p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.394D4, (C2×Q8)⋊3C8, C4.8(C22⋊C8), (C2×C42).16C4, (C2×C4).14M4(2), (C22×Q8).16C4, C22.11(C22×C8), C4.22(C4.10D4), C22⋊C8.156C22, (C22×C4).425C23, (C2×C42).147C22, C23.162(C22×C4), C22.14(C2×M4(2)), C42.12C4.13C2, C4○3(C22.M4(2)), C2.3(C23.C23), C22.M4(2).12C2, (C2×C4×Q8).1C2, (C2×C4).4(C2×C8), (C2×C4⋊C4).31C4, C2.9(C2×C22⋊C8), (C2×C4).1122(C2×D4), (C22×C4).42(C2×C4), C2.2(C2×C4.10D4), (C2×C4⋊C4).734C22, C22.93(C2×C22⋊C4), (C2×C4).311(C22⋊C4), SmallGroup(128,193)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.394D4
G = < a,b,c,d | a4=b4=1, c4=b2, d2=b, ab=ba, ac=ca, ad=da, cbc-1=a2b-1, bd=db, dcd-1=a2bc3 >
Subgroups: 204 in 124 conjugacy classes, 62 normal (18 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, C2×C4, Q8, C23, C42, C42, C4⋊C4, C2×C8, C22×C4, C22×C4, C2×Q8, C2×Q8, C4×C8, C22⋊C8, C4⋊C8, C2×C42, C2×C42, C2×C4⋊C4, C2×C4⋊C4, C4×Q8, C22×Q8, C22.M4(2), C42.12C4, C2×C4×Q8, C42.394D4
Quotients: C1, C2, C4, C22, C8, C2×C4, D4, C23, C22⋊C4, C2×C8, M4(2), C22×C4, C2×D4, C22⋊C8, C4.10D4, C2×C22⋊C4, C22×C8, C2×M4(2), C2×C22⋊C8, C23.C23, C2×C4.10D4, C42.394D4
►On 64 points
(1 19 57 55)(2 20 58 56)(3 21 59 49)(4 22 60 50)(5 23 61 51)(6 24 62 52)(7 17 63 53)(8 18 64 54)(9 41 26 36)(10 42 27 37)(11 43 28 38)(12 44 29 39)(13 45 30 40)(14 46 31 33)(15 47 32 34)(16 48 25 35)
(1 63 5 59)(2 4 6 8)(3 57 7 61)(9 32 13 28)(10 12 14 16)(11 26 15 30)(17 51 21 55)(18 20 22 24)(19 53 23 49)(25 27 29 31)(33 35 37 39)(34 45 38 41)(36 47 40 43)(42 44 46 48)(50 52 54 56)(58 60 62 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 39 63 33 5 35 59 37)(2 40 4 43 6 36 8 47)(3 42 57 44 7 46 61 48)(9 18 32 20 13 22 28 24)(10 19 12 53 14 23 16 49)(11 52 26 54 15 56 30 50)(17 31 51 25 21 27 55 29)(34 58 45 60 38 62 41 64)
G:=sub<Sym(64)| (1,19,57,55)(2,20,58,56)(3,21,59,49)(4,22,60,50)(5,23,61,51)(6,24,62,52)(7,17,63,53)(8,18,64,54)(9,41,26,36)(10,42,27,37)(11,43,28,38)(12,44,29,39)(13,45,30,40)(14,46,31,33)(15,47,32,34)(16,48,25,35), (1,63,5,59)(2,4,6,8)(3,57,7,61)(9,32,13,28)(10,12,14,16)(11,26,15,30)(17,51,21,55)(18,20,22,24)(19,53,23,49)(25,27,29,31)(33,35,37,39)(34,45,38,41)(36,47,40,43)(42,44,46,48)(50,52,54,56)(58,60,62,64), (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), (1,39,63,33,5,35,59,37)(2,40,4,43,6,36,8,47)(3,42,57,44,7,46,61,48)(9,18,32,20,13,22,28,24)(10,19,12,53,14,23,16,49)(11,52,26,54,15,56,30,50)(17,31,51,25,21,27,55,29)(34,58,45,60,38,62,41,64)>;
