p-group, metabelian, nilpotent (class 2), monomial
Aliases: C42.183D4, C23.499C24, C24.350C23, C22.2052- (1+4), C42⋊5C4⋊23C2, C23⋊Q8.12C2, C23.160(C4○D4), (C22×C4).121C23, (C23×C4).411C22, (C2×C42).586C22, C22.329(C22×D4), C23.8Q8.39C2, C23.11D4.24C2, (C22×Q8).444C22, C23.81C23⋊51C2, C23.78C23⋊21C2, C2.72(C22.19C24), C24.C22.40C2, C23.63C23⋊103C2, C2.C42.229C22, C2.50(C22.50C24), C2.28(C23.38C23), C2.73(C22.46C24), (C2×C4×Q8)⋊27C2, (C2×C4).368(C2×D4), (C2×C4).408(C4○D4), (C2×C4⋊C4).339C22, C22.375(C2×C4○D4), (C2×C42⋊C2).43C2, (C2×C22⋊C4).514C22, SmallGroup(128,1331)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 420 in 246 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], Q8 [×8], C23, C23 [×2], C23 [×6], C42 [×4], C42 [×6], C22⋊C4 [×12], C4⋊C4 [×18], C22×C4 [×6], C22×C4 [×8], C22×C4 [×6], C2×Q8 [×6], C24, C2.C42 [×2], C2.C42 [×12], C2×C42 [×2], C2×C42 [×2], C2×C22⋊C4 [×2], C2×C22⋊C4 [×4], C2×C4⋊C4 [×3], C2×C4⋊C4 [×6], C42⋊C2 [×4], C4×Q8 [×4], C23×C4, C22×Q8, C42⋊5C4, C23.8Q8 [×2], C23.63C23 [×4], C24.C22 [×2], C23⋊Q8, C23.78C23, C23.11D4, C23.81C23, C2×C42⋊C2, C2×C4×Q8, C42.183D4
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 [×2], C23.38C23, C22.46C24 [×2], C22.50C24 [×2], C42.183D4
Generators and relations
G = < a,b,c,d | a4=b4=c4=1, d2=a2, ab=ba, cac-1=a-1b2, dad-1=a-1, cbc-1=dbd-1=a2b, dcd-1=a2c-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 16 51 39)(2 13 52 40)(3 14 49 37)(4 15 50 38)(5 19 55 41)(6 20 56 42)(7 17 53 43)(8 18 54 44)(9 32 36 25)(10 29 33 26)(11 30 34 27)(12 31 35 28)(21 57 47 62)(22 58 48 63)(23 59 45 64)(24 60 46 61)
(1 61 31 56)(2 59 32 5)(3 63 29 54)(4 57 30 7)(6 51 60 28)(8 49 58 26)(9 43 40 21)(10 20 37 46)(11 41 38 23)(12 18 39 48)(13 47 36 17)(14 24 33 42)(15 45 34 19)(16 22 35 44)(25 55 52 64)(27 53 50 62)
(1 54 3 56)(2 53 4 55)(5 52 7 50)(6 51 8 49)(9 45 11 47)(10 48 12 46)(13 41 15 43)(14 44 16 42)(17 40 19 38)(18 39 20 37)(21 36 23 34)(22 35 24 33)(25 57 27 59)(26 60 28 58)(29 61 31 63)(30 64 32 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), (1,16,51,39)(2,13,52,40)(3,14,49,37)(4,15,50,38)(5,19,55,41)(6,20,56,42)(7,17,53,43)(8,18,54,44)(9,32,36,25)(10,29,33,26)(11,30,34,27)(12,31,35,28)(21,57,47,62)(22,58,48,63)(23,59,45,64)(24,60,46,61), (1,61,31,56)(2,59,32,5)(3,63,29,54)(4,57,30,7)(6,51,60,28)(8,49,58,26)(9,43,40,21)(10,20,37,46)(11,41,38,23)(12,18,39,48)(13,47,36,17)(14,24,33,42)(15,45,34,19)(16,22,35,44)(25,55,52,64)(27,53,50,62), (1,54,3,56)(2,53,4,55)(5,52,7,50)(6,51,8,49)(9,45,11,47)(10,48,12,46)(13,41,15,43)(14,44,16,42)(17,40,19,38)(18,39,20,37)(21,36,23,34)(22,35,24,33)(25,57,27,59)(26,60,28,58)(29,61,31,63)(30,64,32,62)>;
