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## G = D5×SD32order 320 = 26·5

### Direct product of D5 and SD32

Series: Derived Chief Lower central Upper central

 Derived series C1 — C40 — D5×SD32
 Chief series C1 — C5 — C10 — C20 — C40 — C8×D5 — D5×D8 — D5×SD32
 Lower central C5 — C10 — C20 — C40 — D5×SD32
 Upper central C1 — C2 — C4 — C8 — SD32

Generators and relations for D5×SD32
G = < a,b,c,d | a5=b2=c16=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c7 >

Subgroups: 518 in 90 conjugacy classes, 33 normal (31 characteristic)
C1, C2, C2, C4, C4, C22, C5, C8, C8, C2×C4, D4, Q8, C23, D5, D5, C10, C10, C16, C16, C2×C8, D8, D8, Q16, Q16, C2×D4, C2×Q8, Dic5, Dic5, C20, C20, D10, D10, C2×C10, C2×C16, SD32, SD32, C2×D8, C2×Q16, C52C8, C40, Dic10, C4×D5, C4×D5, D20, C5⋊D4, C5×D4, C5×Q8, C22×D5, C2×SD32, C52C16, C80, C8×D5, D40, Dic20, D4⋊D5, C5⋊Q16, C5×D8, C5×Q16, D4×D5, Q8×D5, D5×C16, C16⋊D5, D8.D5, C5⋊SD32, C5×SD32, D5×D8, D5×Q16, D5×SD32
Quotients: C1, C2, C22, D4, C23, D5, D8, C2×D4, D10, SD32, C2×D8, C22×D5, C2×SD32, D4×D5, D5×D8, D5×SD32

Smallest permutation representation of D5×SD32
On 80 points
Generators in S80
(1 56 30 37 77)(2 57 31 38 78)(3 58 32 39 79)(4 59 17 40 80)(5 60 18 41 65)(6 61 19 42 66)(7 62 20 43 67)(8 63 21 44 68)(9 64 22 45 69)(10 49 23 46 70)(11 50 24 47 71)(12 51 25 48 72)(13 52 26 33 73)(14 53 27 34 74)(15 54 28 35 75)(16 55 29 36 76)
(1 69)(2 70)(3 71)(4 72)(5 73)(6 74)(7 75)(8 76)(9 77)(10 78)(11 79)(12 80)(13 65)(14 66)(15 67)(16 68)(17 25)(18 26)(19 27)(20 28)(21 29)(22 30)(23 31)(24 32)(33 60)(34 61)(35 62)(36 63)(37 64)(38 49)(39 50)(40 51)(41 52)(42 53)(43 54)(44 55)(45 56)(46 57)(47 58)(48 59)
(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)
(2 8)(3 15)(4 6)(5 13)(7 11)(10 16)(12 14)(17 19)(18 26)(20 24)(21 31)(23 29)(25 27)(28 32)(33 41)(34 48)(35 39)(36 46)(38 44)(40 42)(43 47)(49 55)(50 62)(51 53)(52 60)(54 58)(57 63)(59 61)(65 73)(66 80)(67 71)(68 78)(70 76)(72 74)(75 79)

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

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

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

44 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 5A 5B 8A 8B 8C 8D 10A 10B 10C 10D 16A 16B 16C 16D 16E 16F 16G 16H 20A 20B 20C 20D 40A 40B 40C 40D 80A ··· 80H order 1 2 2 2 2 2 4 4 4 4 5 5 8 8 8 8 10 10 10 10 16 16 16 16 16 16 16 16 20 20 20 20 40 40 40 40 80 ··· 80 size 1 1 5 5 8 40 2 8 10 40 2 2 2 2 10 10 2 2 16 16 2 2 2 2 10 10 10 10 4 4 16 16 4 4 4 4 4 ··· 4

44 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 D4 D4 D5 D8 D8 D10 D10 D10 SD32 D4×D5 D5×D8 D5×SD32 kernel D5×SD32 D5×C16 C16⋊D5 D8.D5 C5⋊SD32 C5×SD32 D5×D8 D5×Q16 C5⋊2C8 C4×D5 SD32 Dic5 D10 C16 D8 Q16 D5 C4 C2 C1 # reps 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 8 2 4 8

Matrix representation of D5×SD32 in GL4(𝔽241) generated by

 0 1 0 0 240 189 0 0 0 0 1 0 0 0 0 1
,
 0 1 0 0 1 0 0 0 0 0 240 0 0 0 0 240
,
 1 0 0 0 0 1 0 0 0 0 62 222 0 0 102 97
,
 1 0 0 0 0 1 0 0 0 0 240 54 0 0 0 1
G:=sub<GL(4,GF(241))| [0,240,0,0,1,189,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,240,0,0,0,0,240],[1,0,0,0,0,1,0,0,0,0,62,102,0,0,222,97],[1,0,0,0,0,1,0,0,0,0,240,0,0,0,54,1] >;

D5×SD32 in GAP, Magma, Sage, TeX

D_5\times {\rm SD}_{32}
% in TeX

G:=Group("D5xSD32");
// GroupNames label

G:=SmallGroup(320,540);
// by ID

G=gap.SmallGroup(320,540);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,135,184,346,185,192,851,438,102,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^5=b^2=c^16=d^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^7>;
// generators/relations

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