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G = Dic5.22C24  order 320 = 26·5

22nd non-split extension by Dic5 of C24 acting via C24/C23=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Dic5.22C24, (D4×D5).3C4, D4.F5⋊6C2, C5⋊C8.5C23, Q8.F5⋊6C2, C4○D4.2F5, (Q8×D5).3C4, D5⋊C8⋊4C22, C4○D20.4C4, D4.14(C2×F5), Q8.15(C2×F5), D20.14(C2×C4), C5⋊3(Q8○M4(2)), C4.F5⋊12C22, C2.17(C23×F5), C4.44(C22×F5), D5⋊M4(2)⋊10C2, C20.33(C22×C4), C10.16(C23×C4), (C4×D5).56C23, D10.7(C22×C4), C22.F5⋊6C22, Dic10.15(C2×C4), C22.4(C22×F5), Dic5.7(C22×C4), D4⋊2D5.18C22, Q8⋊2D5.18C22, (C2×Dic5).180C23, (C2×C5⋊C8)⋊5C22, (C2×C4.F5)⋊8C2, (C5×C4○D4).4C4, C5⋊D4.4(C2×C4), (C2×C4).48(C2×F5), (C2×C20).77(C2×C4), (D5×C4○D4).10C2, (C5×D4).14(C2×C4), (C4×D5).39(C2×C4), (C5×Q8).15(C2×C4), (C2×C10).5(C22×C4), (C2×C4×D5).223C22, (C22×D5).65(C2×C4), SmallGroup(320,1602)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C10 — Dic5.22C24
Chief series
C1 — C5 — C10 — Dic5 — C5⋊C8 — C2×C5⋊C8 — D4.F5 — Dic5.22C24
Lower central
C5 — C10 — Dic5.22C24
Upper central
C1 — C2 — C4○D4

Generators and relations for Dic5.22C24
 G = < a,b,c,d,e,f | a10=d2=e2=1, b2=f2=a5, c2=b, bab-1=a-1, cac-1=a3, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, fcf-1=a5c, ede=a5d, df=fd, ef=fe >

Subgroups: 778 in 258 conjugacy classes, 136 normal (18 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, D5, C10, C10, C2×C8, M4(2), C22×C4, C2×D4, C2×Q8, C4○D4, C4○D4, Dic5, Dic5, C20, C20, D10, D10, D10, C2×C10, C2×M4(2), C8○D4, C2×C4○D4, C5⋊C8, Dic10, C4×D5, C4×D5, D20, C2×Dic5, C5⋊D4, C2×C20, C5×D4, C5×Q8, C22×D5, Q8○M4(2), D5⋊C8, C4.F5, C4.F5, C2×C5⋊C8, C22.F5, C2×C4×D5, C4○D20, D4×D5, D4⋊2D5, Q8×D5, Q8⋊2D5, C5×C4○D4, C2×C4.F5, D5⋊M4(2), D4.F5, Q8.F5, D5×C4○D4, Dic5.22C24
Quotients: C1, C2, C4, C22, C2×C4, C23, C22×C4, C24, F5, C23×C4, C2×F5, Q8○M4(2), C22×F5, C23×F5, Dic5.22C24

Smallest permutation representation of Dic5.22C24
►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 16 6 11)(2 15 7 20)(3 14 8 19)(4 13 9 18)(5 12 10 17)(21 34 26 39)(22 33 27 38)(23 32 28 37)(24 31 29 36)(25 40 30 35)(41 54 46 59)(42 53 47 58)(43 52 48 57)(44 51 49 56)(45 60 50 55)(61 74 66 79)(62 73 67 78)(63 72 68 77)(64 71 69 76)(65 80 70 75)

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

(21 26)(22 27)(23 28)(24 29)(25 30)(31 36)(32 37)(33 38)(34 39)(35 40)(61 66)(62 67)(63 68)(64 69)(65 70)(71 76)(72 77)(73 78)(74 79)(75 80)

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

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

44 conjugacy classes

class 1 2A2B2C2D2E2F2G2H4A4B4C4D4E4F4G4H4I 5 8A···8P10A10B10C10D20A20B20C20D20E
order12222222244444444458···8101010102020202020
size1122210101010222255101010410···10488844888

44 irreducible representations

dim1111111111444448
type++++++++++
imageC1C2C2C2C2C2C4C4C4C4F5C2×F5C2×F5C2×F5Q8○M4(2)Dic5.22C24
kernelDic5.22C24C2×C4.F5D5⋊M4(2)D4.F5Q8.F5D5×C4○D4C4○D20D4×D5Q8×D5C5×C4○D4C4○D4C2×C4D4Q8C5C1
# reps1336216622133122

Matrix representation of Dic5.22C24 ►in GL8(𝔽41)

404040400000
10000000
01000000
00100000
000040000
000004000
000000400
000000040
,
10000000
404040400000
00010000
00100000
000032000
000003200
000000320
000000032
,
28021210000
21210280000
2072000000
133434130000
000012050
0000033039
0000350290
000001608
,
10000000
01000000
00100000
00010000
00001000
0000164000
00000010
000040140
,
10000000
01000000
00100000
00010000
00001500
000004000
0000021139
000000040
,
10000000
01000000
00100000
00010000
00009000
00000900
000060320
0000010032

G:=sub<GL(8,GF(41))| [40,1,0,0,0,0,0,0,40,0,1,0,0,0,0,0,40,0,0,1,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40],[1,40,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,40,0,1,0,0,0,0,0,40,1,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,32],[28,21,20,13,0,0,0,0,0,21,7,34,0,0,0,0,21,0,20,34,0,0,0,0,21,28,0,13,0,0,0,0,0,0,0,0,12,0,35,0,0,0,0,0,0,33,0,16,0,0,0,0,5,0,29,0,0,0,0,0,0,39,0,8],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,16,0,4,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,5,40,21,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,39,40],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,9,0,6,0,0,0,0,0,0,9,0,10,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,32] >;
 

Dic5.22C24 in GAP, Magma, Sage, TeX 

{\rm Dic}_5._{22}C_2^4
 
% in TeX
 
G:=Group("Dic5.22C2^4");
 
// GroupNames label
 
G:=SmallGroup(320,1602);
 
// by ID
 
G=gap.SmallGroup(320,1602);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,387,1123,136,102,6278,818]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f|a^10=d^2=e^2=1,b^2=f^2=a^5,c^2=b,b*a*b^-1=a^-1,c*a*c^-1=a^3,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,c*d=d*c,c*e=e*c,f*c*f^-1=a^5*c,e*d*e=a^5*d,d*f=f*d,e*f=f*e>;
 
// generators/relations
 

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