p-group, metabelian, nilpotent (class 3), monomial
Aliases: C4⋊C4.84D4, (C2×D4).10Q8, C2.8(D4.Q8), (C22×C4).143D4, C23.907(C2×D4), C4.31(C22⋊Q8), C2.30(D4⋊D4), C22.4Q16⋊20C2, C4.32(C4.4D4), (C22×C8).67C22, C2.6(C23⋊Q8), C22.209C22≀C2, C22.105(C4○D8), (C2×C42).355C22, (C22×D4).73C22, C22.134(C8⋊C22), C22.7C42⋊10C2, (C22×C4).1441C23, C22.88(C4.4D4), C22.104(C22⋊Q8), C24.3C22.14C2, C2.6(C42.78C22), C2.5(C42.29C22), (C2×C4).281(C2×Q8), (C2×C42.C2)⋊1C2, (C2×C4).1031(C2×D4), (C2×D4⋊C4).12C2, (C2×C4).618(C4○D4), (C2×C4⋊C4).114C22, SmallGroup(128,757)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C4⋊C4.84D4
G = < a,b,c,d | a4=b4=c4=1, d2=a2b2, bab-1=cac-1=a-1, ad=da, cbc-1=ab, dbd-1=b-1, dcd-1=a2b2c-1 >
Subgroups: 352 in 147 conjugacy classes, 50 normal (18 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, C23, C23, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C24, D4⋊C4, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C2×C4⋊C4, C42.C2, C22×C8, C22×D4, C22.7C42, C22.4Q16, C24.3C22, C2×D4⋊C4, C2×C42.C2, C4⋊C4.84D4
Quotients: C1, C2, C22, D4, Q8, C23, C2×D4, C2×Q8, C4○D4, C22≀C2, C22⋊Q8, C4.4D4, C4○D8, C8⋊C22, C23⋊Q8, D4⋊D4, D4.Q8, C42.78C22, C42.29C22, C4⋊C4.84D4
►On 64 points
(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 36 6 42)(2 35 7 41)(3 34 8 44)(4 33 5 43)(9 54 21 47)(10 53 22 46)(11 56 23 45)(12 55 24 48)(13 37 64 32)(14 40 61 31)(15 39 62 30)(16 38 63 29)(17 60 26 50)(18 59 27 49)(19 58 28 52)(20 57 25 51)
(1 32 12 28)(2 31 9 27)(3 30 10 26)(4 29 11 25)(5 38 23 20)(6 37 24 19)(7 40 21 18)(8 39 22 17)(13 54 52 35)(14 53 49 34)(15 56 50 33)(16 55 51 36)(41 64 47 58)(42 63 48 57)(43 62 45 60)(44 61 46 59)
(1 17 8 28)(2 18 5 25)(3 19 6 26)(4 20 7 27)(9 40 23 29)(10 37 24 30)(11 38 21 31)(12 39 22 32)(13 48 62 53)(14 45 63 54)(15 46 64 55)(16 47 61 56)(33 51 41 59)(34 52 42 60)(35 49 43 57)(36 50 44 58)
G:=sub<Sym(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), (1,36,6,42)(2,35,7,41)(3,34,8,44)(4,33,5,43)(9,54,21,47)(10,53,22,46)(11,56,23,45)(12,55,24,48)(13,37,64,32)(14,40,61,31)(15,39,62,30)(16,38,63,29)(17,60,26,50)(18,59,27,49)(19,58,28,52)(20,57,25,51), (1,32,12,28)(2,31,9,27)(3,30,10,26)(4,29,11,25)(5,38,23,20)(6,37,24,19)(7,40,21,18)(8,39,22,17)(13,54,52,35)(14,53,49,34)(15,56,50,33)(16,55,51,36)(41,64,47,58)(42,63,48,57)(43,62,45,60)(44,61,46,59), (1,17,8,28)(2,18,5,25)(3,19,6,26)(4,20,7,27)(9,40,23,29)(10,37,24,30)(11,38,21,31)(12,39,22,32)(13,48,62,53)(14,45,63,54)(15,46,64,55)(16,47,61,56)(33,51,41,59)(34,52,42,60)(35,49,43,57)(36,50,44,58)>;
