Copied to
clipboard

G = C20.29M4(2)  order 320 = 26·5

4th non-split extension by C20 of M4(2) acting via M4(2)/C22=C4

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C20.29M4(2), C23.(C5:C8), (C2xC20).1C8, C5:2C8.21D4, C5:2(C23.C8), C20.C8:6C2, (C22xC4).4F5, (C22xC10).4C8, (C22xC20).14C4, C4.43(C22:F5), C20.41(C22:C4), C10.10(C22:C8), C4.7(C22.F5), C2.3(C23.2F5), (C2xC4).(C5:C8), C22.2(C2xC5:C8), (C2xC5:2C8).6C4, (C2xC10).27(C2xC8), (C2xC4).129(C2xF5), (C2xC20).144(C2xC4), (C2xC4.Dic5).29C2, (C2xC5:2C8).214C22, SmallGroup(320,250)

Series: Derived Chief Lower central Upper central

C1C2xC10 — C20.29M4(2)
C1C5C10C20C5:2C8C2xC5:2C8C20.C8 — C20.29M4(2)
C5C10C2xC10 — C20.29M4(2)
C1C4C2xC4C22xC4

Generators and relations for C20.29M4(2)
 G = < a,b,c | a20=c2=1, b8=a10, bab-1=a3, ac=ca, cbc=a5b5 >

Subgroups: 162 in 58 conjugacy classes, 26 normal (22 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2xC4, C2xC4, C23, C10, C10, C16, C2xC8, M4(2), C22xC4, C20, C20, C2xC10, C2xC10, M5(2), C2xM4(2), C5:2C8, C5:2C8, C2xC20, C2xC20, C22xC10, C23.C8, C5:C16, C2xC5:2C8, C4.Dic5, C22xC20, C20.C8, C2xC4.Dic5, C20.29M4(2)
Quotients: C1, C2, C4, C22, C8, C2xC4, D4, C22:C4, C2xC8, M4(2), F5, C22:C8, C5:C8, C2xF5, C23.C8, C2xC5:C8, C22.F5, C22:F5, C23.2F5, C20.29M4(2)

Smallest permutation representation of C20.29M4(2)
On 80 points
Generators in S80
(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 80 23 60 6 75 28 55 11 70 33 50 16 65 38 45)(2 67 32 43 7 62 37 58 12 77 22 53 17 72 27 48)(3 74 21 46 8 69 26 41 13 64 31 56 18 79 36 51)(4 61 30 49 9 76 35 44 14 71 40 59 19 66 25 54)(5 68 39 52 10 63 24 47 15 78 29 42 20 73 34 57)
(21 31)(22 32)(23 33)(24 34)(25 35)(26 36)(27 37)(28 38)(29 39)(30 40)(61 71)(62 72)(63 73)(64 74)(65 75)(66 76)(67 77)(68 78)(69 79)(70 80)

G:=sub<Sym(80)| (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), (1,80,23,60,6,75,28,55,11,70,33,50,16,65,38,45)(2,67,32,43,7,62,37,58,12,77,22,53,17,72,27,48)(3,74,21,46,8,69,26,41,13,64,31,56,18,79,36,51)(4,61,30,49,9,76,35,44,14,71,40,59,19,66,25,54)(5,68,39,52,10,63,24,47,15,78,29,42,20,73,34,57), (21,31)(22,32)(23,33)(24,34)(25,35)(26,36)(27,37)(28,38)(29,39)(30,40)(61,71)(62,72)(63,73)(64,74)(65,75)(66,76)(67,77)(68,78)(69,79)(70,80)>;

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)(61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80), (1,80,23,60,6,75,28,55,11,70,33,50,16,65,38,45)(2,67,32,43,7,62,37,58,12,77,22,53,17,72,27,48)(3,74,21,46,8,69,26,41,13,64,31,56,18,79,36,51)(4,61,30,49,9,76,35,44,14,71,40,59,19,66,25,54)(5,68,39,52,10,63,24,47,15,78,29,42,20,73,34,57), (21,31)(22,32)(23,33)(24,34)(25,35)(26,36)(27,37)(28,38)(29,39)(30,40)(61,71)(62,72)(63,73)(64,74)(65,75)(66,76)(67,77)(68,78)(69,79)(70,80) );

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),(61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80)], [(1,80,23,60,6,75,28,55,11,70,33,50,16,65,38,45),(2,67,32,43,7,62,37,58,12,77,22,53,17,72,27,48),(3,74,21,46,8,69,26,41,13,64,31,56,18,79,36,51),(4,61,30,49,9,76,35,44,14,71,40,59,19,66,25,54),(5,68,39,52,10,63,24,47,15,78,29,42,20,73,34,57)], [(21,31),(22,32),(23,33),(24,34),(25,35),(26,36),(27,37),(28,38),(29,39),(30,40),(61,71),(62,72),(63,73),(64,74),(65,75),(66,76),(67,77),(68,78),(69,79),(70,80)]])

38 conjugacy classes

class 1 2A2B2C4A4B4C4D 5 8A8B8C8D8E8F10A···10G16A···16H20A···20H
order12224444588888810···1016···1620···20
size1124112441010101020204···420···204···4

38 irreducible representations

dim11111112244444444
type+++++-+--+
imageC1C2C2C4C4C8C8D4M4(2)F5C5:C8C2xF5C5:C8C23.C8C22.F5C22:F5C20.29M4(2)
kernelC20.29M4(2)C20.C8C2xC4.Dic5C2xC5:2C8C22xC20C2xC20C22xC10C5:2C8C20C22xC4C2xC4C2xC4C23C5C4C4C1
# reps12122442211112228

Matrix representation of C20.29M4(2) in GL4(F241) generated by

216000
5413500
3060
160040
,
6202390
224079240
15411790
12701970
,
1000
16224000
0010
4400240
G:=sub<GL(4,GF(241))| [216,54,3,16,0,135,0,0,0,0,6,0,0,0,0,40],[62,224,154,127,0,0,1,0,239,79,179,197,0,240,0,0],[1,162,0,44,0,240,0,0,0,0,1,0,0,0,0,240] >;

C20.29M4(2) in GAP, Magma, Sage, TeX

C_{20}._{29}M_4(2)
% in TeX

G:=Group("C20.29M4(2)");
// GroupNames label

G:=SmallGroup(320,250);
// by ID

G=gap.SmallGroup(320,250);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,141,387,100,1123,102,6278,3156]);
// Polycyclic

G:=Group<a,b,c|a^20=c^2=1,b^8=a^10,b*a*b^-1=a^3,a*c=c*a,c*b*c=a^5*b^5>;
// generators/relations

׿
x
:
Z
F
o
wr
Q
<