p-group, metabelian, nilpotent (class 2), monomial
Aliases: C42.188D4, C23.529C24, C24.368C23, C22.2242- (1+4), C22.3062+ (1+4), C42⋊9C4⋊31C2, (C22×C4).60Q8, C23.68(C2×Q8), C4.87(C22⋊Q8), (C2×C42).606C22, (C22×C4).139C23, (C23×C4).431C22, C22.354(C22×D4), C23.Q8.19C2, C23.7Q8.59C2, C22.134(C22×Q8), C23.81C23⋊61C2, C23.65C23⋊105C2, C2.C42.254C22, C2.26(C22.34C24), C2.27(C22.31C24), C2.21(C23.41C23), C2.26(C22.35C24), (C2×C4).388(C2×D4), (C2×C4).131(C2×Q8), C2.44(C2×C22⋊Q8), (C2×C42.C2)⋊16C2, (C2×C4).661(C4○D4), (C2×C4⋊C4).891C22, C22.401(C2×C4○D4), (C2×C42⋊C2).48C2, (C2×C22⋊C4).218C22, SmallGroup(128,1361)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 420 in 236 conjugacy classes, 108 normal (18 characteristic)
C1, C2 [×3], C2 [×4], C2 [×2], C4 [×4], C4 [×16], C22 [×3], C22 [×4], C22 [×10], C2×C4 [×12], C2×C4 [×44], C23, C23 [×2], C23 [×6], C42 [×4], C42 [×2], C22⋊C4 [×8], C4⋊C4 [×28], C22×C4 [×2], C22×C4 [×16], C22×C4 [×4], C24, C2.C42 [×8], C2×C42 [×2], C2×C22⋊C4 [×6], C2×C4⋊C4 [×18], C42⋊C2 [×4], C42.C2 [×4], C23×C4, C23.7Q8 [×2], C42⋊9C4, C23.65C23 [×2], C23.Q8 [×4], C23.81C23 [×4], C2×C42⋊C2, C2×C42.C2, C42.188D4
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], Q8 [×4], C23 [×15], C2×D4 [×6], C2×Q8 [×6], C4○D4 [×2], C24, C22⋊Q8 [×4], C22×D4, C22×Q8, C2×C4○D4, 2+ (1+4) [×2], 2- (1+4) [×2], C2×C22⋊Q8, C22.31C24 [×2], C22.34C24, C22.35C24, C23.41C23 [×2], C42.188D4
Generators and relations
G = < a,b,c,d | a4=b4=c4=d2=1, ab=ba, cac-1=a-1, dad=ab2, cbc-1=b-1, bd=db, dcd=a2b2c-1 >
(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 41 47)(2 16 42 48)(3 13 43 45)(4 14 44 46)(5 37 62 35)(6 38 63 36)(7 39 64 33)(8 40 61 34)(9 49 23 17)(10 50 24 18)(11 51 21 19)(12 52 22 20)(25 55 31 57)(26 56 32 58)(27 53 29 59)(28 54 30 60)
(1 51 53 6)(2 50 54 5)(3 49 55 8)(4 52 56 7)(9 31 34 13)(10 30 35 16)(11 29 36 15)(12 32 33 14)(17 57 61 43)(18 60 62 42)(19 59 63 41)(20 58 64 44)(21 27 38 47)(22 26 39 46)(23 25 40 45)(24 28 37 48)
(2 42)(4 44)(5 52)(6 17)(7 50)(8 19)(9 38)(10 33)(11 40)(12 35)(14 46)(16 48)(18 64)(20 62)(21 34)(22 37)(23 36)(24 39)(26 32)(28 30)(49 63)(51 61)(54 60)(56 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), (1,15,41,47)(2,16,42,48)(3,13,43,45)(4,14,44,46)(5,37,62,35)(6,38,63,36)(7,39,64,33)(8,40,61,34)(9,49,23,17)(10,50,24,18)(11,51,21,19)(12,52,22,20)(25,55,31,57)(26,56,32,58)(27,53,29,59)(28,54,30,60), (1,51,53,6)(2,50,54,5)(3,49,55,8)(4,52,56,7)(9,31,34,13)(10,30,35,16)(11,29,36,15)(12,32,33,14)(17,57,61,43)(18,60,62,42)(19,59,63,41)(20,58,64,44)(21,27,38,47)(22,26,39,46)(23,25,40,45)(24,28,37,48), (2,42)(4,44)(5,52)(6,17)(7,50)(8,19)(9,38)(10,33)(11,40)(12,35)(14,46)(16,48)(18,64)(20,62)(21,34)(22,37)(23,36)(24,39)(26,32)(28,30)(49,63)(51,61)(54,60)(56,58)>;
