p-group, metabelian, nilpotent (class 2), monomial
Aliases: C42.185D4, C23.511C24, C24.358C23, C22.2112- (1+4), C42⋊4C4⋊29C2, C23.164(C4○D4), (C22×C4).126C23, (C23×C4).415C22, (C2×C42).598C22, C22.337(C22×D4), C23.4Q8.14C2, C23.8Q8.40C2, C23.11D4.25C2, C23.81C23⋊55C2, C23.83C23⋊56C2, C2.80(C22.19C24), C24.C22.42C2, C23.65C23⋊101C2, C23.63C23⋊111C2, C2.C42.240C22, C2.49(C22.26C24), C2.30(C23.38C23), C2.78(C22.46C24), (C2×C4).372(C2×D4), (C2×C42.C2)⋊15C2, (C2×C4).413(C4○D4), (C2×C4⋊C4).350C22, C22.387(C2×C4○D4), (C2×C42⋊C2).44C2, (C2×C22⋊C4).518C22, SmallGroup(128,1343)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 404 in 238 conjugacy classes, 100 normal (34 characteristic)
C1, C2 [×3], C2 [×4], C2 [×2], C4 [×20], C22 [×3], C22 [×4], C22 [×10], C2×C4 [×12], C2×C4 [×44], C23, C23 [×2], C23 [×6], C42 [×4], C42 [×6], C22⋊C4 [×12], C4⋊C4 [×24], C22×C4 [×6], C22×C4 [×8], C22×C4 [×6], C24, C2.C42 [×2], C2.C42 [×10], C2×C42 [×2], C2×C42 [×2], C2×C22⋊C4 [×2], C2×C22⋊C4 [×4], C2×C4⋊C4 [×4], C2×C4⋊C4 [×8], C42⋊C2 [×4], C42.C2 [×4], C23×C4, C42⋊4C4, C23.8Q8 [×2], C23.63C23 [×2], C24.C22 [×2], C23.65C23 [×2], C23.11D4, C23.81C23, C23.4Q8, C23.83C23, C2×C42⋊C2, C2×C42.C2, C42.185D4
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×8], C24, C22×D4, C2×C4○D4 [×4], 2- (1+4) [×2], C22.19C24, C22.26C24, C23.38C23, C22.46C24 [×4], C42.185D4
Generators and relations
G = < a,b,c,d | a4=b4=c4=1, d2=a2b2, ab=ba, cac-1=a-1b2, dad-1=a-1, cbc-1=dbd-1=a2b-1, dcd-1=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 50 29 16)(2 51 30 13)(3 52 31 14)(4 49 32 15)(5 45 61 44)(6 46 62 41)(7 47 63 42)(8 48 64 43)(9 36 28 37)(10 33 25 38)(11 34 26 39)(12 35 27 40)(17 53 24 58)(18 54 21 59)(19 55 22 60)(20 56 23 57)
(1 54 12 63)(2 58 9 6)(3 56 10 61)(4 60 11 8)(5 31 57 25)(7 29 59 27)(13 19 37 48)(14 21 38 42)(15 17 39 46)(16 23 40 44)(18 33 47 52)(20 35 45 50)(22 36 43 51)(24 34 41 49)(26 64 32 55)(28 62 30 53)
(1 5 31 63)(2 8 32 62)(3 7 29 61)(4 6 30 64)(9 60 26 53)(10 59 27 56)(11 58 28 55)(12 57 25 54)(13 46 49 43)(14 45 50 42)(15 48 51 41)(16 47 52 44)(17 34 22 37)(18 33 23 40)(19 36 24 39)(20 35 21 38)
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), (1,50,29,16)(2,51,30,13)(3,52,31,14)(4,49,32,15)(5,45,61,44)(6,46,62,41)(7,47,63,42)(8,48,64,43)(9,36,28,37)(10,33,25,38)(11,34,26,39)(12,35,27,40)(17,53,24,58)(18,54,21,59)(19,55,22,60)(20,56,23,57), (1,54,12,63)(2,58,9,6)(3,56,10,61)(4,60,11,8)(5,31,57,25)(7,29,59,27)(13,19,37,48)(14,21,38,42)(15,17,39,46)(16,23,40,44)(18,33,47,52)(20,35,45,50)(22,36,43,51)(24,34,41,49)(26,64,32,55)(28,62,30,53), (1,5,31,63)(2,8,32,62)(3,7,29,61)(4,6,30,64)(9,60,26,53)(10,59,27,56)(11,58,28,55)(12,57,25,54)(13,46,49,43)(14,45,50,42)(15,48,51,41)(16,47,52,44)(17,34,22,37)(18,33,23,40)(19,36,24,39)(20,35,21,38)>;
