p-group, metabelian, nilpotent (class 2), monomial
Aliases: C42⋊21D4, C24.324C23, C23.444C24, C22.2322+ (1+4), (C2×D4).212D4, C23.50(C2×D4), C4.58(C4⋊D4), C2.70(D4⋊5D4), C23.10D4⋊41C2, C23.23D4⋊55C2, (C2×C42).549C22, (C22×C4).536C23, (C23×C4).113C22, C22.295(C22×D4), C24.3C22⋊55C2, (C22×D4).165C22, (C22×Q8).130C22, C23.67C23⋊58C2, C2.22(C22.29C24), C2.C42.182C22, C2.31(C22.26C24), C2.16(C22.53C24), (C2×C4×D4)⋊47C2, (C4×C4⋊C4)⋊88C2, (C2×C4).353(C2×D4), C2.36(C2×C4⋊D4), (C2×C4.4D4)⋊15C2, (C2×C4⋊1D4).16C2, (C2×C4).147(C4○D4), (C2×C4⋊C4).868C22, C22.321(C2×C4○D4), (C2×C22⋊C4).178C22, SmallGroup(128,1276)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 756 in 352 conjugacy classes, 112 normal (26 characteristic)
C1, C2 [×3], C2 [×4], C2 [×6], C4 [×4], C4 [×14], C22 [×3], C22 [×4], C22 [×34], C2×C4 [×14], C2×C4 [×34], D4 [×28], Q8 [×4], C23, C23 [×4], C23 [×26], C42 [×4], C42 [×4], C22⋊C4 [×24], C4⋊C4 [×6], C22×C4 [×3], C22×C4 [×8], C22×C4 [×8], C2×D4 [×4], C2×D4 [×30], C2×Q8 [×6], C24 [×4], C2.C42 [×6], C2×C42 [×3], C2×C22⋊C4 [×14], C2×C4⋊C4, C2×C4⋊C4 [×2], C4×D4 [×4], C4.4D4 [×8], C4⋊1D4 [×4], C23×C4 [×2], C22×D4 [×2], C22×D4 [×4], C22×Q8, C4×C4⋊C4, C23.23D4 [×4], C24.3C22, C23.67C23, C23.10D4 [×4], C2×C4×D4, C2×C4.4D4 [×2], C2×C4⋊1D4, C42⋊21D4
Quotients:
C1, C2 [×15], C22 [×35], D4 [×8], C23 [×15], C2×D4 [×12], C4○D4 [×6], C24, C4⋊D4 [×4], C22×D4 [×2], C2×C4○D4 [×3], 2+ (1+4) [×2], C2×C4⋊D4, C22.26C24, C22.29C24, D4⋊5D4 [×2], C22.53C24 [×2], C42⋊21D4
Generators and relations
G = < a,b,c,d | a4=b4=c4=d2=1, ab=ba, cac-1=a-1, dad=a-1b2, bc=cb, dbd=b-1, dcd=c-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 21 11)(2 16 22 12)(3 13 23 9)(4 14 24 10)(5 56 38 26)(6 53 39 27)(7 54 40 28)(8 55 37 25)(17 30 50 60)(18 31 51 57)(19 32 52 58)(20 29 49 59)(33 47 61 44)(34 48 62 41)(35 45 63 42)(36 46 64 43)
(1 42 37 59)(2 41 38 58)(3 44 39 57)(4 43 40 60)(5 32 22 48)(6 31 23 47)(7 30 24 46)(8 29 21 45)(9 61 53 51)(10 64 54 50)(11 63 55 49)(12 62 56 52)(13 33 27 18)(14 36 28 17)(15 35 25 20)(16 34 26 19)
(1 37)(2 7)(3 39)(4 5)(6 23)(8 21)(9 27)(10 56)(11 25)(12 54)(13 53)(14 26)(15 55)(16 28)(17 19)(18 51)(20 49)(22 40)(24 38)(30 58)(32 60)(33 61)(34 36)(35 63)(41 46)(43 48)(50 52)(62 64)
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), (1,15,21,11)(2,16,22,12)(3,13,23,9)(4,14,24,10)(5,56,38,26)(6,53,39,27)(7,54,40,28)(8,55,37,25)(17,30,50,60)(18,31,51,57)(19,32,52,58)(20,29,49,59)(33,47,61,44)(34,48,62,41)(35,45,63,42)(36,46,64,43), (1,42,37,59)(2,41,38,58)(3,44,39,57)(4,43,40,60)(5,32,22,48)(6,31,23,47)(7,30,24,46)(8,29,21,45)(9,61,53,51)(10,64,54,50)(11,63,55,49)(12,62,56,52)(13,33,27,18)(14,36,28,17)(15,35,25,20)(16,34,26,19), (1,37)(2,7)(3,39)(4,5)(6,23)(8,21)(9,27)(10,56)(11,25)(12,54)(13,53)(14,26)(15,55)(16,28)(17,19)(18,51)(20,49)(22,40)(24,38)(30,58)(32,60)(33,61)(34,36)(35,63)(41,46)(43,48)(50,52)(62,64)>;
