p-group, metabelian, nilpotent (class 2), monomial
Aliases: C43⋊5C2, C24.127C23, C23.766C24, (C2×C42).1020C22, (C22×C4).1268C23, C24.C22.86C2, C23.63C23⋊207C2, C2.C42.461C22, C2.119(C23.36C23), (C2×C4).535(C4○D4), (C2×C4⋊C4).569C22, C22.607(C2×C4○D4), (C2×C22⋊C4).372C22, SmallGroup(128,1598)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 356 in 209 conjugacy classes, 96 normal (5 characteristic)
C1, C2 [×7], C2, C4 [×21], C22 [×7], C22 [×7], C2×C4 [×14], C2×C4 [×35], C23, C23 [×7], C42 [×14], C22⋊C4 [×14], C4⋊C4 [×14], C22×C4 [×14], C24, C2.C42 [×14], C2×C42 [×7], C2×C22⋊C4 [×7], C2×C4⋊C4 [×7], C43, C23.63C23 [×7], C24.C22 [×7], C43⋊5C2
Quotients:
C1, C2 [×15], C22 [×35], C23 [×15], C4○D4 [×14], C24, C2×C4○D4 [×7], C23.36C23 [×7], C43⋊5C2
Generators and relations
G = < a,b,c,d | a4=b4=c4=d2=1, ab=ba, ac=ca, dad=ac2, bc=cb, dbd=b-1c2, dcd=a2b2c >
►On 64 points
Generators in S64
(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 18 10 7)(2 19 11 8)(3 20 12 5)(4 17 9 6)(13 36 28 21)(14 33 25 22)(15 34 26 23)(16 35 27 24)(29 60 47 37)(30 57 48 38)(31 58 45 39)(32 59 46 40)(41 64 56 51)(42 61 53 52)(43 62 54 49)(44 63 55 50)
(1 28 55 31)(2 25 56 32)(3 26 53 29)(4 27 54 30)(5 34 61 37)(6 35 62 38)(7 36 63 39)(8 33 64 40)(9 16 43 48)(10 13 44 45)(11 14 41 46)(12 15 42 47)(17 24 49 57)(18 21 50 58)(19 22 51 59)(20 23 52 60)
(2 56)(4 54)(5 52)(6 17)(7 50)(8 19)(9 43)(11 41)(13 26)(14 30)(15 28)(16 32)(18 63)(20 61)(21 60)(22 24)(23 58)(25 48)(27 46)(29 45)(31 47)(33 35)(34 39)(36 37)(38 40)(49 62)(51 64)(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), (1,18,10,7)(2,19,11,8)(3,20,12,5)(4,17,9,6)(13,36,28,21)(14,33,25,22)(15,34,26,23)(16,35,27,24)(29,60,47,37)(30,57,48,38)(31,58,45,39)(32,59,46,40)(41,64,56,51)(42,61,53,52)(43,62,54,49)(44,63,55,50), (1,28,55,31)(2,25,56,32)(3,26,53,29)(4,27,54,30)(5,34,61,37)(6,35,62,38)(7,36,63,39)(8,33,64,40)(9,16,43,48)(10,13,44,45)(11,14,41,46)(12,15,42,47)(17,24,49,57)(18,21,50,58)(19,22,51,59)(20,23,52,60), (2,56)(4,54)(5,52)(6,17)(7,50)(8,19)(9,43)(11,41)(13,26)(14,30)(15,28)(16,32)(18,63)(20,61)(21,60)(22,24)(23,58)(25,48)(27,46)(29,45)(31,47)(33,35)(34,39)(36,37)(38,40)(49,62)(51,64)(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), (1,18,10,7)(2,19,11,8)(3,20,12,5)(4,17,9,6)(13,36,28,21)(14,33,25,22)(15,34,26,23)(16,35,27,24)(29,60,47,37)(30,57,48,38)(31,58,45,39)(32,59,46,40)(41,64,56,51)(42,61,53,52)(43,62,54,49)(44,63,55,50), (1,28,55,31)(2,25,56,32)(3,26,53,29)(4,27,54,30)(5,34,61,37)(6,35,62,38)(7,36,63,39)(8,33,64,40)(9,16,43,48)(10,13,44,45)(11,14,41,46)(12,15,42,47)(17,24,49,57)(18,21,50,58)(19,22,51,59)(20,23,52,60), (2,56)(4,54)(5,52)(6,17)(7,50)(8,19)(9,43)(11,41)(13,26)(14,30)(15,28)(16,32)(18,63)(20,61)(21,60)(22,24)(23,58)(25,48)(27,46)(29,45)(31,47)(33,35)(34,39)(36,37)(38,40)(49,62)(51,64)(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)], [(1,18,10,7),(2,19,11,8),(3,20,12,5),(4,17,9,6),(13,36,28,21),(14,33,25,22),(15,34,26,23),(16,35,27,24),(29,60,47,37),(30,57,48,38),(31,58,45,39),(32,59,46,40),(41,64,56,51),(42,61,53,52),(43,62,54,49),(44,63,55,50)], [(1,28,55,31),(2,25,56,32),(3,26,53,29),(4,27,54,30),(5,34,61,37),(6,35,62,38),(7,36,63,39),(8,33,64,40),(9,16,43,48),(10,13,44,45),(11,14,41,46),(12,15,42,47),(17,24,49,57),(18,21,50,58),(19,22,51,59),(20,23,52,60)], [(2,56),(4,54),(5,52),(6,17),(7,50),(8,19),(9,43),(11,41),(13,26),(14,30),(15,28),(16,32),(18,63),(20,61),(21,60),(22,24),(23,58),(25,48),(27,46),(29,45),(31,47),(33,35),(34,39),(36,37),(38,40),(49,62),(51,64),(57,59)])
Matrix representation ►G ⊆ GL6(𝔽5)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 0 | 0 | 0 |
| 0 | 0 | 0 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 2 |
| 0 | 0 | 0 | 0 | 4 | 4 |
| 0 | 3 | 0 | 0 | 0 | 0 |
| 3 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 0 | 0 | 2 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 3 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 0 | 0 | 3 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 4 | 4 |
G:=sub<GL(6,GF(5))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,3,0,0,0,0,0,0,3,0,0,0,0,0,0,1,4,0,0,0,0,2,4],[0,3,0,0,0,0,3,0,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,2,0,0,0,0,0,0,2],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,3,4,0,0,0,0,0,0,3,0,0,0,0,0,0,3],[1,0,0,0,0,0,0,4,0,0,0,0,0,0,1,1,0,0,0,0,0,4,0,0,0,0,0,0,1,4,0,0,0,0,0,4] >;
44 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 4A | ··· | 4AB | 4AC | ··· | 4AI |
| order | 1 | 2 | ··· | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
| size | 1 | 1 | ··· | 1 | 8 | 2 | ··· | 2 | 8 | ··· | 8 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 |
| type | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C4○D4 |
| kernel | C43⋊5C2 | C43 | C23.63C23 | C24.C22 | C2×C4 |
| # reps | 1 | 1 | 7 | 7 | 28 |
In GAP, Magma, Sage, TeX
C_4^3\rtimes_5C_2
% in TeX
G:=Group("C4^3:5C2"); // GroupNames label
G:=SmallGroup(128,1598);
// by ID
G=gap.SmallGroup(128,1598);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,2,253,792,758,268,2019,80]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^4=d^2=1,a*b=b*a,a*c=c*a,d*a*d=a*c^2,b*c=c*b,d*b*d=b^-1*c^2,d*c*d=a^2*b^2*c>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