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## G = (C2×D4)⋊6F5order 320 = 26·5

### 4th semidirect product of C2×D4 and F5 acting via F5/D5=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — (C2×D4)⋊6F5
 Chief series C1 — C5 — C10 — Dic5 — C4×D5 — C4.F5 — D5⋊M4(2) — (C2×D4)⋊6F5
 Lower central C5 — C10 — C20 — (C2×D4)⋊6F5
 Upper central C1 — C2 — C2×C4 — C2×D4

Generators and relations for (C2×D4)⋊6F5
G = < a,b,c,d,e | a2=b4=c2=d5=e4=1, ab=ba, ac=ca, ad=da, eae-1=ab2, cbc=ebe-1=b-1, bd=db, cd=dc, ece-1=abc, ede-1=d3 >

Subgroups: 634 in 154 conjugacy classes, 48 normal (30 characteristic)
C1, C2, C2 [×5], C4 [×2], C4 [×6], C22, C22 [×7], C5, C8 [×2], C2×C4, C2×C4 [×12], D4 [×2], D4 [×5], Q8 [×3], C23 [×2], D5 [×2], C10, C10 [×3], C42 [×2], C22⋊C4, C4⋊C4, C2×C8, M4(2) [×3], C22×C4 [×2], C2×D4, C2×D4, C2×Q8, C4○D4 [×6], Dic5 [×2], Dic5 [×2], C20 [×2], F5 [×2], D10 [×2], D10, C2×C10, C2×C10 [×4], C4≀C2 [×4], C42⋊C2, C2×M4(2), C2×C4○D4, C5⋊C8 [×2], Dic10 [×2], Dic10, C4×D5 [×4], C2×Dic5, C2×Dic5 [×5], C5⋊D4 [×4], C2×C20, C5×D4 [×2], C5×D4, C2×F5 [×2], C22×D5, C22×C10, C42⋊C22, D5⋊C8, C4.F5 [×2], C4×F5 [×2], C4⋊F5, C22.F5, C22⋊F5, C2×Dic10, C2×C4×D5, D42D5 [×4], D42D5 [×2], C22×Dic5, C2×C5⋊D4, D4×C10, D4⋊F5 [×4], D5⋊M4(2), D10.C23, C2×D42D5, (C2×D4)⋊6F5
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, C22⋊C4 [×4], C22×C4, C2×D4 [×2], F5, C2×C22⋊C4, C2×F5 [×3], C42⋊C22, C22⋊F5 [×2], C22×F5, C2×C22⋊F5, (C2×D4)⋊6F5

Smallest permutation representation of (C2×D4)⋊6F5
On 80 points
Generators in S80
(1 51)(2 52)(3 53)(4 54)(5 55)(6 56)(7 57)(8 58)(9 59)(10 60)(11 46)(12 47)(13 48)(14 49)(15 50)(16 41)(17 42)(18 43)(19 44)(20 45)(21 76)(22 77)(23 78)(24 79)(25 80)(26 71)(27 72)(28 73)(29 74)(30 75)(31 61)(32 62)(33 63)(34 64)(35 65)(36 66)(37 67)(38 68)(39 69)(40 70)
(1 46 6 41)(2 47 7 42)(3 48 8 43)(4 49 9 44)(5 50 10 45)(11 56 16 51)(12 57 17 52)(13 58 18 53)(14 59 19 54)(15 60 20 55)(21 66 26 61)(22 67 27 62)(23 68 28 63)(24 69 29 64)(25 70 30 65)(31 76 36 71)(32 77 37 72)(33 78 38 73)(34 79 39 74)(35 80 40 75)
(1 36)(2 37)(3 38)(4 39)(5 40)(6 31)(7 32)(8 33)(9 34)(10 35)(11 21)(12 22)(13 23)(14 24)(15 25)(16 26)(17 27)(18 28)(19 29)(20 30)(41 71)(42 72)(43 73)(44 74)(45 75)(46 76)(47 77)(48 78)(49 79)(50 80)(51 66)(52 67)(53 68)(54 69)(55 70)(56 61)(57 62)(58 63)(59 64)(60 65)
(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 46)(2 48 5 49)(3 50 4 47)(6 41)(7 43 10 44)(8 45 9 42)(11 56)(12 58 15 59)(13 60 14 57)(16 51)(17 53 20 54)(18 55 19 52)(21 76 26 71)(22 78 30 74)(23 80 29 72)(24 77 28 75)(25 79 27 73)(31 61 36 66)(32 63 40 69)(33 65 39 67)(34 62 38 70)(35 64 37 68)

