p-group, metabelian, nilpotent (class 2), monomial
Aliases: C42.186D4, C23.521C24, C24.364C23, C22.2182- 1+4, C42⋊8C4⋊51C2, C23.193(C2×D4), (C22×C4).400D4, C4.101(C4⋊D4), (C23×C4).424C22, (C22×C4).131C23, (C2×C42).602C22, C22.346(C22×D4), C23.11D4.26C2, (C22×Q8).152C22, C23.81C23⋊57C2, C23.67C23⋊71C2, C2.C42.248C22, C2.35(C23.38C23), C2.24(C22.35C24), (C2×C4⋊Q8)⋊16C2, (C2×C4).381(C2×D4), C2.45(C2×C4⋊D4), (C2×C22⋊Q8).37C2, (C2×C4).657(C4○D4), (C2×C4⋊C4).888C22, C22.393(C2×C4○D4), (C2×C42⋊C2).45C2, (C2×C22⋊C4).213C22, SmallGroup(128,1353)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.186D4
G = < a,b,c,d | a4=b4=c4=d2=1, ab=ba, cac-1=a-1b2, dad=ab2, cbc-1=b-1, bd=db, dcd=b2c-1 >
Subgroups: 468 in 264 conjugacy classes, 108 normal (16 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C2×C4, C2×C4, Q8, C23, C23, C23, C42, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×Q8, C24, C2.C42, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C42⋊C2, C22⋊Q8, C4⋊Q8, C23×C4, C22×Q8, C42⋊8C4, C23.67C23, C23.11D4, C23.81C23, C2×C42⋊C2, C2×C22⋊Q8, C2×C4⋊Q8, C42.186D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C4⋊D4, C22×D4, C2×C4○D4, 2- 1+4, C2×C4⋊D4, C23.38C23, C22.35C24, C42.186D4
►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 43 39 11)(2 44 40 12)(3 41 37 9)(4 42 38 10)(5 13 17 45)(6 14 18 46)(7 15 19 47)(8 16 20 48)(21 53 49 25)(22 54 50 26)(23 55 51 27)(24 56 52 28)(29 36 57 61)(30 33 58 62)(31 34 59 63)(32 35 60 64)
(1 47 23 59)(2 14 24 30)(3 45 21 57)(4 16 22 32)(5 25 61 9)(6 56 62 44)(7 27 63 11)(8 54 64 42)(10 20 26 35)(12 18 28 33)(13 49 29 37)(15 51 31 39)(17 53 36 41)(19 55 34 43)(38 48 50 60)(40 46 52 58)
(2 40)(4 38)(5 36)(6 62)(7 34)(8 64)(10 42)(12 44)(13 57)(14 30)(15 59)(16 32)(17 61)(18 33)(19 63)(20 35)(22 50)(24 52)(26 54)(28 56)(29 45)(31 47)(46 58)(48 60)
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), (1,43,39,11)(2,44,40,12)(3,41,37,9)(4,42,38,10)(5,13,17,45)(6,14,18,46)(7,15,19,47)(8,16,20,48)(21,53,49,25)(22,54,50,26)(23,55,51,27)(24,56,52,28)(29,36,57,61)(30,33,58,62)(31,34,59,63)(32,35,60,64), (1,47,23,59)(2,14,24,30)(3,45,21,57)(4,16,22,32)(5,25,61,9)(6,56,62,44)(7,27,63,11)(8,54,64,42)(10,20,26,35)(12,18,28,33)(13,49,29,37)(15,51,31,39)(17,53,36,41)(19,55,34,43)(38,48,50,60)(40,46,52,58), (2,40)(4,38)(5,36)(6,62)(7,34)(8,64)(10,42)(12,44)(13,57)(14,30)(15,59)(16,32)(17,61)(18,33)(19,63)(20,35)(22,50)(24,52)(26,54)(28,56)(29,45)(31,47)(46,58)(48,60)>;
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), (1,43,39,11)(2,44,40,12)(3,41,37,9)(4,42,38,10)(5,13,17,45)(6,14,18,46)(7,15,19,47)(8,16,20,48)(21,53,49,25)(22,54,50,26)(23,55,51,27)(24,56,52,28)(29,36,57,61)(30,33,58,62)(31,34,59,63)(32,35,60,64), (1,47,23,59)(2,14,24,30)(3,45,21,57)(4,16,22,32)(5,25,61,9)(6,56,62,44)(7,27,63,11)(8,54,64,42)(10,20,26,35)(12,18,28,33)(13,49,29,37)(15,51,31,39)(17,53,36,41)(19,55,34,43)(38,48,50,60)(40,46,52,58), (2,40)(4,38)(5,36)(6,62)(7,34)(8,64)(10,42)(12,44)(13,57)(14,30)(15,59)(16,32)(17,61)(18,33)(19,63)(20,35)(22,50)(24,52)(26,54)(28,56)(29,45)(31,47)(46,58)(48,60) );
