p-group, metabelian, nilpotent (class 2), monomial
Aliases: C42.439D4, C24.597C23, C23.751C24, C42⋊8C4⋊75C2, (C22×C4).98Q8, C4.88(C22⋊Q8), C23.104(C2×Q8), (C22×C42).26C2, C22⋊1(C42.C2), C23.251(C4○D4), (C23×C4).651C22, (C22×C4).261C23, C22.461(C22×D4), C23.8Q8.70C2, C23.7Q8.78C2, C22.179(C22×Q8), (C2×C42).1012C22, C2.94(C22.19C24), C23.63C23⋊202C2, C23.65C23⋊167C2, C2.C42.448C22, C2.45(C23.37C23), C2.109(C23.36C23), (C2×C4).171(C2×Q8), C2.46(C2×C22⋊Q8), (C2×C4).1205(C2×D4), (C2×C42.C2)⋊29C2, C2.21(C2×C42.C2), (C2×C4).669(C4○D4), (C2×C4⋊C4).554C22, C22.592(C2×C4○D4), (C2×C22⋊C4).361C22, SmallGroup(128,1583)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.439D4
G = < a,b,c,d | a4=b4=c4=1, d2=b2, ab=ba, cac-1=dad-1=ab2, cbc-1=dbd-1=a2b, dcd-1=b2c-1 >
Subgroups: 452 in 278 conjugacy classes, 120 normal (18 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C2×C4, C2×C4, C23, C23, C23, C42, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C24, C2.C42, C2×C42, C2×C42, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C42.C2, C23×C4, C23×C4, C23.7Q8, C42⋊8C4, C23.8Q8, C23.63C23, C23.65C23, C22×C42, C2×C42.C2, C42.439D4
Quotients: C1, C2, C22, D4, Q8, C23, C2×D4, C2×Q8, C4○D4, C24, C22⋊Q8, C42.C2, C22×D4, C22×Q8, C2×C4○D4, C2×C22⋊Q8, C22.19C24, C2×C42.C2, C23.36C23, C23.37C23, C42.439D4
►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 5 23 11)(2 6 24 12)(3 7 21 9)(4 8 22 10)(13 29 25 37)(14 30 26 38)(15 31 27 39)(16 32 28 40)(17 44 62 60)(18 41 63 57)(19 42 64 58)(20 43 61 59)(33 51 46 56)(34 52 47 53)(35 49 48 54)(36 50 45 55)
(1 57 15 45)(2 42 16 33)(3 59 13 47)(4 44 14 35)(5 20 31 53)(6 62 32 49)(7 18 29 55)(8 64 30 51)(9 63 37 50)(10 19 38 56)(11 61 39 52)(12 17 40 54)(21 43 25 34)(22 60 26 48)(23 41 27 36)(24 58 28 46)
(1 34 23 47)(2 48 24 35)(3 36 21 45)(4 46 22 33)(5 50 11 55)(6 56 12 51)(7 52 9 53)(8 54 10 49)(13 41 25 57)(14 58 26 42)(15 43 27 59)(16 60 28 44)(17 38 62 30)(18 31 63 39)(19 40 64 32)(20 29 61 37)
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), (1,5,23,11)(2,6,24,12)(3,7,21,9)(4,8,22,10)(13,29,25,37)(14,30,26,38)(15,31,27,39)(16,32,28,40)(17,44,62,60)(18,41,63,57)(19,42,64,58)(20,43,61,59)(33,51,46,56)(34,52,47,53)(35,49,48,54)(36,50,45,55), (1,57,15,45)(2,42,16,33)(3,59,13,47)(4,44,14,35)(5,20,31,53)(6,62,32,49)(7,18,29,55)(8,64,30,51)(9,63,37,50)(10,19,38,56)(11,61,39,52)(12,17,40,54)(21,43,25,34)(22,60,26,48)(23,41,27,36)(24,58,28,46), (1,34,23,47)(2,48,24,35)(3,36,21,45)(4,46,22,33)(5,50,11,55)(6,56,12,51)(7,52,9,53)(8,54,10,49)(13,41,25,57)(14,58,26,42)(15,43,27,59)(16,60,28,44)(17,38,62,30)(18,31,63,39)(19,40,64,32)(20,29,61,37)>;