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), (1,15,41,47)(2,16,42,48)(3,13,43,45)(4,14,44,46)(5,37,62,35)(6,38,63,36)(7,39,64,33)(8,40,61,34)(9,49,23,17)(10,50,24,18)(11,51,21,19)(12,52,22,20)(25,55,31,57)(26,56,32,58)(27,53,29,59)(28,54,30,60), (1,51,53,6)(2,50,54,5)(3,49,55,8)(4,52,56,7)(9,31,34,13)(10,30,35,16)(11,29,36,15)(12,32,33,14)(17,57,61,43)(18,60,62,42)(19,59,63,41)(20,58,64,44)(21,27,38,47)(22,26,39,46)(23,25,40,45)(24,28,37,48), (2,42)(4,44)(5,52)(6,17)(7,50)(8,19)(9,38)(10,33)(11,40)(12,35)(14,46)(16,48)(18,64)(20,62)(21,34)(22,37)(23,36)(24,39)(26,32)(28,30)(49,63)(51,61)(54,60)(56,58) );
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)], [(1,15,41,47),(2,16,42,48),(3,13,43,45),(4,14,44,46),(5,37,62,35),(6,38,63,36),(7,39,64,33),(8,40,61,34),(9,49,23,17),(10,50,24,18),(11,51,21,19),(12,52,22,20),(25,55,31,57),(26,56,32,58),(27,53,29,59),(28,54,30,60)], [(1,51,53,6),(2,50,54,5),(3,49,55,8),(4,52,56,7),(9,31,34,13),(10,30,35,16),(11,29,36,15),(12,32,33,14),(17,57,61,43),(18,60,62,42),(19,59,63,41),(20,58,64,44),(21,27,38,47),(22,26,39,46),(23,25,40,45),(24,28,37,48)], [(2,42),(4,44),(5,52),(6,17),(7,50),(8,19),(9,38),(10,33),(11,40),(12,35),(14,46),(16,48),(18,64),(20,62),(21,34),(22,37),(23,36),(24,39),(26,32),(28,30),(49,63),(51,61),(54,60),(56,58)])
Matrix representation ►G ⊆ GL8(𝔽5)
| 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 2 | 0 |
| 0 | 0 | 0 | 0 | 1 | 4 | 0 | 2 |
| 0 | 0 | 0 | 0 | 4 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 | 4 | 1 |
| 1 | 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 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 3 | 2 | 2 |
| 0 | 0 | 0 | 0 | 3 | 4 | 0 | 3 |
| 0 | 4 | 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 | 0 | 0 | 3 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 0 | 2 | 2 |
| 0 | 0 | 0 | 0 | 0 | 3 | 1 | 3 |
| 1 | 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 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 4 | 1 | 0 | 4 |
G:=sub<GL(8,GF(5))| [4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,1,1,4,0,0,0,0,0,0,4,0,4,0,0,0,0,2,0,4,4,0,0,0,0,0,2,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,2,0,1,3,0,0,0,0,2,3,3,4,0,0,0,0,0,0,2,0,0,0,0,0,0,0,2,3],[0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,3,4,3,0,0,0,0,0,3,2,0,3,0,0,0,0,0,0,2,1,0,0,0,0,0,0,2,3],[1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,4,4,0,0,0,0,0,1,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4] >;
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 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | Q8 | C4○D4 | 2+ (1+4) | 2- (1+4) |
| kernel | C42.188D4 | C23.7Q8 | C42⋊9C4 | C23.65C23 | C23.Q8 | C23.81C23 | C2×C42⋊C2 | C2×C42.C2 | C42 | C22×C4 | C2×C4 | C22 | C22 |
| # reps | 1 | 2 | 1 | 2 | 4 | 4 | 1 | 1 | 4 | 4 | 4 | 2 | 2 |
In GAP, Magma, Sage, TeX
C_4^2._{188}D_4 % in TeX
G:=Group("C4^2.188D4"); // GroupNames label
G:=SmallGroup(128,1361);
// by ID
G=gap.SmallGroup(128,1361);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,2,336,253,758,723,184,185,80]);
// 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,d*a*d=a*b^2,c*b*c^-1=b^-1,b*d=d*b,d*c*d=a^2*b^2*c^-1>;
// generators/relations