G:=Group( (1,19,57,55)(2,20,58,56)(3,21,59,49)(4,22,60,50)(5,23,61,51)(6,24,62,52)(7,17,63,53)(8,18,64,54)(9,41,26,36)(10,42,27,37)(11,43,28,38)(12,44,29,39)(13,45,30,40)(14,46,31,33)(15,47,32,34)(16,48,25,35), (1,63,5,59)(2,4,6,8)(3,57,7,61)(9,32,13,28)(10,12,14,16)(11,26,15,30)(17,51,21,55)(18,20,22,24)(19,53,23,49)(25,27,29,31)(33,35,37,39)(34,45,38,41)(36,47,40,43)(42,44,46,48)(50,52,54,56)(58,60,62,64), (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), (1,39,63,33,5,35,59,37)(2,40,4,43,6,36,8,47)(3,42,57,44,7,46,61,48)(9,18,32,20,13,22,28,24)(10,19,12,53,14,23,16,49)(11,52,26,54,15,56,30,50)(17,31,51,25,21,27,55,29)(34,58,45,60,38,62,41,64) );
G=PermutationGroup([[(1,19,57,55),(2,20,58,56),(3,21,59,49),(4,22,60,50),(5,23,61,51),(6,24,62,52),(7,17,63,53),(8,18,64,54),(9,41,26,36),(10,42,27,37),(11,43,28,38),(12,44,29,39),(13,45,30,40),(14,46,31,33),(15,47,32,34),(16,48,25,35)], [(1,63,5,59),(2,4,6,8),(3,57,7,61),(9,32,13,28),(10,12,14,16),(11,26,15,30),(17,51,21,55),(18,20,22,24),(19,53,23,49),(25,27,29,31),(33,35,37,39),(34,45,38,41),(36,47,40,43),(42,44,46,48),(50,52,54,56),(58,60,62,64)], [(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)], [(1,39,63,33,5,35,59,37),(2,40,4,43,6,36,8,47),(3,42,57,44,7,46,61,48),(9,18,32,20,13,22,28,24),(10,19,12,53,14,23,16,49),(11,52,26,54,15,56,30,50),(17,31,51,25,21,27,55,29),(34,58,45,60,38,62,41,64)]])
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | ··· | 4N | 4O | ··· | 4V | 8A | ··· | 8P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | - | ||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C4 | C8 | D4 | M4(2) | C4.10D4 | C23.C23 |
| kernel | C42.394D4 | C22.M4(2) | C42.12C4 | C2×C4×Q8 | C2×C42 | C2×C4⋊C4 | C22×Q8 | C2×Q8 | C42 | C2×C4 | C4 | C2 |
| # reps | 1 | 4 | 2 | 1 | 4 | 2 | 2 | 16 | 4 | 4 | 2 | 2 |
Matrix representation of C42.394D4 ►in GL6(𝔽17)
| 13 | 0 | 0 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 11 | 7 | 1 | 15 |
| 0 | 0 | 2 | 9 | 1 | 16 |
| 0 | 8 | 0 | 0 | 0 | 0 |
| 8 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 10 |
| 0 | 0 | 14 | 13 | 7 | 10 |
| 0 | 0 | 3 | 10 | 4 | 11 |
| 0 | 0 | 5 | 5 | 0 | 1 |
| 0 | 9 | 0 | 0 | 0 | 0 |
| 8 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 14 | 1 | 0 |
| 0 | 0 | 14 | 14 | 1 | 15 |
| 0 | 0 | 10 | 13 | 13 | 11 |
| 0 | 0 | 8 | 7 | 12 | 0 |
G:=sub<GL(6,GF(17))| [13,0,0,0,0,0,0,13,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,16,11,2,0,0,1,0,7,9,0,0,0,0,1,1,0,0,0,0,15,16],[0,8,0,0,0,0,8,0,0,0,0,0,0,0,16,14,3,5,0,0,0,13,10,5,0,0,0,7,4,0,0,0,10,10,11,1],[0,8,0,0,0,0,9,0,0,0,0,0,0,0,7,14,10,8,0,0,14,14,13,7,0,0,1,1,13,12,0,0,0,15,11,0] >;
C42.394D4 in GAP, Magma, Sage, TeX
C_4^2._{394}D_4 % in TeX
G:=Group("C4^2.394D4"); // GroupNames label
G:=SmallGroup(128,193);
// by ID
G=gap.SmallGroup(128,193);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,232,520,1123,851,172]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^4=b^2,d^2=b,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=a^2*b^-1,b*d=d*b,d*c*d^-1=a^2*b*c^3>;
// generators/relations