p-group, metabelian, nilpotent (class 2), monomial
Aliases: C42⋊42D4, C24.533C23, C23.172C24, (C4×D4)⋊17C4, C42⋊14(C2×C4), C4.180(C4×D4), C42⋊4C4⋊9C2, (C22×C42)⋊5C2, C4○(C23.8Q8), C4○(C23.23D4), C22.63(C23×C4), C22.70(C22×D4), C22⋊1(C42⋊C2), C23.215(C4○D4), C4○(C24.C22), (C23×C4).646C22, C23.117(C22×C4), C23.8Q8⋊149C2, C4○(C23.63C23), (C22×C4).1237C23, (C2×C42).1003C22, C2.2(C22.19C24), C23.23D4.81C2, (C22×D4).463C22, C24.C22⋊188C2, C23.63C23⋊208C2, C2.C42.508C22, C2.2(C23.36C23), C2.9(C2×C4×D4), (C4×C4⋊C4)⋊15C2, C4⋊C4⋊37(C2×C4), (C2×C4×D4).25C2, C2.5(C4×C4○D4), (C4×C22⋊C4)⋊5C2, C22⋊C4⋊36(C2×C4), (C22×C4)⋊17(C2×C4), (C2×C42⋊C2)⋊5C2, (C2×D4).208(C2×C4), (C2×C4).1186(C2×D4), C22.64(C2×C4○D4), (C2×C4).513(C4○D4), (C2×C4⋊C4).788C22, (C2×C4).206(C22×C4), C2.15(C2×C42⋊C2), (C2×C22⋊C4).415C22, SmallGroup(128,1022)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42⋊42D4
G = < a,b,c,d | a4=b4=c4=d2=1, ab=ba, cac-1=dad=ab2, bc=cb, bd=db, dcd=c-1 >
Subgroups: 556 in 348 conjugacy classes, 160 normal (34 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C2×C4, C2×C4, D4, C23, C23, C23, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C24, C2.C42, C2×C42, C2×C42, C2×C42, C2×C22⋊C4, C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C42⋊C2, C4×D4, C23×C4, C23×C4, C22×D4, C42⋊4C4, C4×C22⋊C4, C4×C4⋊C4, C23.8Q8, C23.23D4, C23.63C23, C24.C22, C22×C42, C2×C42⋊C2, C2×C4×D4, C42⋊42D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22×C4, C2×D4, C4○D4, C24, C42⋊C2, C4×D4, C23×C4, C22×D4, C2×C4○D4, C2×C42⋊C2, C2×C4×D4, C4×C4○D4, C22.19C24, C23.36C23, C42⋊42D4
►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 11 43)(2 16 12 44)(3 13 9 41)(4 14 10 42)(5 18 38 46)(6 19 39 47)(7 20 40 48)(8 17 37 45)(21 53 49 25)(22 54 50 26)(23 55 51 27)(24 56 52 28)(29 61 57 35)(30 62 58 36)(31 63 59 33)(32 64 60 34)
(1 47 51 59)(2 20 52 32)(3 45 49 57)(4 18 50 30)(5 54 36 42)(6 27 33 15)(7 56 34 44)(8 25 35 13)(9 17 21 29)(10 46 22 58)(11 19 23 31)(12 48 24 60)(14 38 26 62)(16 40 28 64)(37 53 61 41)(39 55 63 43)
(2 12)(4 10)(5 62)(6 33)(7 64)(8 35)(14 42)(16 44)(17 29)(18 58)(19 31)(20 60)(22 50)(24 52)(26 54)(28 56)(30 46)(32 48)(34 40)(36 38)(37 61)(39 63)(45 57)(47 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), (1,15,11,43)(2,16,12,44)(3,13,9,41)(4,14,10,42)(5,18,38,46)(6,19,39,47)(7,20,40,48)(8,17,37,45)(21,53,49,25)(22,54,50,26)(23,55,51,27)(24,56,52,28)(29,61,57,35)(30,62,58,36)(31,63,59,33)(32,64,60,34), (1,47,51,59)(2,20,52,32)(3,45,49,57)(4,18,50,30)(5,54,36,42)(6,27,33,15)(7,56,34,44)(8,25,35,13)(9,17,21,29)(10,46,22,58)(11,19,23,31)(12,48,24,60)(14,38,26,62)(16,40,28,64)(37,53,61,41)(39,55,63,43), (2,12)(4,10)(5,62)(6,33)(7,64)(8,35)(14,42)(16,44)(17,29)(18,58)(19,31)(20,60)(22,50)(24,52)(26,54)(28,56)(30,46)(32,48)(34,40)(36,38)(37,61)(39,63)(45,57)(47,59)>;