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), (1,15,21,11)(2,16,22,12)(3,13,23,9)(4,14,24,10)(5,56,38,26)(6,53,39,27)(7,54,40,28)(8,55,37,25)(17,30,50,60)(18,31,51,57)(19,32,52,58)(20,29,49,59)(33,47,61,44)(34,48,62,41)(35,45,63,42)(36,46,64,43), (1,42,37,59)(2,41,38,58)(3,44,39,57)(4,43,40,60)(5,32,22,48)(6,31,23,47)(7,30,24,46)(8,29,21,45)(9,61,53,51)(10,64,54,50)(11,63,55,49)(12,62,56,52)(13,33,27,18)(14,36,28,17)(15,35,25,20)(16,34,26,19), (1,37)(2,7)(3,39)(4,5)(6,23)(8,21)(9,27)(10,56)(11,25)(12,54)(13,53)(14,26)(15,55)(16,28)(17,19)(18,51)(20,49)(22,40)(24,38)(30,58)(32,60)(33,61)(34,36)(35,63)(41,46)(43,48)(50,52)(62,64) );
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)], [(1,15,21,11),(2,16,22,12),(3,13,23,9),(4,14,24,10),(5,56,38,26),(6,53,39,27),(7,54,40,28),(8,55,37,25),(17,30,50,60),(18,31,51,57),(19,32,52,58),(20,29,49,59),(33,47,61,44),(34,48,62,41),(35,45,63,42),(36,46,64,43)], [(1,42,37,59),(2,41,38,58),(3,44,39,57),(4,43,40,60),(5,32,22,48),(6,31,23,47),(7,30,24,46),(8,29,21,45),(9,61,53,51),(10,64,54,50),(11,63,55,49),(12,62,56,52),(13,33,27,18),(14,36,28,17),(15,35,25,20),(16,34,26,19)], [(1,37),(2,7),(3,39),(4,5),(6,23),(8,21),(9,27),(10,56),(11,25),(12,54),(13,53),(14,26),(15,55),(16,28),(17,19),(18,51),(20,49),(22,40),(24,38),(30,58),(32,60),(33,61),(34,36),(35,63),(41,46),(43,48),(50,52),(62,64)])
Matrix representation ►G ⊆ GL6(𝔽5)
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 4 |
| 0 | 0 | 0 | 0 | 3 | 3 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 2 |
| 0 | 0 | 0 | 0 | 4 | 4 |
| 4 | 2 | 0 | 0 | 0 | 0 |
| 4 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 4 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 4 | 4 |
G:=sub<GL(6,GF(5))| [4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,4,0,0,0,0,1,0,0,0,0,0,0,0,2,3,0,0,0,0,4,3],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,4,0,0,0,0,2,4],[4,4,0,0,0,0,2,1,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[4,4,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,1,4,0,0,0,0,0,4] >;
38 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 2L | 2M | 4A | ··· | 4H | 4I | ··· | 4V | 4W | 4X |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | 4 |
| size | 1 | 1 | ··· | 1 | 4 | 4 | 4 | 4 | 8 | 8 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | 8 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | C4○D4 | 2+ (1+4) |
| kernel | C42⋊21D4 | C4×C4⋊C4 | C23.23D4 | C24.3C22 | C23.67C23 | C23.10D4 | C2×C4×D4 | C2×C4.4D4 | C2×C4⋊1D4 | C42 | C2×D4 | C2×C4 | C22 |
| # reps | 1 | 1 | 4 | 1 | 1 | 4 | 1 | 2 | 1 | 4 | 4 | 12 | 2 |
In GAP, Magma, Sage, TeX
C_4^2\rtimes_{21}D_4 % in TeX
G:=Group("C4^2:21D4"); // GroupNames label
G:=SmallGroup(128,1276);
// by ID
G=gap.SmallGroup(128,1276);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,2,253,456,758,723,184,675,80]);
// 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^-1*b^2,b*c=c*b,d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations