p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.252D4, C42.380C23, C4○(C8⋊Q8), (C2×C8)⋊3Q8, C8⋊Q8⋊36C2, C8.13(C2×Q8), C4.58(C4⋊Q8), C22.5(C4⋊Q8), C4.13(C22×Q8), C4⋊C4.101C23, (C2×C8).271C23, (C2×C4).360C24, (C22×C4).472D4, C23.392(C2×D4), C4⋊Q8.286C22, C8⋊C4.121C22, C4.Q8.132C22, C2.D8.216C22, (C2×C42).859C22, (C22×C8).275C22, C22.620(C22×D4), C2.40(D8⋊C22), (C22×C4).1569C23, C23.25D4.18C2, C42.C2.117C22, C42⋊C2.145C22, C23.37C23.32C2, (C2×C4)○(C8⋊Q8), C2.30(C2×C4⋊Q8), (C2×C4).521(C2×D4), (C2×C4).249(C2×Q8), (C2×C8⋊C4).12C2, SmallGroup(128,1894)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 276 in 174 conjugacy classes, 108 normal (10 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×4], C4 [×12], C22, C22 [×2], C22 [×2], C8 [×8], C2×C4 [×2], C2×C4 [×8], C2×C4 [×10], Q8 [×8], C23, C42 [×4], C42 [×4], C22⋊C4 [×4], C4⋊C4 [×8], C4⋊C4 [×12], C2×C8 [×12], C22×C4, C22×C4 [×2], C2×Q8 [×4], C8⋊C4 [×4], 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×C8⋊C4, C23.25D4 [×4], C8⋊Q8 [×8], C23.37C23 [×2], C42.252D4
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], Q8 [×8], C23 [×15], C2×D4 [×6], C2×Q8 [×12], C24, C4⋊Q8 [×4], C22×D4, C22×Q8 [×2], C2×C4⋊Q8, D8⋊C22 [×2], C42.252D4
Generators and relations
G = < a,b,c,d | a4=b4=1, c4=b2, d2=a2b2, ab=ba, cac-1=ab2, dad-1=a-1b2, bc=cb, bd=db, dcd-1=c3 >
►On 64 points
(1 33 22 32)(2 38 23 29)(3 35 24 26)(4 40 17 31)(5 37 18 28)(6 34 19 25)(7 39 20 30)(8 36 21 27)(9 48 62 54)(10 45 63 51)(11 42 64 56)(12 47 57 53)(13 44 58 50)(14 41 59 55)(15 46 60 52)(16 43 61 49)
(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 53 18 43)(2 56 19 46)(3 51 20 41)(4 54 21 44)(5 49 22 47)(6 52 23 42)(7 55 24 45)(8 50 17 48)(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)| (1,33,22,32)(2,38,23,29)(3,35,24,26)(4,40,17,31)(5,37,18,28)(6,34,19,25)(7,39,20,30)(8,36,21,27)(9,48,62,54)(10,45,63,51)(11,42,64,56)(12,47,57,53)(13,44,58,50)(14,41,59,55)(15,46,60,52)(16,43,61,49), (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), (1,53,18,43)(2,56,19,46)(3,51,20,41)(4,54,21,44)(5,49,22,47)(6,52,23,42)(7,55,24,45)(8,50,17,48)(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( (1,33,22,32)(2,38,23,29)(3,35,24,26)(4,40,17,31)(5,37,18,28)(6,34,19,25)(7,39,20,30)(8,36,21,27)(9,48,62,54)(10,45,63,51)(11,42,64,56)(12,47,57,53)(13,44,58,50)(14,41,59,55)(15,46,60,52)(16,43,61,49), (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), (1,53,18,43)(2,56,19,46)(3,51,20,41)(4,54,21,44)(5,49,22,47)(6,52,23,42)(7,55,24,45)(8,50,17,48)(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([(1,33,22,32),(2,38,23,29),(3,35,24,26),(4,40,17,31),(5,37,18,28),(6,34,19,25),(7,39,20,30),(8,36,21,27),(9,48,62,54),(10,45,63,51),(11,42,64,56),(12,47,57,53),(13,44,58,50),(14,41,59,55),(15,46,60,52),(16,43,61,49)], [(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)], [(1,53,18,43),(2,56,19,46),(3,51,20,41),(4,54,21,44),(5,49,22,47),(6,52,23,42),(7,55,24,45),(8,50,17,48),(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 ⊆ GL6(𝔽17)
| 0 | 1 | 0 | 0 | 0 | 0 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 15 | 0 | 0 |
| 0 | 0 | 8 | 10 | 0 | 0 |
| 0 | 0 | 0 | 0 | 10 | 2 |
| 0 | 0 | 0 | 0 | 9 | 7 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 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 | 13 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 13 | 0 | 0 | 0 |
G:=sub<GL(6,GF(17))| [0,16,0,0,0,0,1,0,0,0,0,0,0,0,7,8,0,0,0,0,15,10,0,0,0,0,0,0,10,9,0,0,0,0,2,7],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,13],[0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,4,0],[4,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,0,13,0,0,0,0,16,0,0,0,0,16,0,0,0,0,4,0,0,0] >;
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | ··· | 4R | 8A | ··· | 8H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 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 | 8 | ··· | 8 | 4 | ··· | 4 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | - | + | |
| image | C1 | C2 | C2 | C2 | C2 | D4 | Q8 | D4 | D8⋊C22 |
| kernel | C42.252D4 | C2×C8⋊C4 | C23.25D4 | C8⋊Q8 | C23.37C23 | C42 | C2×C8 | C22×C4 | C2 |
| # reps | 1 | 1 | 4 | 8 | 2 | 2 | 8 | 2 | 4 |
In GAP, Magma, Sage, TeX
C_4^2._{252}D_4 % in TeX
G:=Group("C4^2.252D4"); // GroupNames label
G:=SmallGroup(128,1894);
// by ID
G=gap.SmallGroup(128,1894);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,112,253,120,758,723,184,248,4037,124]);
// 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,c*a*c^-1=a*b^2,d*a*d^-1=a^-1*b^2,b*c=c*b,b*d=d*b,d*c*d^-1=c^3>;
// generators/relations