metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: (C2×C4).3D20, (C2×D20).4C4, (C2×C20).15D4, (C2×Q8).1D10, C4.10D4⋊5D5, (C4×Dic5).1C4, C20.10D4⋊1C2, C5⋊3(C42.C4), (Q8×C10).1C22, C10.34(C23⋊C4), C20.23D4.1C2, C22.14(D10⋊C4), C2.14(C22.2D20), (C2×C4).3(C4×D5), (C2×C20).3(C2×C4), (C2×C4).3(C5⋊D4), (C5×C4.10D4)⋊11C2, (C2×C10).71(C22⋊C4), SmallGroup(320,35)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for (C2×D20).C4
G = < a,b,c,d | a2=b20=c2=1, d4=b10, ab=ba, ac=ca, dad-1=ab10, cbc=b-1, dbd-1=ab11, dcd-1=b15c >
Subgroups: 334 in 64 conjugacy classes, 21 normal (all characteristic)
C1, C2, C2 [×2], C4 [×4], C22, C22 [×3], C5, C8 [×2], C2×C4 [×3], C2×C4, D4, Q8, C23, D5, C10, C10, C42, C22⋊C4 [×2], M4(2) [×2], C2×D4, C2×Q8, Dic5, C20 [×3], D10 [×3], C2×C10, C4.10D4, C4.10D4, C4.4D4, C5⋊2C8, C40, D20, C2×Dic5, C2×C20 [×3], C5×Q8, C22×D5, C42.C4, C4.Dic5, C4×Dic5, D10⋊C4 [×2], C5×M4(2), C2×D20, Q8×C10, C20.10D4, C5×C4.10D4, C20.23D4, (C2×D20).C4
Quotients: C1, C2 [×3], C4 [×2], C22, C2×C4, D4 [×2], D5, C22⋊C4, D10, C23⋊C4, C4×D5, D20, C5⋊D4, C42.C4, D10⋊C4, C22.2D20, (C2×D20).C4
(1 11)(2 12)(3 13)(4 14)(5 15)(6 16)(7 17)(8 18)(9 19)(10 20)(21 31)(22 32)(23 33)(24 34)(25 35)(26 36)(27 37)(28 38)(29 39)(30 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)(41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60)(61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80)
(1 20)(2 19)(3 18)(4 17)(5 16)(6 15)(7 14)(8 13)(9 12)(10 11)(21 40)(22 39)(23 38)(24 37)(25 36)(26 35)(27 34)(28 33)(29 32)(30 31)(41 55)(42 54)(43 53)(44 52)(45 51)(46 50)(47 49)(56 60)(57 59)(61 65)(62 64)(66 80)(67 79)(68 78)(69 77)(70 76)(71 75)(72 74)
(1 71 21 56 11 61 31 46)(2 72 32 47 12 62 22 57)(3 73 23 58 13 63 33 48)(4 74 34 49 14 64 24 59)(5 75 25 60 15 65 35 50)(6 76 36 51 16 66 26 41)(7 77 27 42 17 67 37 52)(8 78 38 53 18 68 28 43)(9 79 29 44 19 69 39 54)(10 80 40 55 20 70 30 45)
G:=sub<Sym(80)| (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,17)(8,18)(9,19)(10,20)(21,31)(22,32)(23,33)(24,34)(25,35)(26,36)(27,37)(28,38)(29,39)(30,40), (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)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80), (1,20)(2,19)(3,18)(4,17)(5,16)(6,15)(7,14)(8,13)(9,12)(10,11)(21,40)(22,39)(23,38)(24,37)(25,36)(26,35)(27,34)(28,33)(29,32)(30,31)(41,55)(42,54)(43,53)(44,52)(45,51)(46,50)(47,49)(56,60)(57,59)(61,65)(62,64)(66,80)(67,79)(68,78)(69,77)(70,76)(71,75)(72,74), (1,71,21,56,11,61,31,46)(2,72,32,47,12,62,22,57)(3,73,23,58,13,63,33,48)(4,74,34,49,14,64,24,59)(5,75,25,60,15,65,35,50)(6,76,36,51,16,66,26,41)(7,77,27,42,17,67,37,52)(8,78,38,53,18,68,28,43)(9,79,29,44,19,69,39,54)(10,80,40,55,20,70,30,45)>;
G:=Group( (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,17)(8,18)(9,19)(10,20)(21,31)(22,32)(23,33)(24,34)(25,35)(26,36)(27,37)(28,38)(29,39)(30,40), (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)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80), (1,20)(2,19)(3,18)(4,17)(5,16)(6,15)(7,14)(8,13)(9,12)(10,11)(21,40)(22,39)(23,38)(24,37)(25,36)(26,35)(27,34)(28,33)(29,32)(30,31)(41,55)(42,54)(43,53)(44,52)(45,51)(46,50)(47,49)(56,60)(57,59)(61,65)(62,64)(66,80)(67,79)(68,78)(69,77)(70,76)(71,75)(72,74), (1,71,21,56,11,61,31,46)(2,72,32,47,12,62,22,57)(3,73,23,58,13,63,33,48)(4,74,34,49,14,64,24,59)(5,75,25,60,15,65,35,50)(6,76,36,51,16,66,26,41)(7,77,27,42,17,67,37,52)(8,78,38,53,18,68,28,43)(9,79,29,44,19,69,39,54)(10,80,40,55,20,70,30,45) );
