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## G = C4×D5.D5order 400 = 24·52

### Direct product of C4 and D5.D5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C52 — C4×D5.D5
 Chief series C1 — C5 — C52 — C5×D5 — D5×C10 — C2×D5.D5 — C4×D5.D5
 Lower central C52 — C4×D5.D5
 Upper central C1 — C4

Generators and relations for C4×D5.D5
G = < a,b,c,d,e | a4=b5=c2=d5=1, e2=b-1c, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, ebe-1=b2, cd=dc, ece-1=bc, ede-1=d-1 >

Subgroups: 336 in 63 conjugacy classes, 29 normal (19 characteristic)
C1, C2, C2, C4, C4, C22, C5, C5, C2×C4, D5, C10, C10, C42, Dic5, Dic5, C20, C20, F5, D10, C2×C10, C52, C4×D5, C2×Dic5, C2×C20, C2×F5, C5×D5, C5×C10, C4×Dic5, C4×F5, C5×Dic5, C5×C20, D5.D5, D5×C10, D5×C20, C2×D5.D5, C4×D5.D5
Quotients: C1, C2, C4, C22, C2×C4, D5, C42, Dic5, F5, D10, C4×D5, C2×Dic5, C2×F5, C4×Dic5, C4×F5, D5.D5, C2×D5.D5, C4×D5.D5

Smallest permutation representation of C4×D5.D5
On 80 points
Generators in S80
(1 28 8 23)(2 29 9 24)(3 30 10 25)(4 26 6 21)(5 27 7 22)(11 36 16 31)(12 37 17 32)(13 38 18 33)(14 39 19 34)(15 40 20 35)(41 66 46 61)(42 67 47 62)(43 68 48 63)(44 69 49 64)(45 70 50 65)(51 76 56 71)(52 77 57 72)(53 78 58 73)(54 79 59 74)(55 80 60 75)
(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 16)(2 20)(3 19)(4 18)(5 17)(6 13)(7 12)(8 11)(9 15)(10 14)(21 38)(22 37)(23 36)(24 40)(25 39)(26 33)(27 32)(28 31)(29 35)(30 34)(41 58)(42 57)(43 56)(44 60)(45 59)(46 53)(47 52)(48 51)(49 55)(50 54)(61 78)(62 77)(63 76)(64 80)(65 79)(66 73)(67 72)(68 71)(69 75)(70 74)
(1 4 2 5 3)(6 9 7 10 8)(11 13 15 12 14)(16 18 20 17 19)(21 24 22 25 23)(26 29 27 30 28)(31 33 35 32 34)(36 38 40 37 39)(41 45 44 43 42)(46 50 49 48 47)(51 52 53 54 55)(56 57 58 59 60)(61 65 64 63 62)(66 70 69 68 67)(71 72 73 74 75)(76 77 78 79 80)
(1 73 17 67)(2 71 16 69)(3 74 20 66)(4 72 19 68)(5 75 18 70)(6 77 14 63)(7 80 13 65)(8 78 12 62)(9 76 11 64)(10 79 15 61)(21 57 39 43)(22 60 38 45)(23 58 37 42)(24 56 36 44)(25 59 40 41)(26 52 34 48)(27 55 33 50)(28 53 32 47)(29 51 31 49)(30 54 35 46)

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

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

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

52 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C 4D 4E ··· 4L 5A 5B 5C ··· 5G 10A 10B 10C ··· 10G 10H 10I 10J 10K 20A 20B 20C 20D 20E ··· 20N 20O 20P 20Q 20R order 1 2 2 2 4 4 4 4 4 ··· 4 5 5 5 ··· 5 10 10 10 ··· 10 10 10 10 10 20 20 20 20 20 ··· 20 20 20 20 20 size 1 1 5 5 1 1 5 5 25 ··· 25 2 2 4 ··· 4 2 2 4 ··· 4 10 10 10 10 2 2 2 2 4 ··· 4 10 10 10 10

52 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 4 4 4 4 4 4 type + + + + - - + + + image C1 C2 C2 C4 C4 C4 D5 Dic5 Dic5 D10 C4×D5 F5 C2×F5 C4×F5 D5.D5 C2×D5.D5 C4×D5.D5 kernel C4×D5.D5 D5×C20 C2×D5.D5 C5×Dic5 C5×C20 D5.D5 C4×D5 Dic5 C20 D10 D5 C20 C10 C5 C4 C2 C1 # reps 1 1 2 2 2 8 2 2 2 2 8 1 1 2 4 4 8

Matrix representation of C4×D5.D5 in GL4(𝔽41) generated by

 9 0 0 0 0 9 0 0 0 0 9 0 0 0 0 9
,
 37 0 0 0 0 10 0 0 0 0 18 0 0 0 0 16
,
 0 31 0 0 4 0 0 0 0 0 0 25 0 0 23 0
,
 18 0 0 0 0 18 0 0 0 0 16 0 0 0 0 16
,
 0 0 20 0 0 0 0 20 0 2 0 0 2 0 0 0
G:=sub<GL(4,GF(41))| [9,0,0,0,0,9,0,0,0,0,9,0,0,0,0,9],[37,0,0,0,0,10,0,0,0,0,18,0,0,0,0,16],[0,4,0,0,31,0,0,0,0,0,0,23,0,0,25,0],[18,0,0,0,0,18,0,0,0,0,16,0,0,0,0,16],[0,0,0,2,0,0,2,0,20,0,0,0,0,20,0,0] >;

C4×D5.D5 in GAP, Magma, Sage, TeX

C_4\times D_5.D_5
% in TeX

G:=Group("C4xD5.D5");
// GroupNames label

G:=SmallGroup(400,144);
// by ID

G=gap.SmallGroup(400,144);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-5,-5,24,55,1924,8645,2897]);
// Polycyclic

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

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