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## G = C5×D4.10D4order 320 = 26·5

### Direct product of C5 and D4.10D4

direct product, metabelian, nilpotent (class 3), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C4 — C5×D4.10D4
 Chief series C1 — C2 — C22 — C2×C4 — C2×C20 — Q8×C10 — C5×C8.C22 — C5×D4.10D4
 Lower central C1 — C2 — C2×C4 — C5×D4.10D4
 Upper central C1 — C10 — C2×C20 — C5×D4.10D4

Generators and relations for C5×D4.10D4
G = < a,b,c,d,e | a5=b4=c2=1, d4=e2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc=dbd-1=ebe-1=b-1, dcd-1=bc, ece-1=b-1c, ede-1=d3 >

Subgroups: 242 in 142 conjugacy classes, 54 normal (22 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, C2×C4, D4, D4, Q8, Q8, C10, C10, C42, C4⋊C4, M4(2), SD16, Q16, C2×Q8, C2×Q8, C4○D4, C4○D4, C20, C20, C2×C10, C2×C10, C4.10D4, C4≀C2, C4⋊Q8, C8.C22, 2- 1+4, C40, C2×C20, C2×C20, C2×C20, C5×D4, C5×D4, C5×Q8, C5×Q8, D4.10D4, C4×C20, C5×C4⋊C4, C5×M4(2), C5×SD16, C5×Q16, Q8×C10, Q8×C10, C5×C4○D4, C5×C4○D4, C5×C4.10D4, C5×C4≀C2, C5×C4⋊Q8, C5×C8.C22, C5×2- 1+4, C5×D4.10D4
Quotients: C1, C2, C22, C5, D4, C23, C10, C2×D4, C2×C10, C22≀C2, C5×D4, C22×C10, D4.10D4, D4×C10, C5×C22≀C2, C5×D4.10D4

Smallest permutation representation of C5×D4.10D4
On 80 points
Generators in S80
(1 58 17 75 30)(2 59 18 76 31)(3 60 19 77 32)(4 61 20 78 25)(5 62 21 79 26)(6 63 22 80 27)(7 64 23 73 28)(8 57 24 74 29)(9 48 56 69 35)(10 41 49 70 36)(11 42 50 71 37)(12 43 51 72 38)(13 44 52 65 39)(14 45 53 66 40)(15 46 54 67 33)(16 47 55 68 34)
(1 7 5 3)(2 4 6 8)(9 15 13 11)(10 12 14 16)(17 23 21 19)(18 20 22 24)(25 27 29 31)(26 32 30 28)(33 39 37 35)(34 36 38 40)(41 43 45 47)(42 48 46 44)(49 51 53 55)(50 56 54 52)(57 59 61 63)(58 64 62 60)(65 71 69 67)(66 68 70 72)(73 79 77 75)(74 76 78 80)
(1 6)(2 5)(3 8)(4 7)(9 14)(10 13)(11 16)(12 15)(17 22)(18 21)(19 24)(20 23)(25 28)(26 31)(27 30)(29 32)(33 38)(34 37)(35 40)(36 39)(41 44)(42 47)(43 46)(45 48)(49 52)(50 55)(51 54)(53 56)(57 60)(58 63)(59 62)(61 64)(65 70)(66 69)(67 72)(68 71)(73 78)(74 77)(75 80)(76 79)
(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 42 5 46)(2 45 6 41)(3 48 7 44)(4 43 8 47)(9 28 13 32)(10 31 14 27)(11 26 15 30)(12 29 16 25)(17 71 21 67)(18 66 22 70)(19 69 23 65)(20 72 24 68)(33 75 37 79)(34 78 38 74)(35 73 39 77)(36 76 40 80)(49 59 53 63)(50 62 54 58)(51 57 55 61)(52 60 56 64)

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

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

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

80 conjugacy classes

 class 1 2A 2B 2C 2D 4A 4B 4C ··· 4H 4I 5A 5B 5C 5D 8A 8B 10A 10B 10C 10D 10E 10F 10G 10H 10I ··· 10P 20A ··· 20H 20I ··· 20AF 20AG 20AH 20AI 20AJ 40A ··· 40H order 1 2 2 2 2 4 4 4 ··· 4 4 5 5 5 5 8 8 10 10 10 10 10 10 10 10 10 ··· 10 20 ··· 20 20 ··· 20 20 20 20 20 40 ··· 40 size 1 1 2 4 4 2 2 4 ··· 4 8 1 1 1 1 8 8 1 1 1 1 2 2 2 2 4 ··· 4 2 ··· 2 4 ··· 4 8 8 8 8 8 ··· 8

80 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 4 4 type + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C5 C10 C10 C10 C10 C10 D4 D4 D4 C5×D4 C5×D4 C5×D4 D4.10D4 C5×D4.10D4 kernel C5×D4.10D4 C5×C4.10D4 C5×C4≀C2 C5×C4⋊Q8 C5×C8.C22 C5×2- 1+4 D4.10D4 C4.10D4 C4≀C2 C4⋊Q8 C8.C22 2- 1+4 C2×C20 C5×D4 C5×Q8 C2×C4 D4 Q8 C5 C1 # reps 1 1 2 1 2 1 4 4 8 4 8 4 2 2 2 8 8 8 2 8

Matrix representation of C5×D4.10D4 in GL4(𝔽41) generated by

 18 0 0 0 0 18 0 0 0 0 18 0 0 0 0 18
,
 0 1 0 0 40 0 0 0 40 40 40 39 1 0 1 1
,
 14 14 21 28 7 7 34 14 34 14 0 0 27 20 34 20
,
 7 7 34 14 27 27 20 13 7 27 0 0 7 14 14 7
,
 27 34 0 0 34 14 0 0 14 14 21 28 0 27 34 20
G:=sub<GL(4,GF(41))| [18,0,0,0,0,18,0,0,0,0,18,0,0,0,0,18],[0,40,40,1,1,0,40,0,0,0,40,1,0,0,39,1],[14,7,34,27,14,7,14,20,21,34,0,34,28,14,0,20],[7,27,7,7,7,27,27,14,34,20,0,14,14,13,0,7],[27,34,14,0,34,14,14,27,0,0,21,34,0,0,28,20] >;

C5×D4.10D4 in GAP, Magma, Sage, TeX

C_5\times D_4._{10}D_4
% in TeX

G:=Group("C5xD4.10D4");
// GroupNames label

G:=SmallGroup(320,957);
// by ID

G=gap.SmallGroup(320,957);
# by ID

G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,1120,589,1766,856,7004,3511,1768,172,5052]);
// Polycyclic

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

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