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G = D10.11M4(2)  order 320 = 26·5

7th non-split extension by D10 of M4(2) acting via M4(2)/C2xC4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D10.11M4(2), D5:(C22:C8), D10:10(C2xC8), (C22xD5):6C8, C22:2(D5:C8), (C4xD5).119D4, D10:C8:14C2, (C22xC4).14F5, C23.40(C2xF5), (C22xC20).34C4, C10.14(C22xC8), (C23xD5).14C4, C4.45(C22:F5), C20.44(C22:C4), Dic5.104(C2xD4), C2.7(D5:M4(2)), C10.20(C2xM4(2)), C23.2F5:14C2, D10.38(C22:C4), C22.47(C22xF5), (C2xDic5).344C23, (C22xDic5).272C22, C5:2(C2xC22:C8), (C2xC10):3(C2xC8), (C2xC5:C8):7C22, (C2xC4xD5).33C4, (C2xD5:C8):12C2, C2.15(C2xD5:C8), C2.2(C2xC22:F5), C10.4(C2xC22:C4), (C2xC4).107(C2xF5), (D5xC22xC4).34C2, (C2xC20).178(C2xC4), (C2xC4xD5).414C22, (C2xC10).60(C22xC4), (C22xC10).60(C2xC4), (C2xDic5).182(C2xC4), (C22xD5).124(C2xC4), SmallGroup(320,1091)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C10 — D10.11M4(2)
Chief series
C1 — C5 — C10 — Dic5 — C2xDic5 — C2xC5:C8 — C23.2F5 — D10.11M4(2)
Lower central
C5 — C10 — D10.11M4(2)
Upper central
C1 — C2xC4 — C22xC4

Generators and relations for D10.11M4(2)
 G = < a,b,c,d | a10=b2=c8=d2=1, bab=a-1, cac-1=a3, ad=da, cbc-1=a2b, bd=db, dcd=a5c5 >

Subgroups: 762 in 202 conjugacy classes, 70 normal (30 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C5, C8, C2xC4, C2xC4, C23, C23, D5, D5, C10, C10, C2xC8, C22xC4, C22xC4, C24, Dic5, Dic5, C20, C20, D10, D10, C2xC10, C2xC10, C2xC10, C22:C8, C22xC8, C23xC4, C5:C8, C4xD5, C4xD5, C2xDic5, C2xDic5, C2xC20, C2xC20, C22xD5, C22xD5, C22xD5, C22xC10, C2xC22:C8, D5:C8, C2xC5:C8, C2xC4xD5, C2xC4xD5, C22xDic5, C22xC20, C23xD5, D10:C8, C23.2F5, C2xD5:C8, D5xC22xC4, D10.11M4(2)
Quotients: C1, C2, C4, C22, C8, C2xC4, D4, C23, C22:C4, C2xC8, M4(2), C22xC4, C2xD4, F5, C22:C8, C2xC22:C4, C22xC8, C2xM4(2), C2xF5, C2xC22:C8, D5:C8, C22:F5, C22xF5, C2xD5:C8, D5:M4(2), C2xC22:F5, D10.11M4(2)

Smallest permutation representation of D10.11M4(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 5)(2 4)(6 10)(7 9)(12 20)(13 19)(14 18)(15 17)(21 27)(22 26)(23 25)(28 30)(31 37)(32 36)(33 35)(38 40)(41 45)(42 44)(46 50)(47 49)(52 60)(53 59)(54 58)(55 57)(61 63)(64 70)(65 69)(66 68)(71 75)(72 74)(76 80)(77 79)

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

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

56 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I2J2K4A4B4C4D4E4F4G4H4I4J4K4L 5 8A···8P10A···10G20A···20H
order12222222222244444444444458···810···1020···20
size1111225555101011112255551010410···104···44···4

56 irreducible representations

dim11111111122444444
type++++++++++
imageC1C2C2C2C2C4C4C4C8D4M4(2)F5C2xF5C2xF5C22:F5D5:C8D5:M4(2)
kernelD10.11M4(2)D10:C8C23.2F5C2xD5:C8D5xC22xC4C2xC4xD5C22xC20C23xD5C22xD5C4xD5D10C22xC4C2xC4C23C4C22C2
# reps122214221644121444

Matrix representation of D10.11M4(2) ►in GL8(F41)

10000000
01000000
004000000
000400000
00007700
0000344000
00000001
0000004034
,
400000000
040000000
004000000
000400000
00007700
0000403400
00000001
00000010
,
3819000000
03000000
009250000
000320000
00000010
00000001
000040000
00007100
,
10000000
3740000000
004000000
00410000
00001000
00000100
00000010
00000001

G:=sub<GL(8,GF(41))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,7,34,0,0,0,0,0,0,7,40,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,1,34],[40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,7,40,0,0,0,0,0,0,7,34,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0],[38,0,0,0,0,0,0,0,19,3,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,25,32,0,0,0,0,0,0,0,0,0,0,40,7,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[1,37,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,4,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1] >;
 

D10.11M4(2) in GAP, Magma, Sage, TeX 

D_{10}._{11}M_4(2)
 
% in TeX
 
G:=Group("D10.11M4(2)");
 
// GroupNames label
 
G:=SmallGroup(320,1091);
 
// by ID
 
G=gap.SmallGroup(320,1091);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,253,184,136,6278,1595]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^10=b^2=c^8=d^2=1,b*a*b=a^-1,c*a*c^-1=a^3,a*d=d*a,c*b*c^-1=a^2*b,b*d=d*b,d*c*d=a^5*c^5>;
 
// generators/relations
 

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