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## G = C24×F5order 320 = 26·5

### Direct product of C24 and F5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5 — C24×F5
 Chief series C1 — C5 — D5 — F5 — C2×F5 — C22×F5 — C23×F5 — C24×F5
 Lower central C5 — C24×F5
 Upper central C1 — C24

Generators and relations for C24×F5
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, C2×C4, C23, C23, D5, D5, C10, C22×C4, C24, C24, F5, D10, C2×C10, C23×C4, C25, C2×F5, C22×D5, C22×C10, C24×C4, C22×F5, C23×D5, C23×C10, C23×F5, D5×C24, C24×F5
Quotients: C1, C2, C4, C22, C2×C4, C23, C22×C4, C24, F5, C23×C4, C25, C2×F5, C24×C4, C22×F5, C23×F5, C24×F5

Smallest permutation representation of C24×F5
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 ··· 2O 2P ··· 2AE 4A ··· 4AF 5 10A ··· 10O order 1 2 ··· 2 2 ··· 2 4 ··· 4 5 10 ··· 10 size 1 1 ··· 1 5 ··· 5 5 ··· 5 4 4 ··· 4

80 irreducible representations

 dim 1 1 1 1 1 4 4 type + + + + + image C1 C2 C2 C4 C4 F5 C2×F5 kernel C24×F5 C23×F5 D5×C24 C23×D5 C23×C10 C24 C23 # reps 1 30 1 30 2 1 15

Matrix representation of C24×F5 in GL7(𝔽41)

 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 0 0 40 0 0 0 1 0 0 40 0 0 0 0 1 0 40 0 0 0 0 0 1 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 0 40 0 0 0 0 40 0 0 0 0 0 0 0 0 0 40 0 0 0 0 40 0 0

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] >;

C24×F5 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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