p-group, metabelian, nilpotent (class 2), monomial
Aliases: C42⋊46D4, C24.119C23, C23.750C24, C42⋊8C4⋊74C2, (C22×C42)⋊15C2, C23.373(C2×D4), (C22×C4).598D4, C4.102(C4⋊D4), C22⋊1(C4.4D4), C23.250(C4○D4), (C22×C4).260C23, (C23×C4).650C22, C22.460(C22×D4), C23.23D4⋊113C2, C24.3C22⋊99C2, (C2×C42).1011C22, (C22×D4).309C22, (C22×Q8).247C22, C24.C22⋊182C2, C2.93(C22.19C24), C23.67C23⋊108C2, C2.C42.447C22, C2.54(C22.26C24), C2.108(C23.36C23), (C2×C4).685(C2×D4), C2.47(C2×C4⋊D4), (C2×C22⋊Q8)⋊50C2, (C2×C4.4D4)⋊34C2, (C2×C4⋊D4).50C2, C2.33(C2×C4.4D4), (C2×C4).668(C4○D4), (C2×C4⋊C4).553C22, C22.591(C2×C4○D4), (C2×C22⋊C4).360C22, SmallGroup(128,1582)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42⋊46D4
G = < a,b,c,d | a4=b4=c4=d2=1, ab=ba, cac-1=dad=ab2, cbc-1=dbd=a2b, dcd=c-1 >
Subgroups: 660 in 348 conjugacy classes, 120 normal (26 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C2×C4, C2×C4, D4, Q8, C23, C23, C23, C42, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×D4, C2×Q8, C24, C24, C2.C42, C2×C42, C2×C42, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C4⋊D4, C22⋊Q8, C4.4D4, C23×C4, C23×C4, C22×D4, C22×D4, C22×Q8, C42⋊8C4, C23.23D4, C24.C22, C24.3C22, C23.67C23, C22×C42, C2×C4⋊D4, C2×C22⋊Q8, C2×C4.4D4, C42⋊46D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C4⋊D4, C4.4D4, C22×D4, C2×C4○D4, C2×C4⋊D4, C22.19C24, C2×C4.4D4, C23.36C23, C22.26C24, C42⋊46D4
►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 47 39 15)(2 48 40 16)(3 45 37 13)(4 46 38 14)(5 41 17 9)(6 42 18 10)(7 43 19 11)(8 44 20 12)(21 57 49 29)(22 58 50 30)(23 59 51 31)(24 60 52 32)(25 61 53 36)(26 62 54 33)(27 63 55 34)(28 64 56 35)
(1 11 51 55)(2 44 52 28)(3 9 49 53)(4 42 50 26)(5 31 36 47)(6 60 33 16)(7 29 34 45)(8 58 35 14)(10 22 54 38)(12 24 56 40)(13 19 57 63)(15 17 59 61)(18 32 62 48)(20 30 64 46)(21 25 37 41)(23 27 39 43)
(2 40)(4 38)(5 34)(6 64)(7 36)(8 62)(9 53)(10 26)(11 55)(12 28)(13 15)(14 48)(16 46)(17 63)(18 35)(19 61)(20 33)(22 50)(24 52)(25 41)(27 43)(29 31)(30 60)(32 58)(42 54)(44 56)(45 47)(57 59)
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), (1,47,39,15)(2,48,40,16)(3,45,37,13)(4,46,38,14)(5,41,17,9)(6,42,18,10)(7,43,19,11)(8,44,20,12)(21,57,49,29)(22,58,50,30)(23,59,51,31)(24,60,52,32)(25,61,53,36)(26,62,54,33)(27,63,55,34)(28,64,56,35), (1,11,51,55)(2,44,52,28)(3,9,49,53)(4,42,50,26)(5,31,36,47)(6,60,33,16)(7,29,34,45)(8,58,35,14)(10,22,54,38)(12,24,56,40)(13,19,57,63)(15,17,59,61)(18,32,62,48)(20,30,64,46)(21,25,37,41)(23,27,39,43), (2,40)(4,38)(5,34)(6,64)(7,36)(8,62)(9,53)(10,26)(11,55)(12,28)(13,15)(14,48)(16,46)(17,63)(18,35)(19,61)(20,33)(22,50)(24,52)(25,41)(27,43)(29,31)(30,60)(32,58)(42,54)(44,56)(45,47)(57,59)>;
