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## G = D5×C24order 160 = 25·5

### Direct product of C24 and D5

Aliases: D5×C24, C5⋊C25, C10⋊C24, (C2×C10)⋊4C23, (C23×C10)⋊5C2, (C22×C10)⋊8C22, SmallGroup(160,237)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5 — D5×C24
 Chief series C1 — C5 — D5 — D10 — C22×D5 — C23×D5 — D5×C24
 Lower central C5 — D5×C24
 Upper central C1 — C24

Generators and relations for D5×C24
G = < a,b,c,d,e,f | a2=b2=c2=d2=e5=f2=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=e-1 >

Subgroups: 1976 in 748 conjugacy classes, 441 normal (5 characteristic)
C1, C2 [×15], C2 [×16], C22 [×35], C22 [×120], C5, C23 [×15], C23 [×140], D5 [×16], C10 [×15], C24, C24 [×30], D10 [×120], C2×C10 [×35], C25, C22×D5 [×140], C22×C10 [×15], C23×D5 [×30], C23×C10, D5×C24
Quotients: C1, C2 [×31], C22 [×155], C23 [×155], D5, C24 [×31], D10 [×15], C25, C22×D5 [×35], C23×D5 [×15], D5×C24

Smallest permutation representation of D5×C24
On 80 points
Generators in S80
(1 79)(2 80)(3 76)(4 77)(5 78)(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 39)(2 40)(3 36)(4 37)(5 38)(6 31)(7 32)(8 33)(9 34)(10 35)(11 26)(12 27)(13 28)(14 29)(15 30)(16 21)(17 22)(18 23)(19 24)(20 25)(41 76)(42 77)(43 78)(44 79)(45 80)(46 71)(47 72)(48 73)(49 74)(50 75)(51 66)(52 67)(53 68)(54 69)(55 70)(56 61)(57 62)(58 63)(59 64)(60 65)
(1 14)(2 15)(3 11)(4 12)(5 13)(6 16)(7 17)(8 18)(9 19)(10 20)(21 31)(22 32)(23 33)(24 34)(25 35)(26 36)(27 37)(28 38)(29 39)(30 40)(41 51)(42 52)(43 53)(44 54)(45 55)(46 56)(47 57)(48 58)(49 59)(50 60)(61 71)(62 72)(63 73)(64 74)(65 75)(66 76)(67 77)(68 78)(69 79)(70 80)
(1 9)(2 10)(3 6)(4 7)(5 8)(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 58)(2 57)(3 56)(4 60)(5 59)(6 51)(7 55)(8 54)(9 53)(10 52)(11 46)(12 50)(13 49)(14 48)(15 47)(16 41)(17 45)(18 44)(19 43)(20 42)(21 76)(22 80)(23 79)(24 78)(25 77)(26 71)(27 75)(28 74)(29 73)(30 72)(31 66)(32 70)(33 69)(34 68)(35 67)(36 61)(37 65)(38 64)(39 63)(40 62)

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

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

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

D5×C24 is a maximal subgroup of   C24.48D10  C244F5
D5×C24 is a maximal quotient of   C10.C25  D20.37C23  D20.39C23

64 conjugacy classes

 class 1 2A ··· 2O 2P ··· 2AE 5A 5B 10A ··· 10AD order 1 2 ··· 2 2 ··· 2 5 5 10 ··· 10 size 1 1 ··· 1 5 ··· 5 2 2 2 ··· 2

64 irreducible representations

 dim 1 1 1 2 2 type + + + + + image C1 C2 C2 D5 D10 kernel D5×C24 C23×D5 C23×C10 C24 C23 # reps 1 30 1 2 30

Matrix representation of D5×C24 in GL5(𝔽11)

 10 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 10 0 0 0 0 0 10
,
 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 10 0 0 0 0 0 10
,
 10 0 0 0 0 0 1 0 0 0 0 0 10 0 0 0 0 0 10 0 0 0 0 0 10
,
 1 0 0 0 0 0 10 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 10 1 0 0 0 2 8
,
 10 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 9 10

G:=sub<GL(5,GF(11))| [10,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,10,0,0,0,0,0,10],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,10,0,0,0,0,0,10],[10,0,0,0,0,0,1,0,0,0,0,0,10,0,0,0,0,0,10,0,0,0,0,0,10],[1,0,0,0,0,0,10,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,10,2,0,0,0,1,8],[10,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,9,0,0,0,0,10] >;

D5×C24 in GAP, Magma, Sage, TeX

D_5\times C_2^4
% in TeX

G:=Group("D5xC2^4");
// GroupNames label

G:=SmallGroup(160,237);
// by ID

G=gap.SmallGroup(160,237);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-5,4613]);
// Polycyclic

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

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