Copied to
clipboard

## G = C22×D5.D5order 400 = 24·52

### Direct product of C22 and D5.D5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C52 — C22×D5.D5
 Chief series C1 — C5 — C52 — C5×D5 — D5.D5 — C2×D5.D5 — C22×D5.D5
 Lower central C52 — C22×D5.D5
 Upper central C1 — C22

Generators and relations for C22×D5.D5
G = < a,b,c,d,e,f | a2=b2=c5=d2=e5=1, f2=c-1d, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, dcd=c-1, ce=ec, fcf-1=c2, de=ed, fdf-1=cd, fef-1=e-1 >

Subgroups: 568 in 113 conjugacy classes, 53 normal (13 characteristic)
C1, C2, C2, C4, C22, C22, C5, C5, C2×C4, C23, D5, D5, C10, C10, C22×C4, Dic5, F5, D10, C2×C10, C2×C10, C52, C2×Dic5, C2×F5, C22×D5, C22×C10, C5×D5, C5×D5, C5×C10, C22×Dic5, C22×F5, D5.D5, D5×C10, C102, C2×D5.D5, D5×C2×C10, C22×D5.D5
Quotients: C1, C2, C4, C22, C2×C4, C23, D5, C22×C4, Dic5, F5, D10, C2×Dic5, C2×F5, C22×D5, C22×Dic5, C22×F5, D5.D5, C2×D5.D5, C22×D5.D5

Smallest permutation representation of C22×D5.D5
On 80 points
Generators in S80
(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 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 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 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 10)(2 9)(3 8)(4 7)(5 6)(11 18)(12 17)(13 16)(14 20)(15 19)(21 28)(22 27)(23 26)(24 30)(25 29)(31 38)(32 37)(33 36)(34 40)(35 39)(41 48)(42 47)(43 46)(44 50)(45 49)(51 58)(52 57)(53 56)(54 60)(55 59)(61 68)(62 67)(63 66)(64 70)(65 69)(71 78)(72 77)(73 76)(74 80)(75 79)
(1 5 4 3 2)(6 7 8 9 10)(11 15 14 13 12)(16 17 18 19 20)(21 25 24 23 22)(26 27 28 29 30)(31 35 34 33 32)(36 37 38 39 40)(41 43 45 42 44)(46 49 47 50 48)(51 53 55 52 54)(56 59 57 60 58)(61 63 65 62 64)(66 69 67 70 68)(71 73 75 72 74)(76 79 77 80 78)
(1 79 6 71)(2 77 10 73)(3 80 9 75)(4 78 8 72)(5 76 7 74)(11 70 19 65)(12 68 18 62)(13 66 17 64)(14 69 16 61)(15 67 20 63)(21 60 29 55)(22 58 28 52)(23 56 27 54)(24 59 26 51)(25 57 30 53)(31 50 39 45)(32 48 38 42)(33 46 37 44)(34 49 36 41)(35 47 40 43)

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

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

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

52 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A ··· 4H 5A 5B 5C ··· 5G 10A ··· 10F 10G ··· 10U 10V ··· 10AC order 1 2 2 2 2 2 2 2 4 ··· 4 5 5 5 ··· 5 10 ··· 10 10 ··· 10 10 ··· 10 size 1 1 1 1 5 5 5 5 25 ··· 25 2 2 4 ··· 4 2 ··· 2 4 ··· 4 10 ··· 10

52 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 4 4 4 4 type + + + + - + - + + image C1 C2 C2 C4 C4 D5 Dic5 D10 Dic5 F5 C2×F5 D5.D5 C2×D5.D5 kernel C22×D5.D5 C2×D5.D5 D5×C2×C10 D5×C10 C102 C22×D5 D10 D10 C2×C10 C2×C10 C10 C22 C2 # reps 1 6 1 6 2 2 6 6 2 1 3 4 12

Matrix representation of C22×D5.D5 in GL8(𝔽41)

 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 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 37 0 0 0 0 0 0 0 0 10 0 0 0 0 0 0 0 0 18 0 0 0 0 0 37 7 16 16
,
 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 0 10 0 0 0 0 0 0 37 0 0 0 0 0 0 0 16 16 25 2 0 0 0 0 37 7 16 16
,
 6 40 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 6 40 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 10 0 0 0 0 0 0 0 0 10 0 0 0 0 0 0 0 0 37 0 0 0 0 0 11 11 0 37
,
 13 32 0 0 0 0 0 0 28 28 0 0 0 0 0 0 0 0 28 9 0 0 0 0 0 0 13 13 0 0 0 0 0 0 0 0 0 0 37 0 0 0 0 0 37 37 4 20 0 0 0 0 0 10 0 0 0 0 0 0 30 30 0 4

G:=sub<GL(8,GF(41))| [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],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,37,0,0,37,0,0,0,0,0,10,0,7,0,0,0,0,0,0,18,16,0,0,0,0,0,0,0,16],[40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,37,16,37,0,0,0,0,10,0,16,7,0,0,0,0,0,0,25,16,0,0,0,0,0,0,2,16],[6,1,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,6,1,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,10,0,0,11,0,0,0,0,0,10,0,11,0,0,0,0,0,0,37,0,0,0,0,0,0,0,0,37],[13,28,0,0,0,0,0,0,32,28,0,0,0,0,0,0,0,0,28,13,0,0,0,0,0,0,9,13,0,0,0,0,0,0,0,0,0,37,0,30,0,0,0,0,0,37,10,30,0,0,0,0,37,4,0,0,0,0,0,0,0,20,0,4] >;

C22×D5.D5 in GAP, Magma, Sage, TeX

C_2^2\times D_5.D_5
% in TeX

G:=Group("C2^2xD5.D5");
// GroupNames label

G:=SmallGroup(400,215);
// by ID

G=gap.SmallGroup(400,215);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-5,-5,48,1924,8645,1463]);
// Polycyclic

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

׿
×
𝔽