G:=Group( (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), (1,36,6,42)(2,35,7,41)(3,34,8,44)(4,33,5,43)(9,54,21,47)(10,53,22,46)(11,56,23,45)(12,55,24,48)(13,37,64,32)(14,40,61,31)(15,39,62,30)(16,38,63,29)(17,60,26,50)(18,59,27,49)(19,58,28,52)(20,57,25,51), (1,32,12,28)(2,31,9,27)(3,30,10,26)(4,29,11,25)(5,38,23,20)(6,37,24,19)(7,40,21,18)(8,39,22,17)(13,54,52,35)(14,53,49,34)(15,56,50,33)(16,55,51,36)(41,64,47,58)(42,63,48,57)(43,62,45,60)(44,61,46,59), (1,17,8,28)(2,18,5,25)(3,19,6,26)(4,20,7,27)(9,40,23,29)(10,37,24,30)(11,38,21,31)(12,39,22,32)(13,48,62,53)(14,45,63,54)(15,46,64,55)(16,47,61,56)(33,51,41,59)(34,52,42,60)(35,49,43,57)(36,50,44,58) );
G=PermutationGroup([[(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)], [(1,36,6,42),(2,35,7,41),(3,34,8,44),(4,33,5,43),(9,54,21,47),(10,53,22,46),(11,56,23,45),(12,55,24,48),(13,37,64,32),(14,40,61,31),(15,39,62,30),(16,38,63,29),(17,60,26,50),(18,59,27,49),(19,58,28,52),(20,57,25,51)], [(1,32,12,28),(2,31,9,27),(3,30,10,26),(4,29,11,25),(5,38,23,20),(6,37,24,19),(7,40,21,18),(8,39,22,17),(13,54,52,35),(14,53,49,34),(15,56,50,33),(16,55,51,36),(41,64,47,58),(42,63,48,57),(43,62,45,60),(44,61,46,59)], [(1,17,8,28),(2,18,5,25),(3,19,6,26),(4,20,7,27),(9,40,23,29),(10,37,24,30),(11,38,21,31),(12,39,22,32),(13,48,62,53),(14,45,63,54),(15,46,64,55),(16,47,61,56),(33,51,41,59),(34,52,42,60),(35,49,43,57),(36,50,44,58)]])
32 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | ··· | 4N | 8A | ··· | 8H |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
| size | 1 | 1 | ··· | 1 | 8 | 8 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | - | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | Q8 | C4○D4 | C4○D8 | C8⋊C22 |
| kernel | C4⋊C4.84D4 | C22.7C42 | C22.4Q16 | C24.3C22 | C2×D4⋊C4 | C2×C42.C2 | C4⋊C4 | C22×C4 | C2×D4 | C2×C4 | C22 | C22 |
| # reps | 1 | 1 | 2 | 1 | 2 | 1 | 4 | 2 | 2 | 6 | 8 | 2 |
Matrix representation of C4⋊C4.84D4 ►in GL6(𝔽17)
| 16 | 2 | 0 | 0 | 0 | 0 |
| 16 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 10 | 0 | 0 | 0 | 0 |
| 5 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 |
| 4 | 9 | 0 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 15 |
| 0 | 0 | 0 | 0 | 1 | 1 |
| 4 | 9 | 0 | 0 | 0 | 0 |
| 4 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 13 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 15 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(17))| [16,16,0,0,0,0,2,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,5,0,0,0,0,10,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[4,0,0,0,0,0,9,13,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,16,1,0,0,0,0,15,1],[4,4,0,0,0,0,9,13,0,0,0,0,0,0,4,0,0,0,0,0,0,13,0,0,0,0,0,0,16,0,0,0,0,0,15,1] >;
C4⋊C4.84D4 in GAP, Magma, Sage, TeX
C_4\rtimes C_4._{84}D_4 % in TeX
G:=Group("C4:C4.84D4"); // GroupNames label
G:=SmallGroup(128,757);
// by ID
G=gap.SmallGroup(128,757);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,56,141,512,422,387,2804,718,172]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^4=1,d^2=a^2*b^2,b*a*b^-1=c*a*c^-1=a^-1,a*d=d*a,c*b*c^-1=a*b,d*b*d^-1=b^-1,d*c*d^-1=a^2*b^2*c^-1>;
// generators/relations