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), (1,16,51,39)(2,13,52,40)(3,14,49,37)(4,15,50,38)(5,19,55,41)(6,20,56,42)(7,17,53,43)(8,18,54,44)(9,32,36,25)(10,29,33,26)(11,30,34,27)(12,31,35,28)(21,57,47,62)(22,58,48,63)(23,59,45,64)(24,60,46,61), (1,61,31,56)(2,59,32,5)(3,63,29,54)(4,57,30,7)(6,51,60,28)(8,49,58,26)(9,43,40,21)(10,20,37,46)(11,41,38,23)(12,18,39,48)(13,47,36,17)(14,24,33,42)(15,45,34,19)(16,22,35,44)(25,55,52,64)(27,53,50,62), (1,54,3,56)(2,53,4,55)(5,52,7,50)(6,51,8,49)(9,45,11,47)(10,48,12,46)(13,41,15,43)(14,44,16,42)(17,40,19,38)(18,39,20,37)(21,36,23,34)(22,35,24,33)(25,57,27,59)(26,60,28,58)(29,61,31,63)(30,64,32,62) );
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)], [(1,16,51,39),(2,13,52,40),(3,14,49,37),(4,15,50,38),(5,19,55,41),(6,20,56,42),(7,17,53,43),(8,18,54,44),(9,32,36,25),(10,29,33,26),(11,30,34,27),(12,31,35,28),(21,57,47,62),(22,58,48,63),(23,59,45,64),(24,60,46,61)], [(1,61,31,56),(2,59,32,5),(3,63,29,54),(4,57,30,7),(6,51,60,28),(8,49,58,26),(9,43,40,21),(10,20,37,46),(11,41,38,23),(12,18,39,48),(13,47,36,17),(14,24,33,42),(15,45,34,19),(16,22,35,44),(25,55,52,64),(27,53,50,62)], [(1,54,3,56),(2,53,4,55),(5,52,7,50),(6,51,8,49),(9,45,11,47),(10,48,12,46),(13,41,15,43),(14,44,16,42),(17,40,19,38),(18,39,20,37),(21,36,23,34),(22,35,24,33),(25,57,27,59),(26,60,28,58),(29,61,31,63),(30,64,32,62)])
Matrix representation ►G ⊆ GL6(𝔽5)
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 0 | 0 | 0 |
| 0 | 0 | 1 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 3 |
| 0 | 0 | 0 | 0 | 3 | 0 |
| 3 | 0 | 0 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 2 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 3 | 0 | 0 |
| 0 | 0 | 4 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 3 | 0 | 0 |
| 0 | 0 | 0 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 |
| 0 | 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(6,GF(5))| [0,1,0,0,0,0,1,0,0,0,0,0,0,0,2,1,0,0,0,0,0,3,0,0,0,0,0,0,0,3,0,0,0,0,3,0],[3,0,0,0,0,0,0,3,0,0,0,0,0,0,4,2,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[4,0,0,0,0,0,0,1,0,0,0,0,0,0,3,4,0,0,0,0,3,2,0,0,0,0,0,0,0,1,0,0,0,0,4,0],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,3,0,0,0,0,0,3,2,0,0,0,0,0,0,0,1,0,0,0,0,4,0] >;
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 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | C4○D4 | C4○D4 | 2- (1+4) |
| kernel | C42.183D4 | C42⋊5C4 | C23.8Q8 | C23.63C23 | C24.C22 | C23⋊Q8 | C23.78C23 | C23.11D4 | C23.81C23 | C2×C42⋊C2 | C2×C4×Q8 | C42 | C2×C4 | C23 | C22 |
| # reps | 1 | 1 | 2 | 4 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 12 | 4 | 2 |
In GAP, Magma, Sage, TeX
C_4^2._{183}D_4 % in TeX
G:=Group("C4^2.183D4"); // GroupNames label
G:=SmallGroup(128,1331);
// by ID
G=gap.SmallGroup(128,1331);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,2,112,253,120,758,723,675,248]);
// 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=d*b*d^-1=a^2*b,d*c*d^-1=a^2*c^-1>;
// generators/relations