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## G = C4×C22⋊F5order 320 = 26·5

### Direct product of C4 and C22⋊F5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C4×C22⋊F5
 Chief series C1 — C5 — D5 — D10 — C22×D5 — C22×F5 — C2×C22⋊F5 — C4×C22⋊F5
 Lower central C5 — C10 — C4×C22⋊F5
 Upper central C1 — C2×C4 — C22×C4

Generators and relations for C4×C22⋊F5
G = < a,b,c,d,e | a4=b2=c2=d5=e4=1, ab=ba, ac=ca, ad=da, ae=ea, ebe-1=bc=cb, bd=db, cd=dc, ce=ec, ede-1=d3 >

Subgroups: 1002 in 258 conjugacy classes, 86 normal (30 characteristic)
C1, C2 [×3], C2 [×8], C4 [×2], C4 [×12], C22, C22 [×2], C22 [×20], C5, C2×C4 [×2], C2×C4 [×32], C23, C23 [×10], D5 [×4], D5 [×2], C10 [×3], C10 [×2], C42 [×4], C22⋊C4 [×8], C22×C4, C22×C4 [×13], C24, Dic5 [×2], Dic5, C20 [×2], C20, F5 [×8], D10 [×8], D10 [×10], C2×C10, C2×C10 [×2], C2×C10 [×2], C2.C42 [×2], C2×C42 [×2], C2×C22⋊C4 [×2], C23×C4, C4×D5 [×4], C4×D5 [×6], C2×Dic5 [×2], C2×Dic5 [×2], C2×C20 [×2], C2×C20 [×2], C2×F5 [×8], C2×F5 [×8], C22×D5 [×2], C22×D5 [×4], C22×D5 [×4], C22×C10, C4×C22⋊C4, C4×F5 [×4], C22⋊F5 [×8], C2×C4×D5 [×4], C2×C4×D5 [×4], C22×Dic5, C22×C20, C22×F5 [×4], C23×D5, D10.3Q8 [×2], C2×C4×F5 [×2], C2×C22⋊F5 [×2], D5×C22×C4, C4×C22⋊F5
Quotients: C1, C2 [×7], C4 [×12], C22 [×7], C2×C4 [×18], D4 [×4], C23, C42 [×4], C22⋊C4 [×4], C22×C4 [×3], C2×D4 [×2], C4○D4 [×2], F5, C2×C42, C2×C22⋊C4, C42⋊C2, C4×D4 [×4], C2×F5 [×3], C4×C22⋊C4, C4×F5 [×2], C22⋊F5 [×2], C22×F5, C2×C4×F5, D10.C23, C2×C22⋊F5, C4×C22⋊F5

Smallest permutation representation of C4×C22⋊F5
On 80 points
Generators in S80
(1 34 14 24)(2 35 15 25)(3 31 11 21)(4 32 12 22)(5 33 13 23)(6 36 16 26)(7 37 17 27)(8 38 18 28)(9 39 19 29)(10 40 20 30)(41 71 51 61)(42 72 52 62)(43 73 53 63)(44 74 54 64)(45 75 55 65)(46 76 56 66)(47 77 57 67)(48 78 58 68)(49 79 59 69)(50 80 60 70)
(1 54)(2 55)(3 51)(4 52)(5 53)(6 56)(7 57)(8 58)(9 59)(10 60)(11 41)(12 42)(13 43)(14 44)(15 45)(16 46)(17 47)(18 48)(19 49)(20 50)(21 71)(22 72)(23 73)(24 74)(25 75)(26 76)(27 77)(28 78)(29 79)(30 80)(31 61)(32 62)(33 63)(34 64)(35 65)(36 66)(37 67)(38 68)(39 69)(40 70)
(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 14)(2 11 5 12)(3 13 4 15)(6 18 7 20)(8 17 10 16)(9 19)(21 33 22 35)(23 32 25 31)(24 34)(26 38 27 40)(28 37 30 36)(29 39)(41 58 42 60)(43 57 45 56)(44 59)(46 53 47 55)(48 52 50 51)(49 54)(61 78 62 80)(63 77 65 76)(64 79)(66 73 67 75)(68 72 70 71)(69 74)

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

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

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

56 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 2K 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K ··· 4AB 5 10A ··· 10G 20A ··· 20H order 1 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 4 4 4 ··· 4 5 10 ··· 10 20 ··· 20 size 1 1 1 1 2 2 5 5 5 5 10 10 1 1 1 1 2 2 5 5 5 5 10 ··· 10 4 4 ··· 4 4 ··· 4

56 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 4 4 4 4 4 4 type + + + + + + + + + + image C1 C2 C2 C2 C2 C4 C4 C4 C4 D4 C4○D4 F5 C2×F5 C2×F5 C22⋊F5 C4×F5 D10.C23 kernel C4×C22⋊F5 D10.3Q8 C2×C4×F5 C2×C22⋊F5 D5×C22×C4 C22⋊F5 C2×C4×D5 C22×Dic5 C22×C20 C4×D5 D10 C22×C4 C2×C4 C23 C4 C22 C2 # reps 1 2 2 2 1 16 4 2 2 4 4 1 2 1 4 4 4

Matrix representation of C4×C22⋊F5 in GL6(𝔽41)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 32 0 0 0 0 0 0 32 0 0 0 0 0 0 32 0 0 0 0 0 0 32
,
 40 39 0 0 0 0 0 1 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40
,
 40 0 0 0 0 0 0 40 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 40 1 0 0 0 0 40 0 1 0 0 0 40 0 0 1 0 0 40 0 0 0
,
 9 0 0 0 0 0 32 32 0 0 0 0 0 0 40 0 1 0 0 0 0 0 1 40 0 0 0 40 1 0 0 0 0 0 1 0

G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,32,0,0,0,0,0,0,32,0,0,0,0,0,0,32,0,0,0,0,0,0,32],[40,0,0,0,0,0,39,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,40,40,40,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0],[9,32,0,0,0,0,0,32,0,0,0,0,0,0,40,0,0,0,0,0,0,0,40,0,0,0,1,1,1,1,0,0,0,40,0,0] >;

C4×C22⋊F5 in GAP, Magma, Sage, TeX

C_4\times C_2^2\rtimes F_5
% in TeX

G:=Group("C4xC2^2:F5");
// GroupNames label

G:=SmallGroup(320,1101);
// by ID

G=gap.SmallGroup(320,1101);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,477,100,6278,1595]);
// Polycyclic

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

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