p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.364D4, C42.718C23, (C2×C8)⋊13Q8, C4○(C8⋊3Q8), C4○(C8⋊2Q8), C8.23(C2×Q8), C8⋊3Q8⋊29C2, C8⋊2Q8⋊35C2, C4.57(C4⋊Q8), C4○(C8.5Q8), C4.24(C4○D8), C4⋊C4.99C23, C8.5Q8⋊26C2, C22.4(C4⋊Q8), C4.11(C22×Q8), (C2×C8).597C23, (C4×C8).410C22, (C2×C4).358C24, C23.391(C2×D4), (C22×C4).617D4, C4⋊Q8.284C22, C4.Q8.159C22, C2.D8.179C22, (C22×C8).559C22, C23.25D4.7C2, C22.618(C22×D4), (C2×C42).1133C22, (C22×C4).1567C23, C42.C2.115C22, C42⋊C2.144C22, C23.37C23.31C2, (C2×C4×C8).48C2, C2.28(C2×C4⋊Q8), (C2×C4)○(C8⋊2Q8), (C2×C4)○(C8⋊3Q8), C2.33(C2×C4○D8), (C2×C4).862(C2×D4), (C2×C4)○(C8.5Q8), (C2×C4).247(C2×Q8), SmallGroup(128,1892)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 276 in 178 conjugacy classes, 112 normal (14 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×8], C4 [×10], C22, C22 [×2], C22 [×2], C8 [×8], C2×C4 [×2], C2×C4 [×8], C2×C4 [×12], Q8 [×8], C23, C42 [×2], C42 [×2], C42 [×4], C22⋊C4 [×4], C4⋊C4 [×8], C4⋊C4 [×12], C2×C8 [×12], C22×C4, C22×C4 [×2], C2×Q8 [×4], C4×C8 [×2], C4×C8 [×2], C4.Q8 [×8], C2.D8 [×8], C2×C42, C42⋊C2 [×4], C4×Q8 [×4], C22⋊Q8 [×4], C42.C2 [×4], C4⋊Q8 [×4], C22×C8 [×2], C2×C4×C8, C23.25D4 [×4], C8⋊3Q8 [×2], C8.5Q8 [×4], C8⋊2Q8 [×2], C23.37C23 [×2], C42.364D4
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], Q8 [×8], C23 [×15], C2×D4 [×6], C2×Q8 [×12], C24, C4⋊Q8 [×4], C4○D8 [×4], C22×D4, C22×Q8 [×2], C2×C4⋊Q8, C2×C4○D8 [×2], C42.364D4
Generators and relations
G = < a,b,c,d | a4=b4=1, c4=b2, d2=a2b2, ab=ba, ac=ca, dad-1=a-1, bc=cb, bd=db, dcd-1=c3 >
►On 64 points
(1 37 23 28)(2 38 24 29)(3 39 17 30)(4 40 18 31)(5 33 19 32)(6 34 20 25)(7 35 21 26)(8 36 22 27)(9 46 62 51)(10 47 63 52)(11 48 64 53)(12 41 57 54)(13 42 58 55)(14 43 59 56)(15 44 60 49)(16 45 61 50)
(1 7 5 3)(2 8 6 4)(9 11 13 15)(10 12 14 16)(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 43 45 47)(42 44 46 48)(49 51 53 55)(50 52 54 56)(57 59 61 63)(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 54 19 45)(2 49 20 48)(3 52 21 43)(4 55 22 46)(5 50 23 41)(6 53 24 44)(7 56 17 47)(8 51 18 42)(9 40 58 27)(10 35 59 30)(11 38 60 25)(12 33 61 28)(13 36 62 31)(14 39 63 26)(15 34 64 29)(16 37 57 32)
G:=sub<Sym(64)| (1,37,23,28)(2,38,24,29)(3,39,17,30)(4,40,18,31)(5,33,19,32)(6,34,20,25)(7,35,21,26)(8,36,22,27)(9,46,62,51)(10,47,63,52)(11,48,64,53)(12,41,57,54)(13,42,58,55)(14,43,59,56)(15,44,60,49)(16,45,61,50), (1,7,5,3)(2,8,6,4)(9,11,13,15)(10,12,14,16)(17,23,21,19)(18,24,22,20)(25,31,29,27)(26,32,30,28)(33,39,37,35)(34,40,38,36)(41,43,45,47)(42,44,46,48)(49,51,53,55)(50,52,54,56)(57,59,61,63)(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), (1,54,19,45)(2,49,20,48)(3,52,21,43)(4,55,22,46)(5,50,23,41)(6,53,24,44)(7,56,17,47)(8,51,18,42)(9,40,58,27)(10,35,59,30)(11,38,60,25)(12,33,61,28)(13,36,62,31)(14,39,63,26)(15,34,64,29)(16,37,57,32)>;
