p-group, metabelian, nilpotent (class 2), monomial
Aliases: C42⋊28D4, C23.520C24, C24.363C23, C22.2982+ 1+4, C22.2172- 1+4, C42⋊9C4⋊30C2, (C22×C4).399D4, C23.192(C2×D4), C4.100(C4⋊D4), C23.Q8⋊37C2, C23.10D4⋊57C2, (C22×C4).130C23, (C23×C4).423C22, (C2×C42).601C22, C22.345(C22×D4), C24.3C22⋊65C2, (C22×D4).192C22, (C22×Q8).151C22, C23.67C23⋊70C2, C2.35(C22.29C24), C2.C42.247C22, C2.44(C22.36C24), C2.34(C23.38C23), C2.24(C22.31C24), (C2×C4).380(C2×D4), C2.44(C2×C4⋊D4), (C2×C22⋊Q8)⋊27C2, (C2×C4.4D4)⋊20C2, (C2×C4⋊D4).39C2, (C2×C42⋊C2)⋊37C2, (C2×C4).656(C4○D4), (C2×C4⋊C4).887C22, C22.392(C2×C4○D4), (C2×C22⋊C4).212C22, SmallGroup(128,1352)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42⋊28D4
G = < a,b,c,d | a4=b4=c4=d2=1, ab=ba, cac-1=a-1, dad=ab2, cbc-1=b-1, bd=db, dcd=c-1 >
Subgroups: 628 in 306 conjugacy classes, 108 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×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C42⋊C2, C4⋊D4, C22⋊Q8, C4.4D4, C23×C4, C22×D4, C22×D4, C22×Q8, C42⋊9C4, C24.3C22, C23.67C23, C23.10D4, C23.Q8, C2×C42⋊C2, C2×C4⋊D4, C2×C22⋊Q8, C2×C4.4D4, C42⋊28D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C4⋊D4, C22×D4, C2×C4○D4, 2+ 1+4, 2- 1+4, C2×C4⋊D4, C22.29C24, C23.38C23, C22.31C24, C22.36C24, C42⋊28D4
►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 15 41 47)(2 16 42 48)(3 13 43 45)(4 14 44 46)(5 35 62 37)(6 36 63 38)(7 33 64 39)(8 34 61 40)(9 17 23 49)(10 18 24 50)(11 19 21 51)(12 20 22 52)(25 55 31 57)(26 56 32 58)(27 53 29 59)(28 54 30 60)
(1 6 57 17)(2 5 58 20)(3 8 59 19)(4 7 60 18)(9 15 38 25)(10 14 39 28)(11 13 40 27)(12 16 37 26)(21 45 34 29)(22 48 35 32)(23 47 36 31)(24 46 33 30)(41 63 55 49)(42 62 56 52)(43 61 53 51)(44 64 54 50)
(1 17)(2 50)(3 19)(4 52)(5 54)(6 57)(7 56)(8 59)(9 47)(10 16)(11 45)(12 14)(13 21)(15 23)(18 42)(20 44)(22 46)(24 48)(25 36)(26 39)(27 34)(28 37)(29 40)(30 35)(31 38)(32 33)(41 49)(43 51)(53 61)(55 63)(58 64)(60 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), (1,15,41,47)(2,16,42,48)(3,13,43,45)(4,14,44,46)(5,35,62,37)(6,36,63,38)(7,33,64,39)(8,34,61,40)(9,17,23,49)(10,18,24,50)(11,19,21,51)(12,20,22,52)(25,55,31,57)(26,56,32,58)(27,53,29,59)(28,54,30,60), (1,6,57,17)(2,5,58,20)(3,8,59,19)(4,7,60,18)(9,15,38,25)(10,14,39,28)(11,13,40,27)(12,16,37,26)(21,45,34,29)(22,48,35,32)(23,47,36,31)(24,46,33,30)(41,63,55,49)(42,62,56,52)(43,61,53,51)(44,64,54,50), (1,17)(2,50)(3,19)(4,52)(5,54)(6,57)(7,56)(8,59)(9,47)(10,16)(11,45)(12,14)(13,21)(15,23)(18,42)(20,44)(22,46)(24,48)(25,36)(26,39)(27,34)(28,37)(29,40)(30,35)(31,38)(32,33)(41,49)(43,51)(53,61)(55,63)(58,64)(60,62)>;
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), (1,15,41,47)(2,16,42,48)(3,13,43,45)(4,14,44,46)(5,35,62,37)(6,36,63,38)(7,33,64,39)(8,34,61,40)(9,17,23,49)(10,18,24,50)(11,19,21,51)(12,20,22,52)(25,55,31,57)(26,56,32,58)(27,53,29,59)(28,54,30,60), (1,6,57,17)(2,5,58,20)(3,8,59,19)(4,7,60,18)(9,15,38,25)(10,14,39,28)(11,13,40,27)(12,16,37,26)(21,45,34,29)(22,48,35,32)(23,47,36,31)(24,46,33,30)(41,63,55,49)(42,62,56,52)(43,61,53,51)(44,64,54,50), (1,17)(2,50)(3,19)(4,52)(5,54)(6,57)(7,56)(8,59)(9,47)(10,16)(11,45)(12,14)(13,21)(15,23)(18,42)(20,44)(22,46)(24,48)(25,36)(26,39)(27,34)(28,37)(29,40)(30,35)(31,38)(32,33)(41,49)(43,51)(53,61)(55,63)(58,64)(60,62) );
