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## G = C22×C22⋊F5order 320 = 26·5

### Direct product of C22 and C22⋊F5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C22×C22⋊F5
 Chief series C1 — C5 — D5 — D10 — C2×F5 — C22×F5 — C23×F5 — C22×C22⋊F5
 Lower central C5 — C10 — C22×C22⋊F5
 Upper central C1 — C23 — C24

Generators and relations for C22×C22⋊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, fcf-1=cd=dc, ce=ec, de=ed, df=fd, fef-1=e3 >

Subgroups: 2762 in 674 conjugacy classes, 196 normal (13 characteristic)
C1, C2, C2 [×6], C2 [×16], C4 [×8], C22 [×11], C22 [×88], C5, C2×C4 [×32], C23, C23 [×6], C23 [×92], D5 [×8], D5 [×4], C10, C10 [×6], C10 [×4], C22⋊C4 [×16], C22×C4 [×20], C24, C24 [×22], F5 [×8], D10, D10 [×31], D10 [×44], C2×C10 [×11], C2×C10 [×12], C2×C22⋊C4 [×12], C23×C4 [×2], C25, C2×F5 [×8], C2×F5 [×24], C22×D5 [×36], C22×D5 [×52], C22×C10, C22×C10 [×6], C22×C10 [×4], C22×C22⋊C4, C22⋊F5 [×16], C22×F5 [×12], C22×F5 [×8], C23×D5 [×2], C23×D5 [×12], C23×D5 [×8], C23×C10, C2×C22⋊F5 [×12], C23×F5 [×2], D5×C24, C22×C22⋊F5
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], D4 [×8], C23 [×15], C22⋊C4 [×16], C22×C4 [×14], C2×D4 [×12], C24, F5, C2×C22⋊C4 [×12], C23×C4, C22×D4 [×2], C2×F5 [×7], C22×C22⋊C4, C22⋊F5 [×4], C22×F5 [×7], C2×C22⋊F5 [×6], C23×F5, C22×C22⋊F5

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

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

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

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

56 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 2L ··· 2S 2T 2U 2V 2W 4A ··· 4P 5 10A ··· 10O order 1 2 ··· 2 2 2 2 2 2 ··· 2 2 2 2 2 4 ··· 4 5 10 ··· 10 size 1 1 ··· 1 2 2 2 2 5 ··· 5 10 10 10 10 10 ··· 10 4 4 ··· 4

56 irreducible representations

 dim 1 1 1 1 1 1 2 4 4 4 type + + + + + + + + image C1 C2 C2 C2 C4 C4 D4 F5 C2×F5 C22⋊F5 kernel C22×C22⋊F5 C2×C22⋊F5 C23×F5 D5×C24 C23×D5 C23×C10 C22×D5 C24 C23 C22 # reps 1 12 2 1 14 2 8 1 7 8

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

 40 0 0 0 0 0 0 0 0 40 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 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
,
 1 0 0 0 0 0 0 0 14 40 0 0 0 0 0 0 0 0 40 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
,
 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 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
,
 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 34 1 0 0 0 0 0 0 40 0 0 0 0 0 0 0 21 0 0 1 0 0 0 0 21 0 40 6
,
 13 4 0 0 0 0 0 0 40 28 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 0 36 11 22 12 0 0 0 0 36 0 12 22 0 0 0 0 34 38 0 30 0 0 0 0 25 38 5 5

G:=sub<GL(8,GF(41))| [40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,14,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40],[40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,34,40,21,21,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,1,6],[13,40,0,0,0,0,0,0,4,28,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,36,36,34,25,0,0,0,0,11,0,38,38,0,0,0,0,22,12,0,5,0,0,0,0,12,22,30,5] >;

C22×C22⋊F5 in GAP, Magma, Sage, TeX

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

G:=Group("C2^2xC2^2:F5");
// GroupNames label

G:=SmallGroup(320,1607);
// by ID

G=gap.SmallGroup(320,1607);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,1123,6278,818]);
// 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,f*c*f^-1=c*d=d*c,c*e=e*c,d*e=e*d,d*f=f*d,f*e*f^-1=e^3>;
// generators/relations

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