p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.396D4, C4.4(C4.10D4), C22.11(C8○D4), C22⋊C8.126C22, C42.6C4.14C2, C23.171(C22×C4), (C2×C42).152C22, (C22×C4).434C23, C42.12C4.14C2, C2.10(C23.C23), C22.M4(2).11C2, (C2×C4⋊C4).12C4, (C2×C4).1131(C2×D4), (C2×C4⋊C4).10C22, (C22×C4).11(C2×C4), C2.7(C2×C4.10D4), (C2×C42.C2).1C2, (C2×C4).164(C22⋊C4), C2.8((C22×C8)⋊C2), C22.152(C2×C22⋊C4), SmallGroup(128,202)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.396D4
G = < a,b,c,d | a4=b4=1, c4=b2, d2=b, ab=ba, cac-1=ab2, ad=da, cbc-1=a2b, bd=db, dcd-1=a2b-1c3 >
Subgroups: 180 in 102 conjugacy classes, 46 normal (26 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C23, C42, C4⋊C4, C2×C8, C22×C4, C22×C4, C4×C8, C8⋊C4, C22⋊C8, C4⋊C8, C2×C42, C2×C4⋊C4, C2×C4⋊C4, C42.C2, C22.M4(2), C42.12C4, C42.6C4, C2×C42.C2, C42.396D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, C4.10D4, C2×C22⋊C4, C8○D4, (C22×C8)⋊C2, C23.C23, C2×C4.10D4, C42.396D4
►On 64 points
(1 19 63 55)(2 24 64 52)(3 21 57 49)(4 18 58 54)(5 23 59 51)(6 20 60 56)(7 17 61 53)(8 22 62 50)(9 45 30 36)(10 42 31 33)(11 47 32 38)(12 44 25 35)(13 41 26 40)(14 46 27 37)(15 43 28 34)(16 48 29 39)
(1 57 5 61)(2 4 6 8)(3 59 7 63)(9 32 13 28)(10 12 14 16)(11 26 15 30)(17 55 21 51)(18 20 22 24)(19 49 23 53)(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 57 33 5 35 61 37)(2 40 4 43 6 36 8 47)(3 42 59 44 7 46 63 48)(9 22 32 24 13 18 28 20)(10 23 12 53 14 19 16 49)(11 52 26 54 15 56 30 50)(17 27 55 29 21 31 51 25)(34 60 45 62 38 64 41 58)
G:=sub<Sym(64)| (1,19,63,55)(2,24,64,52)(3,21,57,49)(4,18,58,54)(5,23,59,51)(6,20,60,56)(7,17,61,53)(8,22,62,50)(9,45,30,36)(10,42,31,33)(11,47,32,38)(12,44,25,35)(13,41,26,40)(14,46,27,37)(15,43,28,34)(16,48,29,39), (1,57,5,61)(2,4,6,8)(3,59,7,63)(9,32,13,28)(10,12,14,16)(11,26,15,30)(17,55,21,51)(18,20,22,24)(19,49,23,53)(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), (1,39,57,33,5,35,61,37)(2,40,4,43,6,36,8,47)(3,42,59,44,7,46,63,48)(9,22,32,24,13,18,28,20)(10,23,12,53,14,19,16,49)(11,52,26,54,15,56,30,50)(17,27,55,29,21,31,51,25)(34,60,45,62,38,64,41,58)>;
G:=Group( (1,19,63,55)(2,24,64,52)(3,21,57,49)(4,18,58,54)(5,23,59,51)(6,20,60,56)(7,17,61,53)(8,22,62,50)(9,45,30,36)(10,42,31,33)(11,47,32,38)(12,44,25,35)(13,41,26,40)(14,46,27,37)(15,43,28,34)(16,48,29,39), (1,57,5,61)(2,4,6,8)(3,59,7,63)(9,32,13,28)(10,12,14,16)(11,26,15,30)(17,55,21,51)(18,20,22,24)(19,49,23,53)(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), (1,39,57,33,5,35,61,37)(2,40,4,43,6,36,8,47)(3,42,59,44,7,46,63,48)(9,22,32,24,13,18,28,20)(10,23,12,53,14,19,16,49)(11,52,26,54,15,56,30,50)(17,27,55,29,21,31,51,25)(34,60,45,62,38,64,41,58) );
