p-group, metabelian, nilpotent (class 2), monomial
Aliases: C42.184D4, C24.37C23, C23.504C24, C22.2072- 1+4, C42⋊4C4⋊27C2, C23⋊Q8.13C2, (C2×C42).591C22, (C22×C4).849C23, C22.334(C22×D4), C23.4Q8.12C2, (C22×Q8).445C22, C23.65C23⋊99C2, C23.83C23⋊55C2, C23.78C23⋊22C2, C2.77(C22.19C24), C24.C22.41C2, C23.63C23⋊107C2, C2.C42.234C22, C2.47(C22.26C24), C2.29(C23.38C23), C2.75(C22.46C24), C2.52(C22.50C24), (C2×C4×Q8)⋊28C2, (C2×C4).1201(C2×D4), (C2×C4).162(C4○D4), (C2×C4⋊C4).343C22, (C2×C42⋊2C2).9C2, C22.380(C2×C4○D4), (C2×C22⋊C4).204C22, SmallGroup(128,1336)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.184D4
G = < a,b,c,d | a4=b4=c4=1, d2=a2, ab=ba, cac-1=ab2, dad-1=a-1b2, bc=cb, dbd-1=a2b, dcd-1=c-1 >
Subgroups: 404 in 235 conjugacy classes, 100 normal (34 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C2×C4, C2×C4, Q8, C23, C23, C42, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×Q8, C24, C2.C42, C2.C42, C2×C42, C2×C42, C2×C22⋊C4, C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C4×Q8, C42⋊2C2, C22×Q8, C42⋊4C4, C23.63C23, C24.C22, C23.65C23, C23⋊Q8, C23.78C23, C23.4Q8, C23.83C23, C2×C4×Q8, C2×C42⋊2C2, C42.184D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C22×D4, C2×C4○D4, 2- 1+4, C22.19C24, C22.26C24, C23.38C23, C22.46C24, C22.50C24, C42.184D4
►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 57 25 17)(2 58 26 18)(3 59 27 19)(4 60 28 20)(5 33 9 50)(6 34 10 51)(7 35 11 52)(8 36 12 49)(13 29 53 47)(14 30 54 48)(15 31 55 45)(16 32 56 46)(21 38 61 43)(22 39 62 44)(23 40 63 41)(24 37 64 42)
(1 21 13 51)(2 62 14 35)(3 23 15 49)(4 64 16 33)(5 20 37 46)(6 57 38 29)(7 18 39 48)(8 59 40 31)(9 60 42 32)(10 17 43 47)(11 58 44 30)(12 19 41 45)(22 54 52 26)(24 56 50 28)(25 61 53 34)(27 63 55 36)
(1 32 3 30)(2 45 4 47)(5 36 7 34)(6 52 8 50)(9 49 11 51)(10 35 12 33)(13 60 15 58)(14 19 16 17)(18 53 20 55)(21 42 23 44)(22 40 24 38)(25 46 27 48)(26 31 28 29)(37 63 39 61)(41 64 43 62)(54 59 56 57)
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), (1,57,25,17)(2,58,26,18)(3,59,27,19)(4,60,28,20)(5,33,9,50)(6,34,10,51)(7,35,11,52)(8,36,12,49)(13,29,53,47)(14,30,54,48)(15,31,55,45)(16,32,56,46)(21,38,61,43)(22,39,62,44)(23,40,63,41)(24,37,64,42), (1,21,13,51)(2,62,14,35)(3,23,15,49)(4,64,16,33)(5,20,37,46)(6,57,38,29)(7,18,39,48)(8,59,40,31)(9,60,42,32)(10,17,43,47)(11,58,44,30)(12,19,41,45)(22,54,52,26)(24,56,50,28)(25,61,53,34)(27,63,55,36), (1,32,3,30)(2,45,4,47)(5,36,7,34)(6,52,8,50)(9,49,11,51)(10,35,12,33)(13,60,15,58)(14,19,16,17)(18,53,20,55)(21,42,23,44)(22,40,24,38)(25,46,27,48)(26,31,28,29)(37,63,39,61)(41,64,43,62)(54,59,56,57)>;
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), (1,57,25,17)(2,58,26,18)(3,59,27,19)(4,60,28,20)(5,33,9,50)(6,34,10,51)(7,35,11,52)(8,36,12,49)(13,29,53,47)(14,30,54,48)(15,31,55,45)(16,32,56,46)(21,38,61,43)(22,39,62,44)(23,40,63,41)(24,37,64,42), (1,21,13,51)(2,62,14,35)(3,23,15,49)(4,64,16,33)(5,20,37,46)(6,57,38,29)(7,18,39,48)(8,59,40,31)(9,60,42,32)(10,17,43,47)(11,58,44,30)(12,19,41,45)(22,54,52,26)(24,56,50,28)(25,61,53,34)(27,63,55,36), (1,32,3,30)(2,45,4,47)(5,36,7,34)(6,52,8,50)(9,49,11,51)(10,35,12,33)(13,60,15,58)(14,19,16,17)(18,53,20,55)(21,42,23,44)(22,40,24,38)(25,46,27,48)(26,31,28,29)(37,63,39,61)(41,64,43,62)(54,59,56,57) );