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), (1,5,23,11)(2,6,24,12)(3,7,21,9)(4,8,22,10)(13,29,25,37)(14,30,26,38)(15,31,27,39)(16,32,28,40)(17,44,62,60)(18,41,63,57)(19,42,64,58)(20,43,61,59)(33,51,46,56)(34,52,47,53)(35,49,48,54)(36,50,45,55), (1,57,15,45)(2,42,16,33)(3,59,13,47)(4,44,14,35)(5,20,31,53)(6,62,32,49)(7,18,29,55)(8,64,30,51)(9,63,37,50)(10,19,38,56)(11,61,39,52)(12,17,40,54)(21,43,25,34)(22,60,26,48)(23,41,27,36)(24,58,28,46), (1,34,23,47)(2,48,24,35)(3,36,21,45)(4,46,22,33)(5,50,11,55)(6,56,12,51)(7,52,9,53)(8,54,10,49)(13,41,25,57)(14,58,26,42)(15,43,27,59)(16,60,28,44)(17,38,62,30)(18,31,63,39)(19,40,64,32)(20,29,61,37) );
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)], [(1,5,23,11),(2,6,24,12),(3,7,21,9),(4,8,22,10),(13,29,25,37),(14,30,26,38),(15,31,27,39),(16,32,28,40),(17,44,62,60),(18,41,63,57),(19,42,64,58),(20,43,61,59),(33,51,46,56),(34,52,47,53),(35,49,48,54),(36,50,45,55)], [(1,57,15,45),(2,42,16,33),(3,59,13,47),(4,44,14,35),(5,20,31,53),(6,62,32,49),(7,18,29,55),(8,64,30,51),(9,63,37,50),(10,19,38,56),(11,61,39,52),(12,17,40,54),(21,43,25,34),(22,60,26,48),(23,41,27,36),(24,58,28,46)], [(1,34,23,47),(2,48,24,35),(3,36,21,45),(4,46,22,33),(5,50,11,55),(6,56,12,51),(7,52,9,53),(8,54,10,49),(13,41,25,57),(14,58,26,42),(15,43,27,59),(16,60,28,44),(17,38,62,30),(18,31,63,39),(19,40,64,32),(20,29,61,37)]])
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 | Q8 | C4○D4 | C4○D4 |
| kernel | C42.439D4 | C23.7Q8 | C42⋊8C4 | C23.8Q8 | C23.63C23 | C23.65C23 | C22×C42 | C2×C42.C2 | C42 | C22×C4 | C2×C4 | C23 |
| # reps | 1 | 2 | 1 | 4 | 4 | 2 | 1 | 1 | 4 | 4 | 12 | 8 |
Matrix representation of C42.439D4 ►in GL6(𝔽5)
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 0 | 0 | 3 |
| 2 | 0 | 0 | 0 | 0 | 0 |
| 0 | 2 | 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 | 1 |
| 0 | 1 | 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 | 4 |
| 0 | 0 | 0 | 0 | 4 | 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))| [4,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,3,0,0,0,0,0,0,3],[2,0,0,0,0,0,0,2,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,1],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,4,0],[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] >;
C42.439D4 in GAP, Magma, Sage, TeX
C_4^2._{439}D_4 % in TeX
G:=Group("C4^2.439D4"); // GroupNames label
G:=SmallGroup(128,1583);
// by ID
G=gap.SmallGroup(128,1583);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,2,112,253,232,758,100,2019]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^4=1,d^2=b^2,a*b=b*a,c*a*c^-1=d*a*d^-1=a*b^2,c*b*c^-1=d*b*d^-1=a^2*b,d*c*d^-1=b^2*c^-1>;
// generators/relations