G=PermutationGroup([[(1,19,63,55),(2,24,64,52),(3,21,57,49),(4,18,58,54),(5,23,59,51),(6,20,60,56),(7,17,61,53),(8,22,62,50),(9,45,30,36),(10,42,31,33),(11,47,32,38),(12,44,25,35),(13,41,26,40),(14,46,27,37),(15,43,28,34),(16,48,29,39)], [(1,57,5,61),(2,4,6,8),(3,59,7,63),(9,32,13,28),(10,12,14,16),(11,26,15,30),(17,55,21,51),(18,20,22,24),(19,49,23,53),(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)], [(1,39,57,33,5,35,61,37),(2,40,4,43,6,36,8,47),(3,42,59,44,7,46,63,48),(9,22,32,24,13,18,28,20),(10,23,12,53,14,19,16,49),(11,52,26,54,15,56,30,50),(17,27,55,29,21,31,51,25),(34,60,45,62,38,64,41,58)]])
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | ··· | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 8A | ··· | 8H | 8I | 8J | 8K | 8L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 8 | 8 | 8 | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | ··· | 2 | 4 | 4 | 8 | 8 | 8 | 8 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C4 | D4 | C8○D4 | C4.10D4 | C23.C23 |
| kernel | C42.396D4 | C22.M4(2) | C42.12C4 | C42.6C4 | C2×C42.C2 | C2×C4⋊C4 | C42 | C22 | C4 | C2 |
| # reps | 1 | 4 | 1 | 1 | 1 | 8 | 4 | 8 | 2 | 2 |
Matrix representation of C42.396D4 ►in GL6(𝔽17)
| 13 | 4 | 0 | 0 | 0 | 0 |
| 9 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 15 | 0 | 0 |
| 0 | 0 | 1 | 16 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 1 |
| 0 | 0 | 1 | 16 | 16 | 0 |
| 13 | 0 | 0 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 15 | 0 | 0 |
| 0 | 0 | 1 | 16 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 1 |
| 0 | 0 | 1 | 16 | 16 | 0 |
| 13 | 12 | 0 | 0 | 0 | 0 |
| 16 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 0 | 2 | 14 |
| 0 | 0 | 16 | 0 | 11 | 8 |
| 0 | 0 | 10 | 16 | 1 | 7 |
| 0 | 0 | 0 | 10 | 1 | 7 |
| 2 | 15 | 0 | 0 | 0 | 0 |
| 4 | 15 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 15 | 0 |
| 0 | 0 | 0 | 0 | 16 | 1 |
| 0 | 0 | 0 | 1 | 16 | 0 |
| 0 | 0 | 1 | 0 | 16 | 0 |
G:=sub<GL(6,GF(17))| [13,9,0,0,0,0,4,4,0,0,0,0,0,0,1,1,0,1,0,0,15,16,16,16,0,0,0,0,0,16,0,0,0,0,1,0],[13,0,0,0,0,0,0,13,0,0,0,0,0,0,1,1,0,1,0,0,15,16,16,16,0,0,0,0,0,16,0,0,0,0,1,0],[13,16,0,0,0,0,12,4,0,0,0,0,0,0,9,16,10,0,0,0,0,0,16,10,0,0,2,11,1,1,0,0,14,8,7,7],[2,4,0,0,0,0,15,15,0,0,0,0,0,0,1,0,0,1,0,0,0,0,1,0,0,0,15,16,16,16,0,0,0,1,0,0] >;
C42.396D4 in GAP, Magma, Sage, TeX
C_4^2._{396}D_4 % in TeX
G:=Group("C4^2.396D4"); // GroupNames label
G:=SmallGroup(128,202);
// by ID
G=gap.SmallGroup(128,202);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,232,723,184,1123,851,172]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^4=b^2,d^2=b,a*b=b*a,c*a*c^-1=a*b^2,a*d=d*a,c*b*c^-1=a^2*b,b*d=d*b,d*c*d^-1=a^2*b^-1*c^3>;
// generators/relations