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G = C24xF5order 320 = 26·5

Direct product of C24 and F5

direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary

Aliases: C24xF5, D5.C25, D10.21C24, C5:(C24xC4), C10:(C23xC4), D5:(C23xC4), (C23xC10):9C4, (C23xD5):13C4, (D5xC24).6C2, D10:10(C22xC4), (C22xD5).288C23, (C23xD5).141C22, (C2xC10):4(C22xC4), (C22xC10):10(C2xC4), (C22xD5):22(C2xC4), SmallGroup(320,1638)

Series: Derived Chief Lower central Upper central

C1C5 — C24xF5
C1C5D5F5C2xF5C22xF5C23xF5 — C24xF5
C5 — C24xF5
C1C24

Generators and relations for C24xF5
 G = < a,b,c,d,e,f | a2=b2=c2=d2=e5=f4=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=e3 >

Subgroups: 3818 in 1362 conjugacy classes, 748 normal (7 characteristic)
C1, C2, C2, C4, C22, C22, C5, C2xC4, C23, C23, D5, D5, C10, C22xC4, C24, C24, F5, D10, C2xC10, C23xC4, C25, C2xF5, C22xD5, C22xC10, C24xC4, C22xF5, C23xD5, C23xC10, C23xF5, D5xC24, C24xF5
Quotients: C1, C2, C4, C22, C2xC4, C23, C22xC4, C24, F5, C23xC4, C25, C2xF5, C24xC4, C22xF5, C23xF5, C24xF5

Smallest permutation representation of C24xF5
On 80 points
Generators in S80
(1 76)(2 77)(3 78)(4 79)(5 80)(6 71)(7 72)(8 73)(9 74)(10 75)(11 66)(12 67)(13 68)(14 69)(15 70)(16 61)(17 62)(18 63)(19 64)(20 65)(21 56)(22 57)(23 58)(24 59)(25 60)(26 51)(27 52)(28 53)(29 54)(30 55)(31 46)(32 47)(33 48)(34 49)(35 50)(36 41)(37 42)(38 43)(39 44)(40 45)
(1 31)(2 32)(3 33)(4 34)(5 35)(6 36)(7 37)(8 38)(9 39)(10 40)(11 21)(12 22)(13 23)(14 24)(15 25)(16 26)(17 27)(18 28)(19 29)(20 30)(41 71)(42 72)(43 73)(44 74)(45 75)(46 76)(47 77)(48 78)(49 79)(50 80)(51 61)(52 62)(53 63)(54 64)(55 65)(56 66)(57 67)(58 68)(59 69)(60 70)
(1 16)(2 17)(3 18)(4 19)(5 20)(6 11)(7 12)(8 13)(9 14)(10 15)(21 36)(22 37)(23 38)(24 39)(25 40)(26 31)(27 32)(28 33)(29 34)(30 35)(41 56)(42 57)(43 58)(44 59)(45 60)(46 51)(47 52)(48 53)(49 54)(50 55)(61 76)(62 77)(63 78)(64 79)(65 80)(66 71)(67 72)(68 73)(69 74)(70 75)
(1 6)(2 7)(3 8)(4 9)(5 10)(11 16)(12 17)(13 18)(14 19)(15 20)(21 26)(22 27)(23 28)(24 29)(25 30)(31 36)(32 37)(33 38)(34 39)(35 40)(41 46)(42 47)(43 48)(44 49)(45 50)(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)
(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 51)(2 53 5 54)(3 55 4 52)(6 56)(7 58 10 59)(8 60 9 57)(11 41)(12 43 15 44)(13 45 14 42)(16 46)(17 48 20 49)(18 50 19 47)(21 71)(22 73 25 74)(23 75 24 72)(26 76)(27 78 30 79)(28 80 29 77)(31 61)(32 63 35 64)(33 65 34 62)(36 66)(37 68 40 69)(38 70 39 67)

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

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

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

80 conjugacy classes

class 1 2A···2O2P···2AE4A···4AF 5 10A···10O
order12···22···24···4510···10
size11···15···55···544···4

80 irreducible representations

dim1111144
type+++++
imageC1C2C2C4C4F5C2xF5
kernelC24xF5C23xF5D5xC24C23xD5C23xC10C24C23
# reps1301302115

Matrix representation of C24xF5 in GL7(F41)

40000000
04000000
00400000
0001000
0000100
0000010
0000001
,
40000000
04000000
00400000
00040000
00004000
00000400
00000040
,
1000000
04000000
0010000
0001000
0000100
0000010
0000001
,
1000000
0100000
00400000
0001000
0000100
0000010
0000001
,
1000000
0100000
0010000
00000040
00010040
00001040
00000140
,
9000000
03200000
0090000
00000400
00040000
00000040
00004000

G:=sub<GL(7,GF(41))| [40,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[40,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,40,40,40,40],[9,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,40,0,0,0,0,0,0,0,0,40,0] >;

C24xF5 in GAP, Magma, Sage, TeX

C_2^4\times F_5
% in TeX

G:=Group("C2^4xF5");
// GroupNames label

G:=SmallGroup(320,1638);
// by ID

G=gap.SmallGroup(320,1638);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,6278,433]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=e^5=f^4=1,a*b=b*a,a*c=c*a,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,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=e^3>;
// generators/relations

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