p-group, metabelian, nilpotent (class 2), monomial
Aliases: C42.95D4, C24.52(C2×C4), (C22×C4).42Q8, C23.25(C4⋊C4), (C22×C4).264D4, (C22×C8).9C22, C4.180(C4⋊D4), C22.46(C8○D4), C4.109(C22⋊Q8), C23.307(C22×C4), C22.7C42⋊3C2, (C2×C42).244C22, (C23×C4).235C22, C2.7(C23.7Q8), (C22×C4).1618C23, C22.78(C42⋊C2), C2.6(C42.7C22), C2.10(C42.6C22), (C2×C4⋊C8)⋊10C2, (C2×C4⋊C4).50C4, (C2×C4).41(C4⋊C4), C22.89(C2×C4⋊C4), (C2×C4).336(C2×Q8), (C2×C4).1514(C2×D4), (C2×C22⋊C4).36C4, (C2×C22⋊C8).14C2, (C2×C4).926(C4○D4), (C22×C4).112(C2×C4), (C2×C4).122(C22⋊C4), (C2×C42⋊C2).13C2, C22.243(C2×C22⋊C4), C2.25((C22×C8)⋊C2), SmallGroup(128,530)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.95D4
G = < a,b,c,d | a4=b4=1, c4=b2, d2=b, ab=ba, cac-1=a-1b2, dad-1=a-1, bc=cb, bd=db, dcd-1=b-1c3 >
Subgroups: 268 in 156 conjugacy classes, 68 normal (16 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, C23, C23, C23, C42, C42, C22⋊C4, C4⋊C4, C2×C8, C22×C4, C22×C4, C22×C4, C24, C22⋊C8, C4⋊C8, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C42⋊C2, C22×C8, C23×C4, C22.7C42, C2×C22⋊C8, C2×C4⋊C8, C2×C42⋊C2, C42.95D4
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, C8○D4, C23.7Q8, (C22×C8)⋊C2, C42.6C22, C42.7C22, C42.95D4
►On 64 points
(1 21 30 51)(2 56 31 18)(3 23 32 53)(4 50 25 20)(5 17 26 55)(6 52 27 22)(7 19 28 49)(8 54 29 24)(9 35 43 59)(10 64 44 40)(11 37 45 61)(12 58 46 34)(13 39 47 63)(14 60 48 36)(15 33 41 57)(16 62 42 38)
(1 11 5 15)(2 12 6 16)(3 13 7 9)(4 14 8 10)(17 33 21 37)(18 34 22 38)(19 35 23 39)(20 36 24 40)(25 48 29 44)(26 41 30 45)(27 42 31 46)(28 43 32 47)(49 59 53 63)(50 60 54 64)(51 61 55 57)(52 62 56 58)
(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 10 11 4 5 14 15 8)(2 3 12 13 6 7 16 9)(17 36 33 24 21 40 37 20)(18 23 34 39 22 19 38 35)(25 26 48 41 29 30 44 45)(27 28 42 43 31 32 46 47)(49 62 59 56 53 58 63 52)(50 55 60 57 54 51 64 61)
G:=sub<Sym(64)| (1,21,30,51)(2,56,31,18)(3,23,32,53)(4,50,25,20)(5,17,26,55)(6,52,27,22)(7,19,28,49)(8,54,29,24)(9,35,43,59)(10,64,44,40)(11,37,45,61)(12,58,46,34)(13,39,47,63)(14,60,48,36)(15,33,41,57)(16,62,42,38), (1,11,5,15)(2,12,6,16)(3,13,7,9)(4,14,8,10)(17,33,21,37)(18,34,22,38)(19,35,23,39)(20,36,24,40)(25,48,29,44)(26,41,30,45)(27,42,31,46)(28,43,32,47)(49,59,53,63)(50,60,54,64)(51,61,55,57)(52,62,56,58), (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), (1,10,11,4,5,14,15,8)(2,3,12,13,6,7,16,9)(17,36,33,24,21,40,37,20)(18,23,34,39,22,19,38,35)(25,26,48,41,29,30,44,45)(27,28,42,43,31,32,46,47)(49,62,59,56,53,58,63,52)(50,55,60,57,54,51,64,61)>;
G:=Group( (1,21,30,51)(2,56,31,18)(3,23,32,53)(4,50,25,20)(5,17,26,55)(6,52,27,22)(7,19,28,49)(8,54,29,24)(9,35,43,59)(10,64,44,40)(11,37,45,61)(12,58,46,34)(13,39,47,63)(14,60,48,36)(15,33,41,57)(16,62,42,38), (1,11,5,15)(2,12,6,16)(3,13,7,9)(4,14,8,10)(17,33,21,37)(18,34,22,38)(19,35,23,39)(20,36,24,40)(25,48,29,44)(26,41,30,45)(27,42,31,46)(28,43,32,47)(49,59,53,63)(50,60,54,64)(51,61,55,57)(52,62,56,58), (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), (1,10,11,4,5,14,15,8)(2,3,12,13,6,7,16,9)(17,36,33,24,21,40,37,20)(18,23,34,39,22,19,38,35)(25,26,48,41,29,30,44,45)(27,28,42,43,31,32,46,47)(49,62,59,56,53,58,63,52)(50,55,60,57,54,51,64,61) );