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), (1,50,29,16)(2,51,30,13)(3,52,31,14)(4,49,32,15)(5,45,61,44)(6,46,62,41)(7,47,63,42)(8,48,64,43)(9,36,28,37)(10,33,25,38)(11,34,26,39)(12,35,27,40)(17,53,24,58)(18,54,21,59)(19,55,22,60)(20,56,23,57), (1,54,12,63)(2,58,9,6)(3,56,10,61)(4,60,11,8)(5,31,57,25)(7,29,59,27)(13,19,37,48)(14,21,38,42)(15,17,39,46)(16,23,40,44)(18,33,47,52)(20,35,45,50)(22,36,43,51)(24,34,41,49)(26,64,32,55)(28,62,30,53), (1,5,31,63)(2,8,32,62)(3,7,29,61)(4,6,30,64)(9,60,26,53)(10,59,27,56)(11,58,28,55)(12,57,25,54)(13,46,49,43)(14,45,50,42)(15,48,51,41)(16,47,52,44)(17,34,22,37)(18,33,23,40)(19,36,24,39)(20,35,21,38) );
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)], [(1,50,29,16),(2,51,30,13),(3,52,31,14),(4,49,32,15),(5,45,61,44),(6,46,62,41),(7,47,63,42),(8,48,64,43),(9,36,28,37),(10,33,25,38),(11,34,26,39),(12,35,27,40),(17,53,24,58),(18,54,21,59),(19,55,22,60),(20,56,23,57)], [(1,54,12,63),(2,58,9,6),(3,56,10,61),(4,60,11,8),(5,31,57,25),(7,29,59,27),(13,19,37,48),(14,21,38,42),(15,17,39,46),(16,23,40,44),(18,33,47,52),(20,35,45,50),(22,36,43,51),(24,34,41,49),(26,64,32,55),(28,62,30,53)], [(1,5,31,63),(2,8,32,62),(3,7,29,61),(4,6,30,64),(9,60,26,53),(10,59,27,56),(11,58,28,55),(12,57,25,54),(13,46,49,43),(14,45,50,42),(15,48,51,41),(16,47,52,44),(17,34,22,37),(18,33,23,40),(19,36,24,39),(20,35,21,38)])
Matrix representation ►G ⊆ GL6(𝔽5)
| 3 | 4 | 0 | 0 | 0 | 0 |
| 0 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 4 | 0 |
| 1 | 3 | 0 | 0 | 0 | 0 |
| 0 | 4 | 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 | 2 |
| 1 | 3 | 0 | 0 | 0 | 0 |
| 1 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 |
| 1 | 3 | 0 | 0 | 0 | 0 |
| 1 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(5))| [3,0,0,0,0,0,4,2,0,0,0,0,0,0,0,1,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,1,0],[1,0,0,0,0,0,3,4,0,0,0,0,0,0,2,0,0,0,0,0,0,2,0,0,0,0,0,0,2,0,0,0,0,0,0,2],[1,1,0,0,0,0,3,4,0,0,0,0,0,0,0,1,0,0,0,0,4,0,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[1,1,0,0,0,0,3,4,0,0,0,0,0,0,0,4,0,0,0,0,4,0,0,0,0,0,0,0,4,0,0,0,0,0,0,1] >;
38 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 4A | ··· | 4H | 4I | ··· | 4X | 4Y | 4Z | 4AA | 4AB |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 |
| size | 1 | 1 | ··· | 1 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | C4○D4 | C4○D4 | 2- (1+4) |
| kernel | C42.185D4 | C42⋊4C4 | C23.8Q8 | C23.63C23 | C24.C22 | C23.65C23 | C23.11D4 | C23.81C23 | C23.4Q8 | C23.83C23 | C2×C42⋊C2 | C2×C42.C2 | C42 | C2×C4 | C23 | C22 |
| # reps | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 12 | 4 | 2 |
In GAP, Magma, Sage, TeX
C_4^2._{185}D_4 % in TeX
G:=Group("C4^2.185D4"); // GroupNames label
G:=SmallGroup(128,1343);
// by ID
G=gap.SmallGroup(128,1343);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,2,336,253,568,758,723,675,248]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^4=1,d^2=a^2*b^2,a*b=b*a,c*a*c^-1=a^-1*b^2,d*a*d^-1=a^-1,c*b*c^-1=d*b*d^-1=a^2*b^-1,d*c*d^-1=a^2*b^2*c^-1>;
// generators/relations