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, C23.392(C2×D4), (C22×C4).472D4, C4⋊Q8.286C22, C8⋊C4.121C22, C4.Q8.132C22, C2.D8.216C22, (C22×C8).275C22, (C2×C42).859C22, 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
Generators and relations for C42.252D4
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 >
Subgroups: 276 in 174 conjugacy classes, 108 normal (10 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, Q8, C23, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C22×C4, C22×C4, C2×Q8, C8⋊C4, C4.Q8, C2.D8, C2×C42, C42⋊C2, C4×Q8, C22⋊Q8, C42.C2, C4⋊Q8, C22×C8, C2×C8⋊C4, C23.25D4, C8⋊Q8, C23.37C23, C42.252D4
Quotients: C1, C2, C22, D4, Q8, C23, C2×D4, C2×Q8, C24, C4⋊Q8, C22×D4, C22×Q8, C2×C4⋊Q8, D8⋊C22, C42.252D4
►On 64 points
(1 33 22 28)(2 38 23 25)(3 35 24 30)(4 40 17 27)(5 37 18 32)(6 34 19 29)(7 39 20 26)(8 36 21 31)(9 44 60 54)(10 41 61 51)(11 46 62 56)(12 43 63 53)(13 48 64 50)(14 45 57 55)(15 42 58 52)(16 47 59 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 47)(2 56 19 42)(3 51 20 45)(4 54 21 48)(5 49 22 43)(6 52 23 46)(7 55 24 41)(8 50 17 44)(9 40 64 31)(10 35 57 26)(11 38 58 29)(12 33 59 32)(13 36 60 27)(14 39 61 30)(15 34 62 25)(16 37 63 28)
G:=sub<Sym(64)| (1,33,22,28)(2,38,23,25)(3,35,24,30)(4,40,17,27)(5,37,18,32)(6,34,19,29)(7,39,20,26)(8,36,21,31)(9,44,60,54)(10,41,61,51)(11,46,62,56)(12,43,63,53)(13,48,64,50)(14,45,57,55)(15,42,58,52)(16,47,59,49), (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), (1,53,18,47)(2,56,19,42)(3,51,20,45)(4,54,21,48)(5,49,22,43)(6,52,23,46)(7,55,24,41)(8,50,17,44)(9,40,64,31)(10,35,57,26)(11,38,58,29)(12,33,59,32)(13,36,60,27)(14,39,61,30)(15,34,62,25)(16,37,63,28)>;
G:=Group( (1,33,22,28)(2,38,23,25)(3,35,24,30)(4,40,17,27)(5,37,18,32)(6,34,19,29)(7,39,20,26)(8,36,21,31)(9,44,60,54)(10,41,61,51)(11,46,62,56)(12,43,63,53)(13,48,64,50)(14,45,57,55)(15,42,58,52)(16,47,59,49), (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), (1,53,18,47)(2,56,19,42)(3,51,20,45)(4,54,21,48)(5,49,22,43)(6,52,23,46)(7,55,24,41)(8,50,17,44)(9,40,64,31)(10,35,57,26)(11,38,58,29)(12,33,59,32)(13,36,60,27)(14,39,61,30)(15,34,62,25)(16,37,63,28) );
G=PermutationGroup([[(1,33,22,28),(2,38,23,25),(3,35,24,30),(4,40,17,27),(5,37,18,32),(6,34,19,29),(7,39,20,26),(8,36,21,31),(9,44,60,54),(10,41,61,51),(11,46,62,56),(12,43,63,53),(13,48,64,50),(14,45,57,55),(15,42,58,52),(16,47,59,49)], [(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)], [(1,53,18,47),(2,56,19,42),(3,51,20,45),(4,54,21,48),(5,49,22,43),(6,52,23,46),(7,55,24,41),(8,50,17,44),(9,40,64,31),(10,35,57,26),(11,38,58,29),(12,33,59,32),(13,36,60,27),(14,39,61,30),(15,34,62,25),(16,37,63,28)]])
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 |
Matrix representation of C42.252D4 ►in 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] >;
C42.252D4 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