 25)(10 32 62 47)(11 46 63 31)(12 30 64 45)(13 44 57 29)(14 28 58 43)(15 42 59 27)(16 26 60 41)
(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)
G:=sub<Sym(64)| (1,59,51,47)(2,44,52,64)(3,61,53,41)(4,46,54,58)(5,63,55,43)(6,48,56,60)(7,57,49,45)(8,42,50,62)(9,33,26,22)(10,19,27,38)(11,35,28,24)(12,21,29,40)(13,37,30,18)(14,23,31,34)(15,39,32,20)(16,17,25,36), (1,7,5,3)(2,8,6,4)(9,15,13,11)(10,16,14,12)(17,23,21,19)(18,24,22,20)(25,31,29,27)(26,32,30,28)(33,39,37,35)(34,40,38,36)(41,47,45,43)(42,48,46,44)(49,55,53,51)(50,56,54,52)(57,63,61,59)(58,64,62,60), (1,34,20,54)(2,53,21,33)(3,40,22,52)(4,51,23,39)(5,38,24,50)(6,49,17,37)(7,36,18,56)(8,55,19,35)(9,48,61,25)(10,32,62,47)(11,46,63,31)(12,30,64,45)(13,44,57,29)(14,28,58,43)(15,42,59,27)(16,26,60,41), (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)>;
G:=Group( (1,59,51,47)(2,44,52,64)(3,61,53,41)(4,46,54,58)(5,63,55,43)(6,48,56,60)(7,57,49,45)(8,42,50,62)(9,33,26,22)(10,19,27,38)(11,35,28,24)(12,21,29,40)(13,37,30,18)(14,23,31,34)(15,39,32,20)(16,17,25,36), (1,7,5,3)(2,8,6,4)(9,15,13,11)(10,16,14,12)(17,23,21,19)(18,24,22,20)(25,31,29,27)(26,32,30,28)(33,39,37,35)(34,40,38,36)(41,47,45,43)(42,48,46,44)(49,55,53,51)(50,56,54,52)(57,63,61,59)(58,64,62,60), (1,34,20,54)(2,53,21,33)(3,40,22,52)(4,51,23,39)(5,38,24,50)(6,49,17,37)(7,36,18,56)(8,55,19,35)(9,48,61,25)(10,32,62,47)(11,46,63,31)(12,30,64,45)(13,44,57,29)(14,28,58,43)(15,42,59,27)(16,26,60,41), (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) );
G=PermutationGroup([[(1,59,51,47),(2,44,52,64),(3,61,53,41),(4,46,54,58),(5,63,55,43),(6,48,56,60),(7,57,49,45),(8,42,50,62),(9,33,26,22),(10,19,27,38),(11,35,28,24),(12,21,29,40),(13,37,30,18),(14,23,31,34),(15,39,32,20),(16,17,25,36)], [(1,7,5,3),(2,8,6,4),(9,15,13,11),(10,16,14,12),(17,23,21,19),(18,24,22,20),(25,31,29,27),(26,32,30,28),(33,39,37,35),(34,40,38,36),(41,47,45,43),(42,48,46,44),(49,55,53,51),(50,56,54,52),(57,63,61,59),(58,64,62,60)], [(1,34,20,54),(2,53,21,33),(3,40,22,52),(4,51,23,39),(5,38,24,50),(6,49,17,37),(7,36,18,56),(8,55,19,35),(9,48,61,25),(10,32,62,47),(11,46,63,31),(12,30,64,45),(13,44,57,29),(14,28,58,43),(15,42,59,27),(16,26,60,41)], [(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)]])
38 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 4A | ··· | 4H | 4I | ··· | 4N | 4O | 4P | 4Q | 4R | 8A | ··· | 8H |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 8 | ··· | 8 |
| size | 1 | 1 | ··· | 1 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 8 | 4 | ··· | 4 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | D4 | D4 | Q8 | C4○D4 | C4○D8 | C8⋊C22 |
| kernel | C42.100D4 | C22.4Q16 | C42⋊8C4 | C2×D4⋊C4 | C2×C4⋊C8 | C2×C4×D4 | C4×D4 | C42 | C22×C4 | C2×D4 | C2×D4 | C2×C4 | C22 | C22 |
| # reps | 1 | 2 | 1 | 2 | 1 | 1 | 8 | 2 | 2 | 2 | 2 | 4 | 8 | 2 |
Matrix representation of C42.100D4 ►in GL6(𝔽17)
| 13 | 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 | 16 | 13 |
| 0 | 0 | 0 | 0 | 9 | 1 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 16 | 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 |
| 14 | 3 | 0 | 0 | 0 | 0 |
| 3 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 12 | 0 | 0 |
| 0 | 0 | 12 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 3 |
| 0 | 0 | 0 | 0 | 11 | 0 |
| 14 | 3 | 0 | 0 | 0 | 0 |
| 14 | 14 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 12 | 0 | 0 |
| 0 | 0 | 5 | 5 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 14 |
| 0 | 0 | 0 | 0 | 6 | 0 |
G:=sub<GL(6,GF(17))| [13,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,16,9,0,0,0,0,13,1],[0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,16,0,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[14,3,0,0,0,0,3,3,0,0,0,0,0,0,5,12,0,0,0,0,12,12,0,0,0,0,0,0,0,11,0,0,0,0,3,0],[14,14,0,0,0,0,3,14,0,0,0,0,0,0,5,5,0,0,0,0,12,5,0,0,0,0,0,0,0,6,0,0,0,0,14,0] >;
C42.100D4 in GAP, Magma, Sage, TeX
C_4^2._{100}D_4 % in TeX
G:=Group("C4^2.100D4"); // GroupNames label
G:=SmallGroup(128,536);
// by ID
G=gap.SmallGroup(128,536);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,512,422,2804,718,172]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^4=1,d^2=c*b*c^-1=b^-1,a*b=b*a,c*a*c^-1=d*a*d^-1=a^-1*b^2,b*d=d*b,d*c*d^-1=b^-1*c^-1>;
// generators/relations