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## G = C2×D5×C4○D4order 320 = 26·5

### Direct product of C2, D5 and C4○D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C2×D5×C4○D4
 Chief series C1 — C5 — C10 — D10 — C22×D5 — C23×D5 — D5×C22×C4 — C2×D5×C4○D4
 Lower central C5 — C10 — C2×D5×C4○D4
 Upper central C1 — C2×C4 — C2×C4○D4

Generators and relations for C2×D5×C4○D4
G = < a,b,c,d,e,f | a2=b5=c2=d4=f2=1, e2=d2, 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, de=ed, df=fd, fef=d2e >

Subgroups: 2654 in 890 conjugacy classes, 455 normal (18 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C5, C2×C4, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C23, D5, D5, C10, C10, C10, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C4○D4, C4○D4, C24, Dic5, C20, D10, D10, C2×C10, C2×C10, C2×C10, C23×C4, C22×D4, C22×Q8, C2×C4○D4, C2×C4○D4, Dic10, C4×D5, D20, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C5×D4, C5×Q8, C22×D5, C22×D5, C22×D5, C22×C10, C22×C4○D4, C2×Dic10, C2×C4×D5, C2×C4×D5, C2×D20, C4○D20, D4×D5, D42D5, Q8×D5, Q82D5, C22×Dic5, C2×C5⋊D4, C22×C20, D4×C10, Q8×C10, C5×C4○D4, C23×D5, D5×C22×C4, C2×C4○D20, C2×D4×D5, C2×D42D5, C2×Q8×D5, C2×Q82D5, D5×C4○D4, C10×C4○D4, C2×D5×C4○D4
Quotients: C1, C2, C22, C23, D5, C4○D4, C24, D10, C2×C4○D4, C25, C22×D5, C22×C4○D4, C23×D5, D5×C4○D4, D5×C24, C2×D5×C4○D4

Smallest permutation representation of C2×D5×C4○D4
On 80 points
Generators in S80
(1 44)(2 45)(3 41)(4 42)(5 43)(6 46)(7 47)(8 48)(9 49)(10 50)(11 51)(12 52)(13 53)(14 54)(15 55)(16 56)(17 57)(18 58)(19 59)(20 60)(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 48)(2 47)(3 46)(4 50)(5 49)(6 41)(7 45)(8 44)(9 43)(10 42)(11 56)(12 60)(13 59)(14 58)(15 57)(16 51)(17 55)(18 54)(19 53)(20 52)(21 66)(22 70)(23 69)(24 68)(25 67)(26 61)(27 65)(28 64)(29 63)(30 62)(31 76)(32 80)(33 79)(34 78)(35 77)(36 71)(37 75)(38 74)(39 73)(40 72)
(1 19 9 14)(2 20 10 15)(3 16 6 11)(4 17 7 12)(5 18 8 13)(21 36 26 31)(22 37 27 32)(23 38 28 33)(24 39 29 34)(25 40 30 35)(41 56 46 51)(42 57 47 52)(43 58 48 53)(44 59 49 54)(45 60 50 55)(61 76 66 71)(62 77 67 72)(63 78 68 73)(64 79 69 74)(65 80 70 75)
(1 64 9 69)(2 65 10 70)(3 61 6 66)(4 62 7 67)(5 63 8 68)(11 71 16 76)(12 72 17 77)(13 73 18 78)(14 74 19 79)(15 75 20 80)(21 46 26 41)(22 47 27 42)(23 48 28 43)(24 49 29 44)(25 50 30 45)(31 56 36 51)(32 57 37 52)(33 58 38 53)(34 59 39 54)(35 60 40 55)
(21 26)(22 27)(23 28)(24 29)(25 30)(31 36)(32 37)(33 38)(34 39)(35 40)(61 66)(62 67)(63 68)(64 69)(65 70)(71 76)(72 77)(73 78)(74 79)(75 80)

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

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

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

80 conjugacy classes

 class 1 2A 2B 2C 2D ··· 2I 2J 2K 2L 2M 2N ··· 2S 4A 4B 4C 4D 4E ··· 4J 4K 4L 4M 4N 4O ··· 4T 5A 5B 10A ··· 10F 10G ··· 10R 20A ··· 20H 20I ··· 20T order 1 2 2 2 2 ··· 2 2 2 2 2 2 ··· 2 4 4 4 4 4 ··· 4 4 4 4 4 4 ··· 4 5 5 10 ··· 10 10 ··· 10 20 ··· 20 20 ··· 20 size 1 1 1 1 2 ··· 2 5 5 5 5 10 ··· 10 1 1 1 1 2 ··· 2 5 5 5 5 10 ··· 10 2 2 2 ··· 2 4 ··· 4 2 ··· 2 4 ··· 4

80 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 4 type + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 D5 C4○D4 D10 D10 D10 D10 D5×C4○D4 kernel C2×D5×C4○D4 D5×C22×C4 C2×C4○D20 C2×D4×D5 C2×D4⋊2D5 C2×Q8×D5 C2×Q8⋊2D5 D5×C4○D4 C10×C4○D4 C2×C4○D4 D10 C22×C4 C2×D4 C2×Q8 C4○D4 C2 # reps 1 3 3 3 3 1 1 16 1 2 8 6 6 2 16 8

Matrix representation of C2×D5×C4○D4 in GL5(𝔽41)

 40 0 0 0 0 0 40 0 0 0 0 0 40 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 7 1 0 0 0 33 40
,
 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 40 40 0 0 0 0 1
,
 40 0 0 0 0 0 32 0 0 0 0 0 32 0 0 0 0 0 40 0 0 0 0 0 40
,
 1 0 0 0 0 0 0 1 0 0 0 40 0 0 0 0 0 0 1 0 0 0 0 0 1
,
 40 0 0 0 0 0 1 0 0 0 0 0 40 0 0 0 0 0 1 0 0 0 0 0 1

G:=sub<GL(5,GF(41))| [40,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,7,33,0,0,0,1,40],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,40,0,0,0,0,40,1],[40,0,0,0,0,0,32,0,0,0,0,0,32,0,0,0,0,0,40,0,0,0,0,0,40],[1,0,0,0,0,0,0,40,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1],[40,0,0,0,0,0,1,0,0,0,0,0,40,0,0,0,0,0,1,0,0,0,0,0,1] >;

C2×D5×C4○D4 in GAP, Magma, Sage, TeX

C_2\times D_5\times C_4\circ D_4
% in TeX

G:=Group("C2xD5xC4oD4");
// GroupNames label

G:=SmallGroup(320,1618);
// by ID

G=gap.SmallGroup(320,1618);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,136,438,12550]);
// Polycyclic

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

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