direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C5×C23⋊C4, C23⋊C20, (C2×C4)⋊C20, (C2×C20)⋊7C4, C22⋊C4⋊1C10, (C22×C10)⋊1C4, (C2×D4).1C10, (D4×C10).7C2, (C2×C10).21D4, C22.2(C5×D4), C23.1(C2×C10), C22.2(C2×C20), C10.32(C22⋊C4), (C22×C10).1C22, (C5×C22⋊C4)⋊2C2, C2.3(C5×C22⋊C4), (C2×C10).39(C2×C4), SmallGroup(160,49)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×C23⋊C4
G = < a,b,c,d,e | a5=b2=c2=d2=e4=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, ebe-1=bcd, ece-1=cd=dc, de=ed >
Smallest permutation representation of C5×C23⋊C4
►On 40 points
Generators in S40
(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)
(1 36)(2 37)(3 38)(4 39)(5 40)(6 34)(7 35)(8 31)(9 32)(10 33)(11 22)(12 23)(13 24)(14 25)(15 21)(16 27)(17 28)(18 29)(19 30)(20 26)
(1 35)(2 31)(3 32)(4 33)(5 34)(6 40)(7 36)(8 37)(9 38)(10 39)(11 17)(12 18)(13 19)(14 20)(15 16)(21 27)(22 28)(23 29)(24 30)(25 26)
(1 21)(2 22)(3 23)(4 24)(5 25)(6 20)(7 16)(8 17)(9 18)(10 19)(11 37)(12 38)(13 39)(14 40)(15 36)(26 34)(27 35)(28 31)(29 32)(30 33)
(6 14 20 40)(7 15 16 36)(8 11 17 37)(9 12 18 38)(10 13 19 39)(26 34)(27 35)(28 31)(29 32)(30 33)
G:=sub<Sym(40)| (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), (1,36)(2,37)(3,38)(4,39)(5,40)(6,34)(7,35)(8,31)(9,32)(10,33)(11,22)(12,23)(13,24)(14,25)(15,21)(16,27)(17,28)(18,29)(19,30)(20,26), (1,35)(2,31)(3,32)(4,33)(5,34)(6,40)(7,36)(8,37)(9,38)(10,39)(11,17)(12,18)(13,19)(14,20)(15,16)(21,27)(22,28)(23,29)(24,30)(25,26), (1,21)(2,22)(3,23)(4,24)(5,25)(6,20)(7,16)(8,17)(9,18)(10,19)(11,37)(12,38)(13,39)(14,40)(15,36)(26,34)(27,35)(28,31)(29,32)(30,33), (6,14,20,40)(7,15,16,36)(8,11,17,37)(9,12,18,38)(10,13,19,39)(26,34)(27,35)(28,31)(29,32)(30,33)>;
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), (1,36)(2,37)(3,38)(4,39)(5,40)(6,34)(7,35)(8,31)(9,32)(10,33)(11,22)(12,23)(13,24)(14,25)(15,21)(16,27)(17,28)(18,29)(19,30)(20,26), (1,35)(2,31)(3,32)(4,33)(5,34)(6,40)(7,36)(8,37)(9,38)(10,39)(11,17)(12,18)(13,19)(14,20)(15,16)(21,27)(22,28)(23,29)(24,30)(25,26), (1,21)(2,22)(3,23)(4,24)(5,25)(6,20)(7,16)(8,17)(9,18)(10,19)(11,37)(12,38)(13,39)(14,40)(15,36)(26,34)(27,35)(28,31)(29,32)(30,33), (6,14,20,40)(7,15,16,36)(8,11,17,37)(9,12,18,38)(10,13,19,39)(26,34)(27,35)(28,31)(29,32)(30,33) );
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)], [(1,36),(2,37),(3,38),(4,39),(5,40),(6,34),(7,35),(8,31),(9,32),(10,33),(11,22),(12,23),(13,24),(14,25),(15,21),(16,27),(17,28),(18,29),(19,30),(20,26)], [(1,35),(2,31),(3,32),(4,33),(5,34),(6,40),(7,36),(8,37),(9,38),(10,39),(11,17),(12,18),(13,19),(14,20),(15,16),(21,27),(22,28),(23,29),(24,30),(25,26)], [(1,21),(2,22),(3,23),(4,24),(5,25),(6,20),(7,16),(8,17),(9,18),(10,19),(11,37),(12,38),(13,39),(14,40),(15,36),(26,34),(27,35),(28,31),(29,32),(30,33)], [(6,14,20,40),(7,15,16,36),(8,11,17,37),(9,12,18,38),(10,13,19,39),(26,34),(27,35),(28,31),(29,32),(30,33)]])
C5×C23⋊C4 is a maximal subgroup of
C5⋊3C2≀C4 (C2×C20).D4 C23.D20 C23.2D20 C23⋊C4⋊5D5 C23⋊D20 C23.5D20
55 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | ··· | 4E | 5A | 5B | 5C | 5D | 10A | 10B | 10C | 10D | 10E | ··· | 10P | 10Q | 10R | 10S | 10T | 20A | ··· | 20T |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 5 | 5 | 5 | 5 | 10 | 10 | 10 | 10 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 2 | 2 | 2 | 4 | 4 | ··· | 4 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
55 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | |||||||||
| image | C1 | C2 | C2 | C4 | C4 | C5 | C10 | C10 | C20 | C20 | D4 | C5×D4 | C23⋊C4 | C5×C23⋊C4 |
| kernel | C5×C23⋊C4 | C5×C22⋊C4 | D4×C10 | C2×C20 | C22×C10 | C23⋊C4 | C22⋊C4 | C2×D4 | C2×C4 | C23 | C2×C10 | C22 | C5 | C1 |
| # reps | 1 | 2 | 1 | 2 | 2 | 4 | 8 | 4 | 8 | 8 | 2 | 8 | 1 | 4 |
Matrix representation of C5×C23⋊C4 ►in GL4(𝔽41) generated by
| 37 | 0 | 0 | 0 |
| 0 | 37 | 0 | 0 |
| 0 | 0 | 37 | 0 |
| 0 | 0 | 0 | 37 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 40 | 0 | 0 | 0 |
| 39 | 1 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 39 | 1 |
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
| 1 | 40 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 40 | 1 |
| 0 | 0 | 39 | 1 |
G:=sub<GL(4,GF(41))| [37,0,0,0,0,37,0,0,0,0,37,0,0,0,0,37],[0,0,1,0,0,0,0,1,1,0,0,0,0,1,0,0],[40,39,0,0,0,1,0,0,0,0,40,39,0,0,0,1],[40,0,0,0,0,40,0,0,0,0,40,0,0,0,0,40],[1,0,0,0,40,40,0,0,0,0,40,39,0,0,1,1] >;
C5×C23⋊C4 in GAP, Magma, Sage, TeX
C_5\times C_2^3\rtimes C_4
% in TeX
G:=Group("C5xC2^3:C4"); // GroupNames label
G:=SmallGroup(160,49);
// by ID
G=gap.SmallGroup(160,49);
# by ID
G:=PCGroup([6,-2,-2,-5,-2,-2,-2,240,265,2403,1810]);
// Polycyclic
G:=Group<a,b,c,d,e|a^5=b^2=c^2=d^2=e^4=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,e*b*e^-1=b*c*d,e*c*e^-1=c*d=d*c,d*e=e*d>;
// generators/relations
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