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

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

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

32 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 4A 4B 4C 4D 4E 4F ··· 4K 5 8A 8B 8C 8D 10A 10B 10C 10D 10E 10F 10G 20A 20B order 1 2 2 2 2 2 2 4 4 4 4 4 4 ··· 4 5 8 8 8 8 10 10 10 10 10 10 10 20 20 size 1 1 2 4 4 10 10 2 2 5 5 10 20 ··· 20 4 20 20 20 20 4 4 4 8 8 8 8 8 8

32 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 4 4 4 4 4 4 8 type + + + + + + + + + + + + + - image C1 C2 C2 C2 C2 C4 C4 C4 D4 D4 D4 F5 C2×F5 C2×F5 C42⋊C22 C22⋊F5 C22⋊F5 (C2×D4)⋊6F5 kernel (C2×D4)⋊6F5 D4⋊F5 D5⋊M4(2) D10.C23 C2×D4⋊2D5 C2×Dic10 D4⋊2D5 D4×C10 C4×D5 C2×Dic5 C22×D5 C2×D4 C2×C4 D4 C5 C4 C22 C1 # reps 1 4 1 1 1 2 4 2 2 1 1 1 1 2 2 2 2 2

Matrix representation of (C2×D4)⋊6F5 in GL8(𝔽41)

 13 0 4 0 0 0 0 0 7 0 32 32 0 0 0 0 40 0 28 0 0 0 0 0 2 9 7 0 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 40
,
 6 0 5 0 0 0 0 0 2 0 39 1 0 0 0 0 9 0 35 0 0 0 0 0 6 40 19 0 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 40
,
 27 8 0 0 0 0 0 0 32 14 0 0 0 0 0 0 14 15 9 23 0 0 0 0 0 14 9 32 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 40
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 40 0 0 0 0 1 0 0 40 0 0 0 0 0 1 0 40 0 0 0 0 0 0 1 40
,
 35 0 36 0 0 0 0 0 36 0 38 9 0 0 0 0 7 0 6 0 0 0 0 0 29 9 22 0 0 0 0 0 0 0 0 0 0 0 40 0 0 0 0 0 40 0 0 0 0 0 0 0 0 0 0 40 0 0 0 0 0 40 0 0

G:=sub<GL(8,GF(41))| [13,7,40,2,0,0,0,0,0,0,0,9,0,0,0,0,4,32,28,7,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40],[6,2,9,6,0,0,0,0,0,0,0,40,0,0,0,0,5,39,35,19,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40],[27,32,14,0,0,0,0,0,8,14,15,14,0,0,0,0,0,0,9,9,0,0,0,0,0,0,23,32,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,40,40,40,40],[35,36,7,29,0,0,0,0,0,0,0,9,0,0,0,0,36,38,6,22,0,0,0,0,0,9,0,0,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,40,0,0,0,0,40,0,0,0,0,0,0,0,0,0,40,0] >;

(C2×D4)⋊6F5 in GAP, Magma, Sage, TeX

(C_2\times D_4)\rtimes_6F_5
% in TeX

G:=Group("(C2xD4):6F5");
// GroupNames label

G:=SmallGroup(320,1107);
// by ID

G=gap.SmallGroup(320,1107);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,232,422,387,136,1684,438,102,6278,1595]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^4=c^2=d^5=e^4=1,a*b=b*a,a*c=c*a,a*d=d*a,e*a*e^-1=a*b^2,c*b*c=e*b*e^-1=b^-1,b*d=d*b,c*d=d*c,e*c*e^-1=a*b*c,e*d*e^-1=d^3>;
// generators/relations

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