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)], [(1,57,25,17),(2,58,26,18),(3,59,27,19),(4,60,28,20),(5,33,9,50),(6,34,10,51),(7,35,11,52),(8,36,12,49),(13,29,53,47),(14,30,54,48),(15,31,55,45),(16,32,56,46),(21,38,61,43),(22,39,62,44),(23,40,63,41),(24,37,64,42)], [(1,21,13,51),(2,62,14,35),(3,23,15,49),(4,64,16,33),(5,20,37,46),(6,57,38,29),(7,18,39,48),(8,59,40,31),(9,60,42,32),(10,17,43,47),(11,58,44,30),(12,19,41,45),(22,54,52,26),(24,56,50,28),(25,61,53,34),(27,63,55,36)], [(1,32,3,30),(2,45,4,47),(5,36,7,34),(6,52,8,50),(9,49,11,51),(10,35,12,33),(13,60,15,58),(14,19,16,17),(18,53,20,55),(21,42,23,44),(22,40,24,38),(25,46,27,48),(26,31,28,29),(37,63,39,61),(41,64,43,62),(54,59,56,57)]])
38 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 4A | ··· | 4H | 4I | ··· | 4Z | 4AA | 4AB | 4AC |
| order | 1 | 2 | ··· | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | 4 | 4 |
| size | 1 | 1 | ··· | 1 | 8 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | 8 | 8 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | C4○D4 | 2- 1+4 |
| kernel | C42.184D4 | C42⋊4C4 | C23.63C23 | C24.C22 | C23.65C23 | C23⋊Q8 | C23.78C23 | C23.4Q8 | C23.83C23 | C2×C4×Q8 | C2×C42⋊2C2 | C42 | C2×C4 | C22 |
| # reps | 1 | 1 | 2 | 4 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 16 | 2 |
Matrix representation of C42.184D4 ►in GL6(𝔽5)
| 1 | 4 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 2 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 4 |
| 0 | 0 | 0 | 0 | 0 | 2 |
| 2 | 0 | 0 | 0 | 0 | 0 |
| 0 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 0 | 0 | 0 |
| 0 | 0 | 0 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 3 |
| 0 | 0 | 0 | 0 | 0 | 4 |
| 2 | 3 | 0 | 0 | 0 | 0 |
| 4 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 0 | 0 | 0 |
| 0 | 0 | 3 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 1 |
| 0 | 0 | 0 | 0 | 0 | 3 |
| 2 | 3 | 0 | 0 | 0 | 0 |
| 4 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 4 | 0 | 0 |
| 0 | 0 | 3 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 0 | 3 | 2 |
G:=sub<GL(6,GF(5))| [1,0,0,0,0,0,4,4,0,0,0,0,0,0,4,0,0,0,0,0,2,1,0,0,0,0,0,0,3,0,0,0,0,0,4,2],[2,0,0,0,0,0,0,2,0,0,0,0,0,0,2,0,0,0,0,0,0,2,0,0,0,0,0,0,1,0,0,0,0,0,3,4],[2,4,0,0,0,0,3,3,0,0,0,0,0,0,3,3,0,0,0,0,0,2,0,0,0,0,0,0,2,0,0,0,0,0,1,3],[2,4,0,0,0,0,3,3,0,0,0,0,0,0,3,3,0,0,0,0,4,2,0,0,0,0,0,0,3,3,0,0,0,0,0,2] >;
C42.184D4 in GAP, Magma, Sage, TeX
C_4^2._{184}D_4 % in TeX
G:=Group("C4^2.184D4"); // GroupNames label
G:=SmallGroup(128,1336);
// by ID
G=gap.SmallGroup(128,1336);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,2,112,253,120,758,723,184,675,248]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^4=1,d^2=a^2,a*b=b*a,c*a*c^-1=a*b^2,d*a*d^-1=a^-1*b^2,b*c=c*b,d*b*d^-1=a^2*b,d*c*d^-1=c^-1>;
// generators/relations