p-group, metabelian, nilpotent (class 2), monomial
Aliases: C42.189D4, C24.40C23, C23.532C24, C22.3092+ 1+4, C22.2262- 1+4, C42⋊5C4⋊25C2, C23⋊Q8⋊30C2, C23.4Q8⋊29C2, (C22×C4).142C23, (C2×C42).609C22, C22.357(C22×D4), C23.10D4.31C2, (C22×D4).196C22, (C22×Q8).449C22, C24.C22⋊105C2, C23.78C23⋊28C2, C2.41(C22.29C24), C2.83(C22.19C24), C2.C42.257C22, C2.40(C23.38C23), C2.49(C22.36C24), (C2×C4×Q8)⋊29C2, (C2×C4).391(C2×D4), (C2×C4).168(C4○D4), (C2×C4⋊C4).359C22, (C2×C4.4D4).27C2, C22.404(C2×C4○D4), (C2×C22⋊C4).221C22, SmallGroup(128,1364)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.189D4
G = < a,b,c,d | a4=b4=c4=1, d2=a2, ab=ba, cac-1=a-1b2, dad-1=a-1, cbc-1=a2b, bd=db, dcd-1=a2c-1 >
Subgroups: 500 in 250 conjugacy classes, 96 normal (18 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C2×C4, C2×C4, D4, Q8, C23, C23, C42, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×Q8, C24, C2.C42, C2×C42, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C4×Q8, C4.4D4, C22×D4, C22×Q8, C42⋊5C4, C24.C22, C23⋊Q8, C23.10D4, C23.78C23, C23.4Q8, C2×C4×Q8, C2×C4.4D4, C42.189D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C22×D4, C2×C4○D4, 2+ 1+4, 2- 1+4, C22.19C24, C22.29C24, C23.38C23, C22.36C24, C42.189D4
►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 15 42 20)(2 16 43 17)(3 13 44 18)(4 14 41 19)(5 40 28 59)(6 37 25 60)(7 38 26 57)(8 39 27 58)(9 46 51 23)(10 47 52 24)(11 48 49 21)(12 45 50 22)(29 54 61 35)(30 55 62 36)(31 56 63 33)(32 53 64 34)
(1 62 51 38)(2 29 52 60)(3 64 49 40)(4 31 50 58)(5 20 53 46)(6 14 54 22)(7 18 55 48)(8 16 56 24)(9 57 42 30)(10 37 43 61)(11 59 44 32)(12 39 41 63)(13 36 21 26)(15 34 23 28)(17 33 47 27)(19 35 45 25)
(1 8 3 6)(2 7 4 5)(9 33 11 35)(10 36 12 34)(13 37 15 39)(14 40 16 38)(17 57 19 59)(18 60 20 58)(21 61 23 63)(22 64 24 62)(25 42 27 44)(26 41 28 43)(29 46 31 48)(30 45 32 47)(49 54 51 56)(50 53 52 55)
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), (1,15,42,20)(2,16,43,17)(3,13,44,18)(4,14,41,19)(5,40,28,59)(6,37,25,60)(7,38,26,57)(8,39,27,58)(9,46,51,23)(10,47,52,24)(11,48,49,21)(12,45,50,22)(29,54,61,35)(30,55,62,36)(31,56,63,33)(32,53,64,34), (1,62,51,38)(2,29,52,60)(3,64,49,40)(4,31,50,58)(5,20,53,46)(6,14,54,22)(7,18,55,48)(8,16,56,24)(9,57,42,30)(10,37,43,61)(11,59,44,32)(12,39,41,63)(13,36,21,26)(15,34,23,28)(17,33,47,27)(19,35,45,25), (1,8,3,6)(2,7,4,5)(9,33,11,35)(10,36,12,34)(13,37,15,39)(14,40,16,38)(17,57,19,59)(18,60,20,58)(21,61,23,63)(22,64,24,62)(25,42,27,44)(26,41,28,43)(29,46,31,48)(30,45,32,47)(49,54,51,56)(50,53,52,55)>;
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), (1,15,42,20)(2,16,43,17)(3,13,44,18)(4,14,41,19)(5,40,28,59)(6,37,25,60)(7,38,26,57)(8,39,27,58)(9,46,51,23)(10,47,52,24)(11,48,49,21)(12,45,50,22)(29,54,61,35)(30,55,62,36)(31,56,63,33)(32,53,64,34), (1,62,51,38)(2,29,52,60)(3,64,49,40)(4,31,50,58)(5,20,53,46)(6,14,54,22)(7,18,55,48)(8,16,56,24)(9,57,42,30)(10,37,43,61)(11,59,44,32)(12,39,41,63)(13,36,21,26)(15,34,23,28)(17,33,47,27)(19,35,45,25), (1,8,3,6)(2,7,4,5)(9,33,11,35)(10,36,12,34)(13,37,15,39)(14,40,16,38)(17,57,19,59)(18,60,20,58)(21,61,23,63)(22,64,24,62)(25,42,27,44)(26,41,28,43)(29,46,31,48)(30,45,32,47)(49,54,51,56)(50,53,52,55) );