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)], [(1,43,39,11),(2,44,40,12),(3,41,37,9),(4,42,38,10),(5,13,17,45),(6,14,18,46),(7,15,19,47),(8,16,20,48),(21,53,49,25),(22,54,50,26),(23,55,51,27),(24,56,52,28),(29,36,57,61),(30,33,58,62),(31,34,59,63),(32,35,60,64)], [(1,47,23,59),(2,14,24,30),(3,45,21,57),(4,16,22,32),(5,25,61,9),(6,56,62,44),(7,27,63,11),(8,54,64,42),(10,20,26,35),(12,18,28,33),(13,49,29,37),(15,51,31,39),(17,53,36,41),(19,55,34,43),(38,48,50,60),(40,46,52,58)], [(2,40),(4,38),(5,36),(6,62),(7,34),(8,64),(10,42),(12,44),(13,57),(14,30),(15,59),(16,32),(17,61),(18,33),(19,63),(20,35),(22,50),(24,52),(26,54),(28,56),(29,45),(31,47),(46,58),(48,60)]])
32 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | ··· | 4N | 4O | ··· | 4V |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 |
| size | 1 | 1 | ··· | 1 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 8 | ··· | 8 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | C4○D4 | 2- 1+4 |
| kernel | C42.186D4 | C42⋊8C4 | C23.67C23 | C23.11D4 | C23.81C23 | C2×C42⋊C2 | C2×C22⋊Q8 | C2×C4⋊Q8 | C42 | C22×C4 | C2×C4 | C22 |
| # reps | 1 | 1 | 2 | 4 | 4 | 1 | 2 | 1 | 4 | 4 | 4 | 4 |
Matrix representation of C42.186D4 ►in GL8(𝔽5)
| 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 | 4 | 0 | 2 | 0 |
| 0 | 0 | 0 | 0 | 3 | 1 | 2 | 3 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 4 |
| 4 | 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 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 4 | 3 |
| 4 | 2 | 0 | 0 | 0 | 0 | 0 | 0 |
| 4 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 4 | 1 |
| 0 | 0 | 0 | 0 | 1 | 4 | 3 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 4 | 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 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 4 | 1 | 0 | 4 |
G:=sub<GL(8,GF(5))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,4,3,0,0,0,0,0,0,0,1,0,0,0,0,0,0,2,2,1,0,0,0,0,0,0,3,0,4],[4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,2,0,0,1,0,0,0,0,0,3,0,0,0,0,0,0,0,0,2,4,0,0,0,0,0,0,0,3],[4,4,0,0,0,0,0,0,2,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,4,1,1,0,0,0,0,1,0,0,4,0,0,0,0,0,0,4,3,0,0,0,0,0,0,1,1],[1,1,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,1,4,0,0,0,0,0,1,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4] >;
C42.186D4 in GAP, Magma, Sage, TeX
C_4^2._{186}D_4 % in TeX
G:=Group("C4^2.186D4"); // GroupNames label
G:=SmallGroup(128,1353);
// by ID
G=gap.SmallGroup(128,1353);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,2,224,253,758,723,100,185,80]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^4=d^2=1,a*b=b*a,c*a*c^-1=a^-1*b^2,d*a*d=a*b^2,c*b*c^-1=b^-1,b*d=d*b,d*c*d=b^2*c^-1>;
// generators/relations