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## G = D5×C22⋊Q8order 320 = 26·5

### Direct product of D5 and C22⋊Q8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — D5×C22⋊Q8
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C23×D5 — D5×C22×C4 — D5×C22⋊Q8
 Lower central C5 — C2×C10 — D5×C22⋊Q8
 Upper central C1 — C22 — C22⋊Q8

Generators and relations for D5×C22⋊Q8
G = < a,b,c,d,e,f | a5=b2=c2=d2=e4=1, f2=e2, bab=a-1, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, fcf-1=cd=dc, ce=ec, de=ed, df=fd, fef-1=e-1 >

Subgroups: 1150 in 322 conjugacy classes, 121 normal (43 characteristic)
C1, C2 [×3], C2 [×8], C4 [×2], C4 [×12], C22, C22 [×2], C22 [×20], C5, C2×C4 [×2], C2×C4 [×4], C2×C4 [×28], Q8 [×8], C23, C23 [×10], D5 [×4], D5 [×2], C10 [×3], C10 [×2], C22⋊C4 [×2], C22⋊C4 [×6], C4⋊C4, C4⋊C4 [×2], C4⋊C4 [×9], C22×C4, C22×C4 [×13], C2×Q8, C2×Q8 [×7], C24, Dic5 [×2], Dic5 [×5], C20 [×2], C20 [×5], D10 [×8], D10 [×10], C2×C10, C2×C10 [×2], C2×C10 [×2], C2×C22⋊C4 [×2], C2×C4⋊C4 [×3], C22⋊Q8, C22⋊Q8 [×7], C23×C4, C22×Q8, Dic10 [×6], C4×D5 [×4], C4×D5 [×14], C2×Dic5 [×2], C2×Dic5 [×4], C2×Dic5 [×2], C2×C20 [×2], C2×C20 [×4], C2×C20 [×2], C5×Q8 [×2], C22×D5 [×2], C22×D5 [×4], C22×D5 [×4], C22×C10, C2×C22⋊Q8, C10.D4 [×6], C4⋊Dic5, C4⋊Dic5 [×2], D10⋊C4 [×4], C23.D5 [×2], C5×C22⋊C4 [×2], C5×C4⋊C4, C5×C4⋊C4 [×2], C2×Dic10, C2×Dic10 [×2], C2×C4×D5 [×4], C2×C4×D5 [×4], C2×C4×D5 [×4], Q8×D5 [×4], C22×Dic5, C22×C20, Q8×C10, C23×D5, Dic5.14D4 [×2], D5×C22⋊C4 [×2], D5×C4⋊C4, D5×C4⋊C4 [×2], D10⋊Q8 [×2], D102Q8, C20.48D4, D103Q8, C5×C22⋊Q8, D5×C22×C4, C2×Q8×D5, D5×C22⋊Q8
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], Q8 [×4], C23 [×15], D5, C2×D4 [×6], C2×Q8 [×6], C4○D4 [×2], C24, D10 [×7], C22⋊Q8 [×4], C22×D4, C22×Q8, C2×C4○D4, C22×D5 [×7], C2×C22⋊Q8, D4×D5 [×2], Q8×D5 [×2], C23×D5, C2×D4×D5, C2×Q8×D5, D5×C4○D4, D5×C22⋊Q8

Smallest permutation representation of D5×C22⋊Q8
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 18)(2 17)(3 16)(4 20)(5 19)(6 11)(7 15)(8 14)(9 13)(10 12)(21 36)(22 40)(23 39)(24 38)(25 37)(26 31)(27 35)(28 34)(29 33)(30 32)(41 56)(42 60)(43 59)(44 58)(45 57)(46 51)(47 55)(48 54)(49 53)(50 52)(61 76)(62 80)(63 79)(64 78)(65 77)(66 71)(67 75)(68 74)(69 73)(70 72)
(1 9)(2 10)(3 6)(4 7)(5 8)(11 16)(12 17)(13 18)(14 19)(15 20)(21 31)(22 32)(23 33)(24 34)(25 35)(26 36)(27 37)(28 38)(29 39)(30 40)(41 46)(42 47)(43 48)(44 49)(45 50)(51 56)(52 57)(53 58)(54 59)(55 60)(61 71)(62 72)(63 73)(64 74)(65 75)(66 76)(67 77)(68 78)(69 79)(70 80)
(1 19)(2 20)(3 16)(4 17)(5 18)(6 11)(7 12)(8 13)(9 14)(10 15)(21 36)(22 37)(23 38)(24 39)(25 40)(26 31)(27 32)(28 33)(29 34)(30 35)(41 56)(42 57)(43 58)(44 59)(45 60)(46 51)(47 52)(48 53)(49 54)(50 55)(61 76)(62 77)(63 78)(64 79)(65 80)(66 71)(67 72)(68 73)(69 74)(70 75)
(1 59 14 49)(2 60 15 50)(3 56 11 46)(4 57 12 47)(5 58 13 48)(6 51 16 41)(7 52 17 42)(8 53 18 43)(9 54 19 44)(10 55 20 45)(21 61 31 71)(22 62 32 72)(23 63 33 73)(24 64 34 74)(25 65 35 75)(26 66 36 76)(27 67 37 77)(28 68 38 78)(29 69 39 79)(30 70 40 80)
(1 79 14 69)(2 80 15 70)(3 76 11 66)(4 77 12 67)(5 78 13 68)(6 71 16 61)(7 72 17 62)(8 73 18 63)(9 74 19 64)(10 75 20 65)(21 51 31 41)(22 52 32 42)(23 53 33 43)(24 54 34 44)(25 55 35 45)(26 56 36 46)(27 57 37 47)(28 58 38 48)(29 59 39 49)(30 60 40 50)

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

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

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

56 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 2K 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 4M 4N 4O 4P 5A 5B 10A ··· 10F 10G 10H 10I 10J 20A ··· 20H 20I ··· 20P 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 4 4 4 4 5 5 10 ··· 10 10 10 10 10 20 ··· 20 20 ··· 20 size 1 1 1 1 2 2 5 5 5 5 10 10 2 2 2 2 4 4 4 4 10 10 10 10 20 20 20 20 2 2 2 ··· 2 4 4 4 4 4 ··· 4 8 ··· 8

56 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + - + + + + + + - image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 D4 Q8 D5 C4○D4 D10 D10 D10 D10 D4×D5 Q8×D5 D5×C4○D4 kernel D5×C22⋊Q8 Dic5.14D4 D5×C22⋊C4 D5×C4⋊C4 D10⋊Q8 D10⋊2Q8 C20.48D4 D10⋊3Q8 C5×C22⋊Q8 D5×C22×C4 C2×Q8×D5 C4×D5 C22×D5 C22⋊Q8 D10 C22⋊C4 C4⋊C4 C22×C4 C2×Q8 C4 C22 C2 # reps 1 2 2 3 2 1 1 1 1 1 1 4 4 2 4 4 6 2 2 4 4 4

Matrix representation of D5×C22⋊Q8 in GL6(𝔽41)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 35 1 0 0 0 0 5 40 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40 40 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 1 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 39 40
,
 40 0 0 0 0 0 0 40 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 1 0 0 0 0 0 0 1 0 0 0 0 0 0 32 0 0 0 0 0 18 9
,
 32 23 0 0 0 0 9 9 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 9 9 0 0 0 0 0 32

G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,35,5,0,0,0,0,1,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,40,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[40,1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,39,0,0,0,0,0,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,32,18,0,0,0,0,0,9],[32,9,0,0,0,0,23,9,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,9,0,0,0,0,0,9,32] >;

D5×C22⋊Q8 in GAP, Magma, Sage, TeX

D_5\times C_2^2\rtimes Q_8
% in TeX

G:=Group("D5xC2^2:Q8");
// GroupNames label

G:=SmallGroup(320,1298);
// by ID

G=gap.SmallGroup(320,1298);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,100,794,297,136,12550]);
// Polycyclic

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

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