G=PermutationGroup([[(1,21,30,51),(2,56,31,18),(3,23,32,53),(4,50,25,20),(5,17,26,55),(6,52,27,22),(7,19,28,49),(8,54,29,24),(9,35,43,59),(10,64,44,40),(11,37,45,61),(12,58,46,34),(13,39,47,63),(14,60,48,36),(15,33,41,57),(16,62,42,38)], [(1,11,5,15),(2,12,6,16),(3,13,7,9),(4,14,8,10),(17,33,21,37),(18,34,22,38),(19,35,23,39),(20,36,24,40),(25,48,29,44),(26,41,30,45),(27,42,31,46),(28,43,32,47),(49,59,53,63),(50,60,54,64),(51,61,55,57),(52,62,56,58)], [(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)], [(1,10,11,4,5,14,15,8),(2,3,12,13,6,7,16,9),(17,36,33,24,21,40,37,20),(18,23,34,39,22,19,38,35),(25,26,48,41,29,30,44,45),(27,28,42,43,31,32,46,47),(49,62,59,56,53,58,63,52),(50,55,60,57,54,51,64,61)]])
44 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 4A | ··· | 4H | 4I | ··· | 4R | 8A | ··· | 8P |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
| size | 1 | 1 | ··· | 1 | 4 | 4 | 1 | ··· | 1 | 4 | ··· | 4 | 4 | ··· | 4 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | - | ||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | D4 | D4 | Q8 | C4○D4 | C8○D4 |
| kernel | C42.95D4 | C22.7C42 | C2×C22⋊C8 | C2×C4⋊C8 | C2×C42⋊C2 | C2×C22⋊C4 | C2×C4⋊C4 | C42 | C22×C4 | C22×C4 | C2×C4 | C22 |
| # reps | 1 | 2 | 2 | 2 | 1 | 4 | 4 | 4 | 2 | 2 | 4 | 16 |
Matrix representation of C42.95D4 ►in GL6(𝔽17)
| 13 | 11 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 15 |
| 0 | 0 | 0 | 0 | 0 | 13 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 |
| 10 | 8 | 0 | 0 | 0 | 0 |
| 15 | 7 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 15 | 0 | 0 |
| 0 | 0 | 16 | 13 | 0 | 0 |
| 0 | 0 | 0 | 0 | 15 | 0 |
| 0 | 0 | 0 | 0 | 0 | 15 |
| 7 | 10 | 0 | 0 | 0 | 0 |
| 2 | 10 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 15 | 0 | 0 |
| 0 | 0 | 0 | 13 | 0 | 0 |
| 0 | 0 | 0 | 0 | 15 | 0 |
| 0 | 0 | 0 | 0 | 9 | 2 |
G:=sub<GL(6,GF(17))| [13,0,0,0,0,0,11,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,15,13],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[10,15,0,0,0,0,8,7,0,0,0,0,0,0,4,16,0,0,0,0,15,13,0,0,0,0,0,0,15,0,0,0,0,0,0,15],[7,2,0,0,0,0,10,10,0,0,0,0,0,0,4,0,0,0,0,0,15,13,0,0,0,0,0,0,15,9,0,0,0,0,0,2] >;
C42.95D4 in GAP, Magma, Sage, TeX
C_4^2._{95}D_4 % in TeX
G:=Group("C4^2.95D4"); // GroupNames label
G:=SmallGroup(128,530);
// by ID
G=gap.SmallGroup(128,530);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,64,422,723,124]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^4=b^2,d^2=b,a*b=b*a,c*a*c^-1=a^-1*b^2,d*a*d^-1=a^-1,b*c=c*b,b*d=d*b,d*c*d^-1=b^-1*c^3>;
// generators/relations