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)], [(1,15,41,47),(2,16,42,48),(3,13,43,45),(4,14,44,46),(5,35,62,37),(6,36,63,38),(7,33,64,39),(8,34,61,40),(9,17,23,49),(10,18,24,50),(11,19,21,51),(12,20,22,52),(25,55,31,57),(26,56,32,58),(27,53,29,59),(28,54,30,60)], [(1,6,57,17),(2,5,58,20),(3,8,59,19),(4,7,60,18),(9,15,38,25),(10,14,39,28),(11,13,40,27),(12,16,37,26),(21,45,34,29),(22,48,35,32),(23,47,36,31),(24,46,33,30),(41,63,55,49),(42,62,56,52),(43,61,53,51),(44,64,54,50)], [(1,17),(2,50),(3,19),(4,52),(5,54),(6,57),(7,56),(8,59),(9,47),(10,16),(11,45),(12,14),(13,21),(15,23),(18,42),(20,44),(22,46),(24,48),(25,36),(26,39),(27,34),(28,37),(29,40),(30,35),(31,38),(32,33),(41,49),(43,51),(53,61),(55,63),(58,64),(60,62)]])
32 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 4A | 4B | 4C | 4D | 4E | ··· | 4N | 4O | ··· | 4T |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 |
| size | 1 | 1 | ··· | 1 | 4 | 4 | 8 | 8 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 8 | ··· | 8 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | C4○D4 | 2+ 1+4 | 2- 1+4 |
| kernel | C42⋊28D4 | C42⋊9C4 | C24.3C22 | C23.67C23 | C23.10D4 | C23.Q8 | C2×C42⋊C2 | C2×C4⋊D4 | C2×C22⋊Q8 | C2×C4.4D4 | C42 | C22×C4 | C2×C4 | C22 | C22 |
| # reps | 1 | 1 | 1 | 1 | 4 | 4 | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 2 | 2 |
Matrix representation of C42⋊28D4 ►in GL8(𝔽5)
| 1 | 2 | 0 | 0 | 0 | 0 | 0 | 0 |
| 4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 3 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 3 |
| 0 | 0 | 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 4 |
| 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 4 | 0 |
| 2 | 4 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 2 | 4 | 0 | 0 | 0 | 0 | 0 | 0 |
| 3 | 3 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 | 0 | 1 |
G:=sub<GL(8,GF(5))| [1,4,0,0,0,0,0,0,2,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,3,0,4,0,0,0,0,0,0,3,0,4],[4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0],[2,0,0,0,0,0,0,0,4,3,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1],[2,3,0,0,0,0,0,0,4,3,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,4,0,4,0,0,0,0,0,0,4,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1] >;
C42⋊28D4 in GAP, Magma, Sage, TeX
C_4^2\rtimes_{28}D_4 % in TeX
G:=Group("C4^2:28D4"); // GroupNames label
G:=SmallGroup(128,1352);
// by ID
G=gap.SmallGroup(128,1352);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,2,336,253,758,723,185,80]);
// 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*b^2,c*b*c^-1=b^-1,b*d=d*b,d*c*d=c^-1>;
// generators/relations