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)], [(1,15,42,20),(2,16,43,17),(3,13,44,18),(4,14,41,19),(5,40,28,59),(6,37,25,60),(7,38,26,57),(8,39,27,58),(9,46,51,23),(10,47,52,24),(11,48,49,21),(12,45,50,22),(29,54,61,35),(30,55,62,36),(31,56,63,33),(32,53,64,34)], [(1,62,51,38),(2,29,52,60),(3,64,49,40),(4,31,50,58),(5,20,53,46),(6,14,54,22),(7,18,55,48),(8,16,56,24),(9,57,42,30),(10,37,43,61),(11,59,44,32),(12,39,41,63),(13,36,21,26),(15,34,23,28),(17,33,47,27),(19,35,45,25)], [(1,8,3,6),(2,7,4,5),(9,33,11,35),(10,36,12,34),(13,37,15,39),(14,40,16,38),(17,57,19,59),(18,60,20,58),(21,61,23,63),(22,64,24,62),(25,42,27,44),(26,41,28,43),(29,46,31,48),(30,45,32,47),(49,54,51,56),(50,53,52,55)]])
32 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | ··· | 4P | 4Q | ··· | 4V |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 |
| size | 1 | 1 | ··· | 1 | 8 | 8 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 8 | ··· | 8 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | C4○D4 | 2+ 1+4 | 2- 1+4 |
| kernel | C42.189D4 | C42⋊5C4 | C24.C22 | C23⋊Q8 | C23.10D4 | C23.78C23 | C23.4Q8 | C2×C4×Q8 | C2×C4.4D4 | C42 | C2×C4 | C22 | C22 |
| # reps | 1 | 1 | 4 | 2 | 2 | 2 | 2 | 1 | 1 | 4 | 8 | 2 | 2 |
Matrix representation of C42.189D4 ►in GL8(𝔽5)
| 0 | 3 | 0 | 0 | 0 | 0 | 0 | 0 |
| 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 | 3 | 2 |
| 0 | 0 | 0 | 0 | 0 | 4 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 3 | 1 | 0 |
| 0 | 0 | 0 | 0 | 4 | 3 | 0 | 1 |
| 3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 2 | 3 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 3 |
| 3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 | 3 | 2 |
| 0 | 0 | 0 | 0 | 0 | 1 | 4 | 0 |
| 0 | 0 | 0 | 0 | 0 | 2 | 4 | 0 |
| 0 | 0 | 0 | 0 | 4 | 2 | 3 | 1 |
| 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 |
| 3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 0 | 2 | 3 |
| 0 | 0 | 0 | 0 | 2 | 4 | 0 | 3 |
G:=sub<GL(8,GF(5))| [0,2,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,4,0,0,0,0,0,4,3,3,0,0,0,0,3,1,1,0,0,0,0,0,2,0,0,1],[3,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,3,3],[3,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,3,3,0,0,0,0,0,0,0,0,4,0,0,4,0,0,0,0,0,1,2,2,0,0,0,0,3,4,4,3,0,0,0,0,2,0,0,1],[0,3,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,3,1,0,0,0,0,0,0,2,2,0,0,0,0,0,0,0,0,0,1,2,2,0,0,0,0,4,0,0,4,0,0,0,0,0,0,2,0,0,0,0,0,0,0,3,3] >;
C42.189D4 in GAP, Magma, Sage, TeX
C_4^2._{189}D_4 % in TeX
G:=Group("C4^2.189D4"); // GroupNames label
G:=SmallGroup(128,1364);
// by ID
G=gap.SmallGroup(128,1364);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,2,112,253,120,758,723,185,136]);
// 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^-1*b^2,d*a*d^-1=a^-1,c*b*c^-1=a^2*b,b*d=d*b,d*c*d^-1=a^2*c^-1>;
// generators/relations