p-group, metabelian, nilpotent (class 2), monomial
Aliases: C42.440D4, C23.757C24, C24.600C23, C4⋊4(C22⋊Q8), C22⋊1(C4⋊Q8), (C22×C4)⋊22Q8, C42⋊9C4⋊41C2, C23.375(C2×D4), (C22×C4).599D4, C23.106(C2×Q8), C4.103(C4⋊D4), (C22×C42).28C2, (C22×C4).264C23, (C23×C4).684C22, C22.467(C22×D4), C23.7Q8.80C2, C22.181(C22×Q8), (C2×C42).1092C22, (C22×Q8).249C22, C23.65C23⋊170C2, C23.67C23⋊110C2, C2.C42.453C22, C2.57(C22.26C24), C2.47(C23.37C23), (C2×C4⋊Q8)⋊26C2, C2.21(C2×C4⋊Q8), C2.50(C2×C4⋊D4), (C2×C4).173(C2×Q8), C2.48(C2×C22⋊Q8), (C2×C4).1396(C2×D4), (C2×C22⋊Q8).50C2, (C2×C4).673(C4○D4), (C2×C4⋊C4).560C22, C22.598(C2×C4○D4), (C2×C22⋊C4).367C22, SmallGroup(128,1589)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 532 in 322 conjugacy classes, 144 normal (18 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C4 [×12], C4 [×14], C22 [×3], C22 [×8], C22 [×12], C2×C4 [×24], C2×C4 [×54], Q8 [×8], C23, C23 [×6], C23 [×4], C42 [×4], C42 [×6], C22⋊C4 [×8], C4⋊C4 [×28], C22×C4 [×2], C22×C4 [×24], C22×C4 [×12], C2×Q8 [×12], C24, C2.C42 [×8], C2×C42 [×2], C2×C42 [×2], C2×C42 [×4], C2×C22⋊C4 [×4], C2×C4⋊C4 [×14], C22⋊Q8 [×8], C4⋊Q8 [×4], C23×C4, C23×C4 [×2], C22×Q8 [×2], C23.7Q8 [×4], C42⋊9C4, C23.65C23 [×4], C23.67C23 [×2], C22×C42, C2×C22⋊Q8 [×2], C2×C4⋊Q8, C42.440D4
Quotients:
C1, C2 [×15], C22 [×35], D4 [×8], Q8 [×8], C23 [×15], C2×D4 [×12], C2×Q8 [×12], C4○D4 [×6], C24, C4⋊D4 [×4], C22⋊Q8 [×8], C4⋊Q8 [×4], C22×D4 [×2], C22×Q8 [×2], C2×C4○D4 [×3], C2×C4⋊D4, C2×C22⋊Q8 [×2], C2×C4⋊Q8, C22.26C24, C23.37C23 [×2], C42.440D4
Generators and relations
G = < a,b,c,d | a4=b4=c4=1, d2=a2, ab=ba, cac-1=dad-1=a-1, cbc-1=dbd-1=b-1, dcd-1=a2c-1 >
►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 13 25 37)(2 14 26 38)(3 15 27 39)(4 16 28 40)(5 21 58 42)(6 22 59 43)(7 23 60 44)(8 24 57 41)(9 33 49 29)(10 34 50 30)(11 35 51 31)(12 36 52 32)(17 56 48 64)(18 53 45 61)(19 54 46 62)(20 55 47 63)
(1 8 9 55)(2 7 10 54)(3 6 11 53)(4 5 12 56)(13 41 33 20)(14 44 34 19)(15 43 35 18)(16 42 36 17)(21 32 48 40)(22 31 45 39)(23 30 46 38)(24 29 47 37)(25 57 49 63)(26 60 50 62)(27 59 51 61)(28 58 52 64)
(1 61 3 63)(2 64 4 62)(5 52 7 50)(6 51 8 49)(9 59 11 57)(10 58 12 60)(13 45 15 47)(14 48 16 46)(17 40 19 38)(18 39 20 37)(21 36 23 34)(22 35 24 33)(25 53 27 55)(26 56 28 54)(29 43 31 41)(30 42 32 44)
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), (1,13,25,37)(2,14,26,38)(3,15,27,39)(4,16,28,40)(5,21,58,42)(6,22,59,43)(7,23,60,44)(8,24,57,41)(9,33,49,29)(10,34,50,30)(11,35,51,31)(12,36,52,32)(17,56,48,64)(18,53,45,61)(19,54,46,62)(20,55,47,63), (1,8,9,55)(2,7,10,54)(3,6,11,53)(4,5,12,56)(13,41,33,20)(14,44,34,19)(15,43,35,18)(16,42,36,17)(21,32,48,40)(22,31,45,39)(23,30,46,38)(24,29,47,37)(25,57,49,63)(26,60,50,62)(27,59,51,61)(28,58,52,64), (1,61,3,63)(2,64,4,62)(5,52,7,50)(6,51,8,49)(9,59,11,57)(10,58,12,60)(13,45,15,47)(14,48,16,46)(17,40,19,38)(18,39,20,37)(21,36,23,34)(22,35,24,33)(25,53,27,55)(26,56,28,54)(29,43,31,41)(30,42,32,44)>;
