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## G = C2×D5×C22⋊C4order 320 = 26·5

### Direct product of C2, D5 and C22⋊C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C2×D5×C22⋊C4
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C23×D5 — D5×C24 — C2×D5×C22⋊C4
 Lower central C5 — C10 — C2×D5×C22⋊C4
 Upper central C1 — C23 — C2×C22⋊C4

Generators and relations for C2×D5×C22⋊C4
G = < a,b,c,d,e,f | a2=b5=c2=d2=e2=f4=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, cbc=b-1, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, fdf-1=de=ed, ef=fe >

Subgroups: 2654 in 674 conjugacy classes, 207 normal (17 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C5, C2×C4, C2×C4, C23, C23, C23, D5, D5, C10, C10, C10, C22⋊C4, C22⋊C4, C22×C4, C22×C4, C24, C24, Dic5, C20, D10, D10, C2×C10, C2×C10, C2×C10, C2×C22⋊C4, C2×C22⋊C4, C23×C4, C25, C4×D5, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×D5, C22×D5, C22×C10, C22×C10, C22×C10, C22×C22⋊C4, D10⋊C4, C23.D5, C5×C22⋊C4, C2×C4×D5, C2×C4×D5, C22×Dic5, C22×C20, C23×D5, C23×D5, C23×D5, C23×C10, D5×C22⋊C4, C2×D10⋊C4, C2×C23.D5, C10×C22⋊C4, D5×C22×C4, D5×C24, C2×D5×C22⋊C4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D5, C22⋊C4, C22×C4, C2×D4, C24, D10, C2×C22⋊C4, C23×C4, C22×D4, C4×D5, C22×D5, C22×C22⋊C4, C2×C4×D5, D4×D5, C23×D5, D5×C22⋊C4, D5×C22×C4, C2×D4×D5, C2×D5×C22⋊C4

Smallest permutation representation of C2×D5×C22⋊C4
On 80 points
Generators in S80
(1 49)(2 50)(3 46)(4 47)(5 48)(6 41)(7 42)(8 43)(9 44)(10 45)(11 56)(12 57)(13 58)(14 59)(15 60)(16 51)(17 52)(18 53)(19 54)(20 55)(21 61)(22 62)(23 63)(24 64)(25 65)(26 66)(27 67)(28 68)(29 69)(30 70)(31 71)(32 72)(33 73)(34 74)(35 75)(36 76)(37 77)(38 78)(39 79)(40 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 53)(2 52)(3 51)(4 55)(5 54)(6 56)(7 60)(8 59)(9 58)(10 57)(11 41)(12 45)(13 44)(14 43)(15 42)(16 46)(17 50)(18 49)(19 48)(20 47)(21 76)(22 80)(23 79)(24 78)(25 77)(26 71)(27 75)(28 74)(29 73)(30 72)(31 66)(32 70)(33 69)(34 68)(35 67)(36 61)(37 65)(38 64)(39 63)(40 62)
(1 14)(2 15)(3 11)(4 12)(5 13)(6 16)(7 17)(8 18)(9 19)(10 20)(21 36)(22 37)(23 38)(24 39)(25 40)(26 31)(27 32)(28 33)(29 34)(30 35)(41 51)(42 52)(43 53)(44 54)(45 55)(46 56)(47 57)(48 58)(49 59)(50 60)(61 76)(62 77)(63 78)(64 79)(65 80)(66 71)(67 72)(68 73)(69 74)(70 75)
(1 9)(2 10)(3 6)(4 7)(5 8)(11 16)(12 17)(13 18)(14 19)(15 20)(21 26)(22 27)(23 28)(24 29)(25 30)(31 36)(32 37)(33 38)(34 39)(35 40)(41 46)(42 47)(43 48)(44 49)(45 50)(51 56)(52 57)(53 58)(54 59)(55 60)(61 66)(62 67)(63 68)(64 69)(65 70)(71 76)(72 77)(73 78)(74 79)(75 80)
(1 34 14 24)(2 35 15 25)(3 31 11 21)(4 32 12 22)(5 33 13 23)(6 36 16 26)(7 37 17 27)(8 38 18 28)(9 39 19 29)(10 40 20 30)(41 76 51 66)(42 77 52 67)(43 78 53 68)(44 79 54 69)(45 80 55 70)(46 71 56 61)(47 72 57 62)(48 73 58 63)(49 74 59 64)(50 75 60 65)

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

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

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

80 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 2L ··· 2S 2T 2U 2V 2W 4A ··· 4H 4I ··· 4P 5A 5B 10A ··· 10N 10O ··· 10V 20A ··· 20P order 1 2 ··· 2 2 2 2 2 2 ··· 2 2 2 2 2 4 ··· 4 4 ··· 4 5 5 10 ··· 10 10 ··· 10 20 ··· 20 size 1 1 ··· 1 2 2 2 2 5 ··· 5 10 10 10 10 2 ··· 2 10 ··· 10 2 2 2 ··· 2 4 ··· 4 4 ··· 4

80 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 4 type + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C4 D4 D5 D10 D10 D10 C4×D5 D4×D5 kernel C2×D5×C22⋊C4 D5×C22⋊C4 C2×D10⋊C4 C2×C23.D5 C10×C22⋊C4 D5×C22×C4 D5×C24 C23×D5 C22×D5 C2×C22⋊C4 C22⋊C4 C22×C4 C24 C23 C22 # reps 1 8 2 1 1 2 1 16 8 2 8 4 2 16 8

Matrix representation of C2×D5×C22⋊C4 in GL6(𝔽41)

 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 0 1 0 0 0 0 40 34 0 0 0 0 0 0 0 1 0 0 0 0 40 6 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 0 40 0 0 0 0 40 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 40 0 0 0 0 0 0 40
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 1 0 0 0 0 0 32 40
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 40 0 0 0 0 0 0 40
,
 40 0 0 0 0 0 0 40 0 0 0 0 0 0 32 0 0 0 0 0 0 32 0 0 0 0 0 0 32 39 0 0 0 0 40 9

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,40,0,0,0,0,1,34,0,0,0,0,0,0,0,40,0,0,0,0,1,6,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,40,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,32,0,0,0,0,0,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,32,0,0,0,0,0,0,32,0,0,0,0,0,0,32,40,0,0,0,0,39,9] >;

C2×D5×C22⋊C4 in GAP, Magma, Sage, TeX

C_2\times D_5\times C_2^2\rtimes C_4
% in TeX

G:=Group("C2xD5xC2^2:C4");
// GroupNames label

G:=SmallGroup(320,1156);
// by ID

G=gap.SmallGroup(320,1156);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,297,80,12550]);
// Polycyclic

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

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