G=PermutationGroup([(1,11),(2,12),(3,13),(4,14),(5,15),(6,16),(7,17),(8,18),(9,19),(10,20),(21,31),(22,32),(23,33),(24,34),(25,35),(26,36),(27,37),(28,38),(29,39),(30,40)], [(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),(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60),(61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80)], [(1,20),(2,19),(3,18),(4,17),(5,16),(6,15),(7,14),(8,13),(9,12),(10,11),(21,40),(22,39),(23,38),(24,37),(25,36),(26,35),(27,34),(28,33),(29,32),(30,31),(41,55),(42,54),(43,53),(44,52),(45,51),(46,50),(47,49),(56,60),(57,59),(61,65),(62,64),(66,80),(67,79),(68,78),(69,77),(70,76),(71,75),(72,74)], [(1,71,21,56,11,61,31,46),(2,72,32,47,12,62,22,57),(3,73,23,58,13,63,33,48),(4,74,34,49,14,64,24,59),(5,75,25,60,15,65,35,50),(6,76,36,51,16,66,26,41),(7,77,27,42,17,67,37,52),(8,78,38,53,18,68,28,43),(9,79,29,44,19,69,39,54),(10,80,40,55,20,70,30,45)])
35 conjugacy classes
class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 5A | 5B | 8A | 8B | 8C | 8D | 10A | 10B | 10C | 10D | 20A | 20B | 20C | 20D | 20E | 20F | 20G | 20H | 40A | ··· | 40H |
order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 10 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 40 | ··· | 40 |
size | 1 | 1 | 2 | 40 | 4 | 4 | 4 | 20 | 20 | 2 | 2 | 8 | 8 | 40 | 40 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | ··· | 8 |
35 irreducible representations
dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 8 |
type | + | + | + | + | + | + | + | + | + | + | ||||||
image | C1 | C2 | C2 | C2 | C4 | C4 | D4 | D5 | D10 | C4×D5 | D20 | C5⋊D4 | C23⋊C4 | C42.C4 | C22.2D20 | (C2×D20).C4 |
kernel | (C2×D20).C4 | C20.10D4 | C5×C4.10D4 | C20.23D4 | C4×Dic5 | C2×D20 | C2×C20 | C4.10D4 | C2×Q8 | C2×C4 | C2×C4 | C2×C4 | C10 | C5 | C2 | C1 |
# reps | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 1 | 2 | 4 | 2 |
Matrix representation of (C2×D20).C4 ►in GL6(𝔽41)
1 | 0 | 0 | 0 | 0 | 0 |
0 | 1 | 0 | 0 | 0 | 0 |
0 | 0 | 40 | 0 | 0 | 0 |
0 | 0 | 0 | 40 | 0 | 0 |
0 | 0 | 0 | 0 | 1 | 0 |
0 | 0 | 25 | 25 | 0 | 1 |
6 | 40 | 0 | 0 | 0 | 0 |
1 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 1 | 0 | 0 |
0 | 0 | 40 | 0 | 0 | 0 |
0 | 0 | 9 | 9 | 1 | 4 |
0 | 0 | 0 | 16 | 20 | 40 |
6 | 40 | 0 | 0 | 0 | 0 |
35 | 35 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 1 | 0 | 0 |
0 | 0 | 1 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 1 | 0 |
0 | 0 | 16 | 16 | 20 | 40 |
40 | 0 | 0 | 0 | 0 | 0 |
0 | 40 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 1 | 0 |
0 | 0 | 32 | 32 | 40 | 37 |
0 | 0 | 32 | 0 | 0 | 0 |
0 | 0 | 2 | 18 | 0 | 9 |
G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,25,0,0,0,40,0,25,0,0,0,0,1,0,0,0,0,0,0,1],[6,1,0,0,0,0,40,0,0,0,0,0,0,0,0,40,9,0,0,0,1,0,9,16,0,0,0,0,1,20,0,0,0,0,4,40],[6,35,0,0,0,0,40,35,0,0,0,0,0,0,0,1,0,16,0,0,1,0,0,16,0,0,0,0,1,20,0,0,0,0,0,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,32,32,2,0,0,0,32,0,18,0,0,1,40,0,0,0,0,0,37,0,9] >;
(C2×D20).C4 in GAP, Magma, Sage, TeX
(C_2\times D_{20}).C_4
% in TeX
G:=Group("(C2xD20).C4");
// GroupNames label
G:=SmallGroup(320,35);
// by ID
G=gap.SmallGroup(320,35);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,141,36,422,184,1123,794,297,136,851,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^20=c^2=1,d^4=b^10,a*b=b*a,a*c=c*a,d*a*d^-1=a*b^10,c*b*c=b^-1,d*b*d^-1=a*b^11,d*c*d^-1=b^15*c>;
// generators/relations