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## G = (C5×D4).31D4order 320 = 26·5

### 1st non-split extension by C5×D4 of D4 acting via D4/C22=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C20 — (C5×D4).31D4
 Chief series C1 — C5 — C10 — C2×C10 — C2×C20 — C2×Dic10 — C2×D4.D5 — (C5×D4).31D4
 Lower central C5 — C10 — C2×C20 — (C5×D4).31D4
 Upper central C1 — C22 — C22×C4 — C22×D4

Generators and relations for (C5×D4).31D4
G = < a,b,c,d,e | a5=b4=c2=1, d4=e2=b2, ab=ba, ac=ca, dad-1=eae-1=a-1, cbc=ebe-1=b-1, bd=db, dcd-1=bc, ece-1=b-1c, ede-1=d3 >

Subgroups: 574 in 188 conjugacy classes, 51 normal (25 characteristic)
C1, C2 [×3], C2 [×6], C4 [×2], C4 [×3], C22, C22 [×2], C22 [×18], C5, C8 [×2], C2×C4 [×2], C2×C4 [×4], D4 [×4], D4 [×6], Q8 [×2], C23, C23 [×10], C10 [×3], C10 [×6], C22⋊C4, C4⋊C4 [×2], C2×C8 [×2], SD16 [×4], C22×C4, C2×D4 [×2], C2×D4 [×5], C2×Q8, C24, Dic5 [×2], C20 [×2], C20, C2×C10, C2×C10 [×2], C2×C10 [×18], C22⋊C8, D4⋊C4 [×2], C22⋊Q8, C2×SD16 [×2], C22×D4, C52C8 [×2], Dic10 [×2], C2×Dic5 [×2], C2×C20 [×2], C2×C20 [×2], C5×D4 [×4], C5×D4 [×6], C22×C10, C22×C10 [×10], C22⋊SD16, C2×C52C8 [×2], C10.D4, C4⋊Dic5, D4.D5 [×4], C23.D5, C2×Dic10, C22×C20, D4×C10 [×2], D4×C10 [×5], C23×C10, C20.55D4, D4⋊Dic5 [×2], C20.48D4, C2×D4.D5 [×2], D4×C2×C10, (C5×D4).31D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, D5, SD16 [×2], C2×D4 [×3], D10 [×3], C22≀C2, C2×SD16, C8⋊C22, C5⋊D4 [×6], C22×D5, C22⋊SD16, D4.D5 [×2], C2×C5⋊D4 [×3], D4.D10, C2×D4.D5, C242D5, (C5×D4).31D4

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

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

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

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

59 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 4A 4B 4C 4D 4E 5A 5B 8A 8B 8C 8D 10A ··· 10N 10O ··· 10AD 20A ··· 20H order 1 2 2 2 2 2 2 2 2 2 4 4 4 4 4 5 5 8 8 8 8 10 ··· 10 10 ··· 10 20 ··· 20 size 1 1 1 1 2 2 4 4 4 4 2 2 4 40 40 2 2 20 20 20 20 2 ··· 2 4 ··· 4 4 ··· 4

59 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 D4 D4 D4 D5 SD16 D10 D10 C5⋊D4 C5⋊D4 C5⋊D4 C8⋊C22 D4.D5 D4.D10 kernel (C5×D4).31D4 C20.55D4 D4⋊Dic5 C20.48D4 C2×D4.D5 D4×C2×C10 C2×C20 C5×D4 C22×C10 C22×D4 C2×C10 C22×C4 C2×D4 C2×C4 D4 C23 C10 C22 C2 # reps 1 1 2 1 2 1 1 4 1 2 4 2 4 4 16 4 1 4 4

Matrix representation of (C5×D4).31D4 in GL4(𝔽41) generated by

 10 0 0 0 24 37 0 0 0 0 1 0 0 0 0 1
,
 40 0 0 0 0 40 0 0 0 0 25 9 0 0 17 16
,
 40 0 0 0 20 1 0 0 0 0 40 0 0 0 1 1
,
 31 40 0 0 17 10 0 0 0 0 20 29 0 0 32 32
,
 31 40 0 0 17 10 0 0 0 0 32 12 0 0 0 9
G:=sub<GL(4,GF(41))| [10,24,0,0,0,37,0,0,0,0,1,0,0,0,0,1],[40,0,0,0,0,40,0,0,0,0,25,17,0,0,9,16],[40,20,0,0,0,1,0,0,0,0,40,1,0,0,0,1],[31,17,0,0,40,10,0,0,0,0,20,32,0,0,29,32],[31,17,0,0,40,10,0,0,0,0,32,0,0,0,12,9] >;

(C5×D4).31D4 in GAP, Magma, Sage, TeX

(C_5\times D_4)._{31}D_4
% in TeX

G:=Group("(C5xD4).31D4");
// GroupNames label

G:=SmallGroup(320,845);
// by ID

G=gap.SmallGroup(320,845);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,253,254,1684,851,102,12550]);
// 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,d*a*d^-1=e*a*e^-1=a^-1,c*b*c=e*b*e^-1=b^-1,b*d=d*b,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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