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## G = C5×C42⋊2C2order 160 = 25·5

### Direct product of C5 and C42⋊2C2

direct product, metabelian, nilpotent (class 2), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C5×C42⋊2C2
 Chief series C1 — C2 — C22 — C2×C10 — C22×C10 — C5×C22⋊C4 — C5×C42⋊2C2
 Lower central C1 — C22 — C5×C42⋊2C2
 Upper central C1 — C2×C10 — C5×C42⋊2C2

Generators and relations for C5×C422C2
G = < a,b,c,d | a5=b4=c4=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=bc2, dcd=b2c-1 >

Subgroups: 84 in 60 conjugacy classes, 40 normal (10 characteristic)
C1, C2 [×3], C2, C4 [×6], C22, C22 [×3], C5, C2×C4 [×6], C23, C10 [×3], C10, C42, C22⋊C4 [×3], C4⋊C4 [×3], C20 [×6], C2×C10, C2×C10 [×3], C422C2, C2×C20 [×6], C22×C10, C4×C20, C5×C22⋊C4 [×3], C5×C4⋊C4 [×3], C5×C422C2
Quotients: C1, C2 [×7], C22 [×7], C5, C23, C10 [×7], C4○D4 [×3], C2×C10 [×7], C422C2, C22×C10, C5×C4○D4 [×3], C5×C422C2

Smallest permutation representation of C5×C422C2
On 80 points
Generators in S80
(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 67 27 56)(2 68 28 57)(3 69 29 58)(4 70 30 59)(5 66 26 60)(6 54 14 45)(7 55 15 41)(8 51 11 42)(9 52 12 43)(10 53 13 44)(16 47 76 36)(17 48 77 37)(18 49 78 38)(19 50 79 39)(20 46 80 40)(21 75 35 61)(22 71 31 62)(23 72 32 63)(24 73 33 64)(25 74 34 65)
(1 55 35 36)(2 51 31 37)(3 52 32 38)(4 53 33 39)(5 54 34 40)(6 74 80 60)(7 75 76 56)(8 71 77 57)(9 72 78 58)(10 73 79 59)(11 62 17 68)(12 63 18 69)(13 64 19 70)(14 65 20 66)(15 61 16 67)(21 47 27 41)(22 48 28 42)(23 49 29 43)(24 50 30 44)(25 46 26 45)
(6 14)(7 15)(8 11)(9 12)(10 13)(16 76)(17 77)(18 78)(19 79)(20 80)(36 41)(37 42)(38 43)(39 44)(40 45)(46 54)(47 55)(48 51)(49 52)(50 53)(56 75)(57 71)(58 72)(59 73)(60 74)(61 67)(62 68)(63 69)(64 70)(65 66)

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

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

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

C5×C422C2 is a maximal subgroup of
C42.159D10  C42.160D10  C4223D10  C4224D10  C42.189D10  C42.161D10  C42.162D10  C42.163D10  C42.164D10  C4225D10  C42.165D10

70 conjugacy classes

 class 1 2A 2B 2C 2D 4A ··· 4F 4G 4H 4I 5A 5B 5C 5D 10A ··· 10L 10M 10N 10O 10P 20A ··· 20X 20Y ··· 20AJ order 1 2 2 2 2 4 ··· 4 4 4 4 5 5 5 5 10 ··· 10 10 10 10 10 20 ··· 20 20 ··· 20 size 1 1 1 1 4 2 ··· 2 4 4 4 1 1 1 1 1 ··· 1 4 4 4 4 2 ··· 2 4 ··· 4

70 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 type + + + + image C1 C2 C2 C2 C5 C10 C10 C10 C4○D4 C5×C4○D4 kernel C5×C42⋊2C2 C4×C20 C5×C22⋊C4 C5×C4⋊C4 C42⋊2C2 C42 C22⋊C4 C4⋊C4 C10 C2 # reps 1 1 3 3 4 4 12 12 6 24

Matrix representation of C5×C422C2 in GL4(𝔽41) generated by

 10 0 0 0 0 10 0 0 0 0 1 0 0 0 0 1
,
 32 0 0 0 0 32 0 0 0 0 1 0 0 0 18 40
,
 0 1 0 0 1 0 0 0 0 0 32 0 0 0 2 9
,
 1 0 0 0 0 40 0 0 0 0 1 9 0 0 0 40
G:=sub<GL(4,GF(41))| [10,0,0,0,0,10,0,0,0,0,1,0,0,0,0,1],[32,0,0,0,0,32,0,0,0,0,1,18,0,0,0,40],[0,1,0,0,1,0,0,0,0,0,32,2,0,0,0,9],[1,0,0,0,0,40,0,0,0,0,1,0,0,0,9,40] >;

C5×C422C2 in GAP, Magma, Sage, TeX

C_5\times C_4^2\rtimes_2C_2
% in TeX

G:=Group("C5xC4^2:2C2");
// GroupNames label

G:=SmallGroup(160,187);
// by ID

G=gap.SmallGroup(160,187);
# by ID

G:=PCGroup([6,-2,-2,-2,-5,-2,-2,505,727,1514,194]);
// Polycyclic

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

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