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), (1,47,39,15)(2,48,40,16)(3,45,37,13)(4,46,38,14)(5,41,17,9)(6,42,18,10)(7,43,19,11)(8,44,20,12)(21,57,49,29)(22,58,50,30)(23,59,51,31)(24,60,52,32)(25,61,53,36)(26,62,54,33)(27,63,55,34)(28,64,56,35), (1,11,51,55)(2,44,52,28)(3,9,49,53)(4,42,50,26)(5,31,36,47)(6,60,33,16)(7,29,34,45)(8,58,35,14)(10,22,54,38)(12,24,56,40)(13,19,57,63)(15,17,59,61)(18,32,62,48)(20,30,64,46)(21,25,37,41)(23,27,39,43), (2,40)(4,38)(5,34)(6,64)(7,36)(8,62)(9,53)(10,26)(11,55)(12,28)(13,15)(14,48)(16,46)(17,63)(18,35)(19,61)(20,33)(22,50)(24,52)(25,41)(27,43)(29,31)(30,60)(32,58)(42,54)(44,56)(45,47)(57,59) );
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)], [(1,47,39,15),(2,48,40,16),(3,45,37,13),(4,46,38,14),(5,41,17,9),(6,42,18,10),(7,43,19,11),(8,44,20,12),(21,57,49,29),(22,58,50,30),(23,59,51,31),(24,60,52,32),(25,61,53,36),(26,62,54,33),(27,63,55,34),(28,64,56,35)], [(1,11,51,55),(2,44,52,28),(3,9,49,53),(4,42,50,26),(5,31,36,47),(6,60,33,16),(7,29,34,45),(8,58,35,14),(10,22,54,38),(12,24,56,40),(13,19,57,63),(15,17,59,61),(18,32,62,48),(20,30,64,46),(21,25,37,41),(23,27,39,43)], [(2,40),(4,38),(5,34),(6,64),(7,36),(8,62),(9,53),(10,26),(11,55),(12,28),(13,15),(14,48),(16,46),(17,63),(18,35),(19,61),(20,33),(22,50),(24,52),(25,41),(27,43),(29,31),(30,60),(32,58),(42,54),(44,56),(45,47),(57,59)]])
44 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 2L | 2M | 4A | ··· | 4X | 4Y | ··· | 4AD |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 8 | 8 | 2 | ··· | 2 | 8 | ··· | 8 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | C4○D4 | C4○D4 |
| kernel | C42⋊46D4 | C42⋊8C4 | C23.23D4 | C24.C22 | C24.3C22 | C23.67C23 | C22×C42 | C2×C4⋊D4 | C2×C22⋊Q8 | C2×C4.4D4 | C42 | C22×C4 | C2×C4 | C23 |
| # reps | 1 | 1 | 4 | 4 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 4 | 12 | 8 |
Matrix representation of C42⋊46D4 ►in GL6(𝔽5)
| 3 | 0 | 0 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 3 |
| 0 | 0 | 0 | 0 | 3 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 0 | 0 | 0 |
| 0 | 0 | 0 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 3 |
| 0 | 0 | 0 | 0 | 3 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 4 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 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 | 1 |
G:=sub<GL(6,GF(5))| [3,0,0,0,0,0,0,3,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,3,0,0,0,0,3,0],[0,4,0,0,0,0,4,0,0,0,0,0,0,0,2,0,0,0,0,0,0,2,0,0,0,0,0,0,0,3,0,0,0,0,3,0],[0,1,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,1,0],[1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,1] >;
C42⋊46D4 in GAP, Magma, Sage, TeX
C_4^2\rtimes_{46}D_4 % in TeX
G:=Group("C4^2:46D4"); // GroupNames label
G:=SmallGroup(128,1582);
// by ID
G=gap.SmallGroup(128,1582);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,2,253,232,758,100,2019]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^4=d^2=1,a*b=b*a,c*a*c^-1=d*a*d=a*b^2,c*b*c^-1=d*b*d=a^2*b,d*c*d=c^-1>;
// generators/relations