G:=Group( (1,37,23,28)(2,38,24,29)(3,39,17,30)(4,40,18,31)(5,33,19,32)(6,34,20,25)(7,35,21,26)(8,36,22,27)(9,46,62,51)(10,47,63,52)(11,48,64,53)(12,41,57,54)(13,42,58,55)(14,43,59,56)(15,44,60,49)(16,45,61,50), (1,7,5,3)(2,8,6,4)(9,11,13,15)(10,12,14,16)(17,23,21,19)(18,24,22,20)(25,31,29,27)(26,32,30,28)(33,39,37,35)(34,40,38,36)(41,43,45,47)(42,44,46,48)(49,51,53,55)(50,52,54,56)(57,59,61,63)(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), (1,54,19,45)(2,49,20,48)(3,52,21,43)(4,55,22,46)(5,50,23,41)(6,53,24,44)(7,56,17,47)(8,51,18,42)(9,40,58,27)(10,35,59,30)(11,38,60,25)(12,33,61,28)(13,36,62,31)(14,39,63,26)(15,34,64,29)(16,37,57,32) );
G=PermutationGroup([(1,37,23,28),(2,38,24,29),(3,39,17,30),(4,40,18,31),(5,33,19,32),(6,34,20,25),(7,35,21,26),(8,36,22,27),(9,46,62,51),(10,47,63,52),(11,48,64,53),(12,41,57,54),(13,42,58,55),(14,43,59,56),(15,44,60,49),(16,45,61,50)], [(1,7,5,3),(2,8,6,4),(9,11,13,15),(10,12,14,16),(17,23,21,19),(18,24,22,20),(25,31,29,27),(26,32,30,28),(33,39,37,35),(34,40,38,36),(41,43,45,47),(42,44,46,48),(49,51,53,55),(50,52,54,56),(57,59,61,63),(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)], [(1,54,19,45),(2,49,20,48),(3,52,21,43),(4,55,22,46),(5,50,23,41),(6,53,24,44),(7,56,17,47),(8,51,18,42),(9,40,58,27),(10,35,59,30),(11,38,60,25),(12,33,61,28),(13,36,62,31),(14,39,63,26),(15,34,64,29),(16,37,57,32)])
Matrix representation ►G ⊆ GL4(𝔽17) generated by
| 1 | 1 | 0 | 0 |
| 15 | 16 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 |
| 16 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 4 |
| 16 | 16 | 0 | 0 |
| 2 | 1 | 0 | 0 |
| 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 15 |
| 9 | 10 | 0 | 0 |
| 2 | 8 | 0 | 0 |
| 0 | 0 | 0 | 15 |
| 0 | 0 | 9 | 0 |
G:=sub<GL(4,GF(17))| [1,15,0,0,1,16,0,0,0,0,16,0,0,0,0,16],[16,0,0,0,0,16,0,0,0,0,4,0,0,0,0,4],[16,2,0,0,16,1,0,0,0,0,9,0,0,0,0,15],[9,2,0,0,10,8,0,0,0,0,0,9,0,0,15,0] >;
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 | 8 | ··· | 8 | 2 | ··· | 2 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | - | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | Q8 | D4 | C4○D8 |
| kernel | C42.364D4 | C2×C4×C8 | C23.25D4 | C8⋊3Q8 | C8.5Q8 | C8⋊2Q8 | C23.37C23 | C42 | C2×C8 | C22×C4 | C4 |
| # reps | 1 | 1 | 4 | 2 | 4 | 2 | 2 | 2 | 8 | 2 | 16 |
In GAP, Magma, Sage, TeX
C_4^2._{364}D_4 % in TeX
G:=Group("C4^2.364D4"); // GroupNames label
G:=SmallGroup(128,1892);
// by ID
G=gap.SmallGroup(128,1892);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,112,253,120,758,184,248,4037,124]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^4=b^2,d^2=a^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=c^3>;
// generators/relations