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## G = (C22×C4)⋊7F5order 320 = 26·5

### 3rd semidirect product of C22×C4 and F5 acting via F5/D5=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — (C22×C4)⋊7F5
 Chief series C1 — C5 — D5 — D10 — C22×D5 — C22×F5 — C2×C22⋊F5 — (C22×C4)⋊7F5
 Lower central C5 — C2×C10 — (C22×C4)⋊7F5
 Upper central C1 — C22 — C22×C4

Generators and relations for (C22×C4)⋊7F5
G = < a,b,c,d,e | a2=b2=c4=d5=e4=1, ab=ba, ece-1=ac=ca, ad=da, ae=ea, bc=cb, bd=db, ebe-1=bc2, cd=dc, ede-1=d3 >

Subgroups: 954 in 218 conjugacy classes, 60 normal (16 characteristic)
C1, C2, C2 [×2], C2 [×8], C4 [×8], C22, C22 [×2], C22 [×20], C5, C2×C4 [×2], C2×C4 [×26], C23, C23 [×10], D5 [×4], D5 [×2], C10, C10 [×2], C10 [×2], C22⋊C4 [×4], C22×C4, C22×C4 [×13], C24, Dic5 [×2], C20 [×2], F5 [×4], D10 [×8], D10 [×10], C2×C10, C2×C10 [×2], C2×C10 [×2], C2.C42 [×4], C2×C22⋊C4 [×2], C23×C4, C4×D5 [×8], C2×Dic5 [×2], C2×Dic5 [×2], C2×C20 [×2], C2×C20 [×2], C2×F5 [×12], C22×D5 [×2], C22×D5 [×4], C22×D5 [×4], C22×C10, C23.34D4, C22⋊F5 [×4], C2×C4×D5 [×4], C2×C4×D5 [×4], C22×Dic5, C22×C20, C22×F5 [×4], C23×D5, D10.3Q8 [×4], C2×C22⋊F5 [×2], D5×C22×C4, (C22×C4)⋊7F5
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, C22⋊C4 [×4], C22×C4, C2×D4 [×2], C4○D4 [×4], F5, C2×C22⋊C4, C42⋊C2 [×2], C22.D4 [×4], C2×F5 [×3], C23.34D4, C22⋊F5 [×2], C22×F5, D10.C23 [×2], C2×C22⋊F5, (C22×C4)⋊7F5

Smallest permutation representation of (C22×C4)⋊7F5
On 80 points
Generators in S80
(1 11)(2 12)(3 13)(4 14)(5 15)(6 16)(7 17)(8 18)(9 19)(10 20)(21 31)(22 32)(23 33)(24 34)(25 35)(26 36)(27 37)(28 38)(29 39)(30 40)(41 51)(42 52)(43 53)(44 54)(45 55)(46 56)(47 57)(48 58)(49 59)(50 60)(61 71)(62 72)(63 73)(64 74)(65 75)(66 76)(67 77)(68 78)(69 79)(70 80)
(1 41)(2 42)(3 43)(4 44)(5 45)(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 26 6 21)(2 27 7 22)(3 28 8 23)(4 29 9 24)(5 30 10 25)(11 36 16 31)(12 37 17 32)(13 38 18 33)(14 39 19 34)(15 40 20 35)(41 66 46 61)(42 67 47 62)(43 68 48 63)(44 69 49 64)(45 70 50 65)(51 76 56 71)(52 77 57 72)(53 78 58 73)(54 79 59 74)(55 80 60 75)
(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 21 16 36)(2 23 20 39)(3 25 19 37)(4 22 18 40)(5 24 17 38)(6 26 11 31)(7 28 15 34)(8 30 14 32)(9 27 13 35)(10 29 12 33)(41 66 56 71)(42 68 60 74)(43 70 59 72)(44 67 58 75)(45 69 57 73)(46 61 51 76)(47 63 55 79)(48 65 54 77)(49 62 53 80)(50 64 52 78)

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

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

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

44 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 2K 4A 4B 4C 4D 4E 4F 4G 4H 4I ··· 4P 5 10A ··· 10G 20A ··· 20H order 1 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 4 ··· 4 5 10 ··· 10 20 ··· 20 size 1 1 1 1 2 2 5 5 5 5 10 10 2 2 2 2 10 10 10 10 20 ··· 20 4 4 ··· 4 4 ··· 4

44 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 4 4 4 4 4 type + + + + + + + + + image C1 C2 C2 C2 C4 C4 C4 D4 C4○D4 F5 C2×F5 C2×F5 C22⋊F5 D10.C23 kernel (C22×C4)⋊7F5 D10.3Q8 C2×C22⋊F5 D5×C22×C4 C2×C4×D5 C22×Dic5 C22×C20 C22×D5 D10 C22×C4 C2×C4 C23 C22 C2 # reps 1 4 2 1 4 2 2 4 8 1 2 1 4 8

Matrix representation of (C22×C4)⋊7F5 in GL8(𝔽41)

 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 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 1
,
 1 18 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 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
,
 32 2 0 0 0 0 0 0 0 9 0 0 0 0 0 0 0 0 40 2 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 1 0 0 0 0 0 0 0 0 1
,
 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
,
 40 23 0 0 0 0 0 0 32 1 0 0 0 0 0 0 0 0 32 18 0 0 0 0 0 0 32 9 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0

G:=sub<GL(8,GF(41))| [40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,18,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40],[32,0,0,0,0,0,0,0,2,9,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,2,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[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],[40,32,0,0,0,0,0,0,23,1,0,0,0,0,0,0,0,0,32,32,0,0,0,0,0,0,18,9,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0] >;

(C22×C4)⋊7F5 in GAP, Magma, Sage, TeX

(C_2^2\times C_4)\rtimes_7F_5
% in TeX

G:=Group("(C2^2xC4):7F5");
// GroupNames label

G:=SmallGroup(320,1102);
// by ID

G=gap.SmallGroup(320,1102);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,477,422,184,6278,1595]);
// Polycyclic

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

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