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), (1,13,25,37)(2,14,26,38)(3,15,27,39)(4,16,28,40)(5,21,58,42)(6,22,59,43)(7,23,60,44)(8,24,57,41)(9,33,49,29)(10,34,50,30)(11,35,51,31)(12,36,52,32)(17,56,48,64)(18,53,45,61)(19,54,46,62)(20,55,47,63), (1,8,9,55)(2,7,10,54)(3,6,11,53)(4,5,12,56)(13,41,33,20)(14,44,34,19)(15,43,35,18)(16,42,36,17)(21,32,48,40)(22,31,45,39)(23,30,46,38)(24,29,47,37)(25,57,49,63)(26,60,50,62)(27,59,51,61)(28,58,52,64), (1,61,3,63)(2,64,4,62)(5,52,7,50)(6,51,8,49)(9,59,11,57)(10,58,12,60)(13,45,15,47)(14,48,16,46)(17,40,19,38)(18,39,20,37)(21,36,23,34)(22,35,24,33)(25,53,27,55)(26,56,28,54)(29,43,31,41)(30,42,32,44) );
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)], [(1,13,25,37),(2,14,26,38),(3,15,27,39),(4,16,28,40),(5,21,58,42),(6,22,59,43),(7,23,60,44),(8,24,57,41),(9,33,49,29),(10,34,50,30),(11,35,51,31),(12,36,52,32),(17,56,48,64),(18,53,45,61),(19,54,46,62),(20,55,47,63)], [(1,8,9,55),(2,7,10,54),(3,6,11,53),(4,5,12,56),(13,41,33,20),(14,44,34,19),(15,43,35,18),(16,42,36,17),(21,32,48,40),(22,31,45,39),(23,30,46,38),(24,29,47,37),(25,57,49,63),(26,60,50,62),(27,59,51,61),(28,58,52,64)], [(1,61,3,63),(2,64,4,62),(5,52,7,50),(6,51,8,49),(9,59,11,57),(10,58,12,60),(13,45,15,47),(14,48,16,46),(17,40,19,38),(18,39,20,37),(21,36,23,34),(22,35,24,33),(25,53,27,55),(26,56,28,54),(29,43,31,41),(30,42,32,44)])
Matrix representation ►G ⊆ GL6(𝔽5)
| 2 | 0 | 0 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 |
| 2 | 0 | 0 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 0 | 0 | 3 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 4 | 0 | 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 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(6,GF(5))| [2,0,0,0,0,0,0,3,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[2,0,0,0,0,0,0,3,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,2,0,0,0,0,0,0,3],[0,4,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,4,0],[0,4,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;
44 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 4A | ··· | 4X | 4Y | ··· | 4AF |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 8 | ··· | 8 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | Q8 | C4○D4 |
| kernel | C42.440D4 | C23.7Q8 | C42⋊9C4 | C23.65C23 | C23.67C23 | C22×C42 | C2×C22⋊Q8 | C2×C4⋊Q8 | C42 | C22×C4 | C22×C4 | C2×C4 |
| # reps | 1 | 4 | 1 | 4 | 2 | 1 | 2 | 1 | 4 | 4 | 8 | 12 |
In GAP, Magma, Sage, TeX
C_4^2._{440}D_4 % in TeX
G:=Group("C4^2.440D4"); // GroupNames label
G:=SmallGroup(128,1589);
// by ID
G=gap.SmallGroup(128,1589);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,2,112,253,120,758,184,2019]);
// 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=d*a*d^-1=a^-1,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=a^2*c^-1>;
// generators/relations