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## G = C22×C4⋊F5order 320 = 26·5

### Direct product of C22 and C4⋊F5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C22×C4⋊F5
 Chief series C1 — C5 — D5 — D10 — C2×F5 — C22×F5 — C23×F5 — C22×C4⋊F5
 Lower central C5 — C10 — C22×C4⋊F5
 Upper central C1 — C23 — C22×C4

Generators and relations for C22×C4⋊F5
G = < a,b,c,d,e | a2=b2=c4=d5=e4=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece-1=c-1, ede-1=d3 >

Subgroups: 1386 in 418 conjugacy classes, 196 normal (15 characteristic)
C1, C2, C2 [×6], C2 [×8], C4 [×4], C4 [×12], C22 [×7], C22 [×28], C5, C2×C4 [×6], C2×C4 [×54], C23, C23 [×14], D5 [×2], D5 [×6], C10, C10 [×6], C4⋊C4 [×16], C22×C4, C22×C4 [×33], C24, Dic5 [×4], C20 [×4], F5 [×8], D10, D10 [×27], C2×C10 [×7], C2×C4⋊C4 [×12], C23×C4 [×3], C4×D5 [×16], C2×Dic5 [×6], C2×C20 [×6], C2×F5 [×8], C2×F5 [×24], C22×D5 [×14], C22×C10, C22×C4⋊C4, C4⋊F5 [×16], C2×C4×D5 [×12], C22×Dic5, C22×C20, C22×F5 [×12], C22×F5 [×8], C23×D5, C2×C4⋊F5 [×12], D5×C22×C4, C23×F5 [×2], C22×C4⋊F5
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], D4 [×4], Q8 [×4], C23 [×15], C4⋊C4 [×16], C22×C4 [×14], C2×D4 [×6], C2×Q8 [×6], C24, F5, C2×C4⋊C4 [×12], C23×C4, C22×D4, C22×Q8, C2×F5 [×7], C22×C4⋊C4, C4⋊F5 [×4], C22×F5 [×7], C2×C4⋊F5 [×6], C23×F5, C22×C4⋊F5

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

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

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

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

56 conjugacy classes

 class 1 2A ··· 2G 2H ··· 2O 4A 4B 4C 4D 4E ··· 4X 5 10A ··· 10G 20A ··· 20H order 1 2 ··· 2 2 ··· 2 4 4 4 4 4 ··· 4 5 10 ··· 10 20 ··· 20 size 1 1 ··· 1 5 ··· 5 2 2 2 2 10 ··· 10 4 4 ··· 4 4 ··· 4

56 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 4 4 4 4 type + + + + + - + + + image C1 C2 C2 C2 C4 C4 C4 D4 Q8 F5 C2×F5 C2×F5 C4⋊F5 kernel C22×C4⋊F5 C2×C4⋊F5 D5×C22×C4 C23×F5 C2×C4×D5 C22×Dic5 C22×C20 C22×D5 C22×D5 C22×C4 C2×C4 C23 C22 # reps 1 12 1 2 12 2 2 4 4 1 6 1 8

Matrix representation of C22×C4⋊F5 in GL8(𝔽41)

 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 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 0 0 0 0 0 0 0 0 1 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 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 40
,
 40 39 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 2 0 0 0 0 0 0 40 40 0 0 0 0 0 0 0 0 34 14 0 27 0 0 0 0 0 7 14 27 0 0 0 0 27 14 7 0 0 0 0 0 27 0 14 34
,
 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 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
,
 40 39 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 32 23 0 0 0 0 0 0 0 9 0 0 0 0 0 0 0 0 34 0 7 14 0 0 0 0 0 14 34 7 0 0 0 0 27 7 34 14 0 0 0 0 27 14 7 0

G:=sub<GL(8,GF(41))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40],[40,1,0,0,0,0,0,0,39,1,0,0,0,0,0,0,0,0,1,40,0,0,0,0,0,0,2,40,0,0,0,0,0,0,0,0,34,0,27,27,0,0,0,0,14,7,14,0,0,0,0,0,0,14,7,14,0,0,0,0,27,27,0,34],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,40,40,40,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0],[40,0,0,0,0,0,0,0,39,1,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,23,9,0,0,0,0,0,0,0,0,34,0,27,27,0,0,0,0,0,14,7,14,0,0,0,0,7,34,34,7,0,0,0,0,14,7,14,0] >;

C22×C4⋊F5 in GAP, Magma, Sage, TeX

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

G:=Group("C2^2xC4:F5");
// GroupNames label

G:=SmallGroup(320,1591);
// by ID

G=gap.SmallGroup(320,1591);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,1123,136,6278,818]);
// Polycyclic

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

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