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), (1,15,11,43)(2,16,12,44)(3,13,9,41)(4,14,10,42)(5,18,38,46)(6,19,39,47)(7,20,40,48)(8,17,37,45)(21,53,49,25)(22,54,50,26)(23,55,51,27)(24,56,52,28)(29,61,57,35)(30,62,58,36)(31,63,59,33)(32,64,60,34), (1,47,51,59)(2,20,52,32)(3,45,49,57)(4,18,50,30)(5,54,36,42)(6,27,33,15)(7,56,34,44)(8,25,35,13)(9,17,21,29)(10,46,22,58)(11,19,23,31)(12,48,24,60)(14,38,26,62)(16,40,28,64)(37,53,61,41)(39,55,63,43), (2,12)(4,10)(5,62)(6,33)(7,64)(8,35)(14,42)(16,44)(17,29)(18,58)(19,31)(20,60)(22,50)(24,52)(26,54)(28,56)(30,46)(32,48)(34,40)(36,38)(37,61)(39,63)(45,57)(47,59) );
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)], [(1,15,11,43),(2,16,12,44),(3,13,9,41),(4,14,10,42),(5,18,38,46),(6,19,39,47),(7,20,40,48),(8,17,37,45),(21,53,49,25),(22,54,50,26),(23,55,51,27),(24,56,52,28),(29,61,57,35),(30,62,58,36),(31,63,59,33),(32,64,60,34)], [(1,47,51,59),(2,20,52,32),(3,45,49,57),(4,18,50,30),(5,54,36,42),(6,27,33,15),(7,56,34,44),(8,25,35,13),(9,17,21,29),(10,46,22,58),(11,19,23,31),(12,48,24,60),(14,38,26,62),(16,40,28,64),(37,53,61,41),(39,55,63,43)], [(2,12),(4,10),(5,62),(6,33),(7,64),(8,35),(14,42),(16,44),(17,29),(18,58),(19,31),(20,60),(22,50),(24,52),(26,54),(28,56),(30,46),(32,48),(34,40),(36,38),(37,61),(39,63),(45,57),(47,59)]])
56 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 2L | 2M | 4A | ··· | 4H | 4I | ··· | 4AB | 4AC | ··· | 4AP |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | ··· | 4 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | C4○D4 | C4○D4 |
| kernel | C42⋊42D4 | C42⋊4C4 | C4×C22⋊C4 | C4×C4⋊C4 | C23.8Q8 | C23.23D4 | C23.63C23 | C24.C22 | C22×C42 | C2×C42⋊C2 | C2×C4×D4 | C4×D4 | C42 | C2×C4 | C23 |
| # reps | 1 | 1 | 2 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 16 | 4 | 12 | 8 |
Matrix representation of C42⋊42D4 ►in GL5(𝔽5)
| 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 2 |
| 0 | 0 | 0 | 0 | 1 |
| 4 | 0 | 0 | 0 | 0 |
| 0 | 2 | 0 | 0 | 0 |
| 0 | 0 | 2 | 0 | 0 |
| 0 | 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 0 | 3 |
| 4 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 4 | 2 |
| 0 | 0 | 0 | 4 | 1 |
| 4 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 | 4 |
G:=sub<GL(5,GF(5))| [3,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,2,1],[4,0,0,0,0,0,2,0,0,0,0,0,2,0,0,0,0,0,3,0,0,0,0,0,3],[4,0,0,0,0,0,1,0,0,0,0,0,4,0,0,0,0,0,4,4,0,0,0,2,1],[4,0,0,0,0,0,1,0,0,0,0,0,4,0,0,0,0,0,1,1,0,0,0,0,4] >;
C42⋊42D4 in GAP, Magma, Sage, TeX
C_4^2\rtimes_{42}D_4 % in TeX
G:=Group("C4^2:42D4"); // GroupNames label
G:=SmallGroup(128,1022);
// by ID
G=gap.SmallGroup(128,1022);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,456,758,80]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^4=d^2=1,a*b=b*a,c*a*c^-1=d*a*d=a*b^2,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations