p-group, metabelian, nilpotent (class 2), monomial
Aliases: C42.188D4, C23.529C24, C24.368C23, C22.3062+ 1+4, C22.2242- 1+4, C42⋊9C4⋊31C2, (C22×C4).60Q8, C23.68(C2×Q8), C4.87(C22⋊Q8), (C23×C4).431C22, (C22×C4).139C23, (C2×C42).606C22, C22.354(C22×D4), C23.7Q8.59C2, C23.Q8.19C2, C22.134(C22×Q8), C23.81C23⋊61C2, C23.65C23⋊105C2, C2.C42.254C22, C2.26(C22.35C24), C2.21(C23.41C23), C2.27(C22.31C24), C2.26(C22.34C24), (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
Generators and relations for C42.188D4
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 >
Subgroups: 420 in 236 conjugacy classes, 108 normal (18 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C2×C4, C2×C4, C23, C23, C23, C42, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C24, C2.C42, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C42⋊C2, C42.C2, C23×C4, C23.7Q8, C42⋊9C4, C23.65C23, C23.Q8, C23.81C23, C2×C42⋊C2, C2×C42.C2, C42.188D4
Quotients: C1, C2, C22, D4, Q8, C23, C2×D4, C2×Q8, C4○D4, C24, C22⋊Q8, C22×D4, C22×Q8, C2×C4○D4, 2+ 1+4, 2- 1+4, C2×C22⋊Q8, C22.31C24, C22.34C24, C22.35C24, C23.41C23, C42.188D4
►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 9 47)(2 16 10 48)(3 13 11 45)(4 14 12 46)(5 35 37 62)(6 36 38 63)(7 33 39 64)(8 34 40 61)(17 23 49 41)(18 24 50 42)(19 21 51 43)(20 22 52 44)(25 57 55 31)(26 58 56 32)(27 59 53 29)(28 60 54 30)
(1 19 53 36)(2 18 54 35)(3 17 55 34)(4 20 56 33)(5 16 42 30)(6 15 43 29)(7 14 44 32)(8 13 41 31)(9 51 27 63)(10 50 28 62)(11 49 25 61)(12 52 26 64)(21 59 38 47)(22 58 39 46)(23 57 40 45)(24 60 37 48)
(2 10)(4 12)(5 44)(6 23)(7 42)(8 21)(14 46)(16 48)(17 63)(18 33)(19 61)(20 35)(22 37)(24 39)(26 56)(28 54)(30 60)(32 58)(34 51)(36 49)(38 41)(40 43)(50 64)(52 62)
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), (1,15,9,47)(2,16,10,48)(3,13,11,45)(4,14,12,46)(5,35,37,62)(6,36,38,63)(7,33,39,64)(8,34,40,61)(17,23,49,41)(18,24,50,42)(19,21,51,43)(20,22,52,44)(25,57,55,31)(26,58,56,32)(27,59,53,29)(28,60,54,30), (1,19,53,36)(2,18,54,35)(3,17,55,34)(4,20,56,33)(5,16,42,30)(6,15,43,29)(7,14,44,32)(8,13,41,31)(9,51,27,63)(10,50,28,62)(11,49,25,61)(12,52,26,64)(21,59,38,47)(22,58,39,46)(23,57,40,45)(24,60,37,48), (2,10)(4,12)(5,44)(6,23)(7,42)(8,21)(14,46)(16,48)(17,63)(18,33)(19,61)(20,35)(22,37)(24,39)(26,56)(28,54)(30,60)(32,58)(34,51)(36,49)(38,41)(40,43)(50,64)(52,62)>;
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), (1,15,9,47)(2,16,10,48)(3,13,11,45)(4,14,12,46)(5,35,37,62)(6,36,38,63)(7,33,39,64)(8,34,40,61)(17,23,49,41)(18,24,50,42)(19,21,51,43)(20,22,52,44)(25,57,55,31)(26,58,56,32)(27,59,53,29)(28,60,54,30), (1,19,53,36)(2,18,54,35)(3,17,55,34)(4,20,56,33)(5,16,42,30)(6,15,43,29)(7,14,44,32)(8,13,41,31)(9,51,27,63)(10,50,28,62)(11,49,25,61)(12,52,26,64)(21,59,38,47)(22,58,39,46)(23,57,40,45)(24,60,37,48), (2,10)(4,12)(5,44)(6,23)(7,42)(8,21)(14,46)(16,48)(17,63)(18,33)(19,61)(20,35)(22,37)(24,39)(26,56)(28,54)(30,60)(32,58)(34,51)(36,49)(38,41)(40,43)(50,64)(52,62) );
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)], [(1,15,9,47),(2,16,10,48),(3,13,11,45),(4,14,12,46),(5,35,37,62),(6,36,38,63),(7,33,39,64),(8,34,40,61),(17,23,49,41),(18,24,50,42),(19,21,51,43),(20,22,52,44),(25,57,55,31),(26,58,56,32),(27,59,53,29),(28,60,54,30)], [(1,19,53,36),(2,18,54,35),(3,17,55,34),(4,20,56,33),(5,16,42,30),(6,15,43,29),(7,14,44,32),(8,13,41,31),(9,51,27,63),(10,50,28,62),(11,49,25,61),(12,52,26,64),(21,59,38,47),(22,58,39,46),(23,57,40,45),(24,60,37,48)], [(2,10),(4,12),(5,44),(6,23),(7,42),(8,21),(14,46),(16,48),(17,63),(18,33),(19,61),(20,35),(22,37),(24,39),(26,56),(28,54),(30,60),(32,58),(34,51),(36,49),(38,41),(40,43),(50,64),(52,62)]])
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 |
Matrix representation of C42.188D4 ►in 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] >;
C42.188D4 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