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## G = C2×C23.F5order 320 = 26·5

### Direct product of C2 and C23.F5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C2×C23.F5
 Chief series C1 — C5 — C10 — Dic5 — C2×Dic5 — C22.F5 — C2×C22.F5 — C2×C23.F5
 Lower central C5 — C10 — C2×C10 — C2×C23.F5
 Upper central C1 — C22 — C23 — C24

Generators and relations for C2×C23.F5
G = < a,b,c,d,e,f | a2=b2=c2=d2=e5=1, f4=d, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, fbf-1=bcd, fcf-1=cd=dc, ce=ec, de=ed, df=fd, fef-1=e3 >

Subgroups: 842 in 186 conjugacy classes, 52 normal (20 characteristic)
C1, C2, C2 [×2], C2 [×6], C4 [×4], C22 [×3], C22 [×18], C5, C8 [×4], C2×C4 [×6], D4 [×8], C23, C23 [×2], C23 [×10], D5 [×2], C10, C10 [×2], C10 [×4], C2×C8 [×2], M4(2) [×6], C22×C4, C2×D4 [×8], C24, C24, Dic5 [×4], D10 [×8], C2×C10 [×3], C2×C10 [×10], C4.D4 [×4], C2×M4(2) [×2], C22×D4, C5⋊C8 [×4], C2×Dic5 [×2], C2×Dic5 [×4], C5⋊D4 [×8], C22×D5 [×2], C22×D5 [×4], C22×C10, C22×C10 [×2], C22×C10 [×4], C2×C4.D4, C2×C5⋊C8 [×2], C22.F5 [×4], C22.F5 [×2], C22×Dic5, C2×C5⋊D4 [×4], C2×C5⋊D4 [×4], C23×D5, C23×C10, C23.F5 [×4], C2×C22.F5 [×2], C22×C5⋊D4, C2×C23.F5
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, C22⋊C4 [×4], C22×C4, C2×D4 [×2], F5, C4.D4 [×2], C2×C22⋊C4, C2×F5 [×3], C2×C4.D4, C22⋊F5 [×2], C22×F5, C23.F5 [×2], C2×C22⋊F5, C2×C23.F5

Smallest permutation representation of C2×C23.F5
On 80 points
Generators in S80
(1 28)(2 29)(3 30)(4 31)(5 32)(6 25)(7 26)(8 27)(9 33)(10 34)(11 35)(12 36)(13 37)(14 38)(15 39)(16 40)(17 48)(18 41)(19 42)(20 43)(21 44)(22 45)(23 46)(24 47)(49 59)(50 60)(51 61)(52 62)(53 63)(54 64)(55 57)(56 58)(65 77)(66 78)(67 79)(68 80)(69 73)(70 74)(71 75)(72 76)
(1 7)(2 4)(3 5)(6 8)(9 69)(10 66)(11 67)(12 72)(13 65)(14 70)(15 71)(16 68)(17 59)(18 64)(19 57)(20 62)(21 63)(22 60)(23 61)(24 58)(25 27)(26 28)(29 31)(30 32)(33 73)(34 78)(35 79)(36 76)(37 77)(38 74)(39 75)(40 80)(41 54)(42 55)(43 52)(44 53)(45 50)(46 51)(47 56)(48 49)
(2 6)(4 8)(10 14)(12 16)(18 22)(20 24)(25 29)(27 31)(34 38)(36 40)(41 45)(43 47)(50 54)(52 56)(58 62)(60 64)(66 70)(68 72)(74 78)(76 80)
(1 5)(2 6)(3 7)(4 8)(9 13)(10 14)(11 15)(12 16)(17 21)(18 22)(19 23)(20 24)(25 29)(26 30)(27 31)(28 32)(33 37)(34 38)(35 39)(36 40)(41 45)(42 46)(43 47)(44 48)(49 53)(50 54)(51 55)(52 56)(57 61)(58 62)(59 63)(60 64)(65 69)(66 70)(67 71)(68 72)(73 77)(74 78)(75 79)(76 80)
(1 57 33 75 21)(2 76 58 22 34)(3 23 77 35 59)(4 36 24 60 78)(5 61 37 79 17)(6 80 62 18 38)(7 19 73 39 63)(8 40 20 64 74)(9 71 44 28 55)(10 29 72 56 45)(11 49 30 46 65)(12 47 50 66 31)(13 67 48 32 51)(14 25 68 52 41)(15 53 26 42 69)(16 43 54 70 27)
(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)

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

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

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

38 conjugacy classes

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

38 irreducible representations

 dim 1 1 1 1 1 1 1 2 4 4 4 4 4 type + + + + + + + + + image C1 C2 C2 C2 C4 C4 C4 D4 F5 C4.D4 C2×F5 C22⋊F5 C23.F5 kernel C2×C23.F5 C23.F5 C2×C22.F5 C22×C5⋊D4 C2×C5⋊D4 C23×D5 C23×C10 C2×Dic5 C24 C10 C23 C22 C2 # reps 1 4 2 1 4 2 2 4 1 2 3 4 8

Matrix representation of C2×C23.F5 in GL6(𝔽41)

 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40
,
 0 40 0 0 0 0 40 0 0 0 0 0 0 0 18 40 0 0 0 0 36 23 0 0 0 0 9 0 17 40 0 0 28 32 1 24
,
 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 14 2 40 0 0 0 14 2 0 40
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 40 1 0 0 0 0 5 35 0 0 0 0 13 9 40 34 0 0 23 28 7 7
,
 32 0 0 0 0 0 0 9 0 0 0 0 0 0 0 0 40 1 0 0 22 9 39 34 0 0 4 30 32 0 0 0 25 6 32 0

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[0,40,0,0,0,0,40,0,0,0,0,0,0,0,18,36,9,28,0,0,40,23,0,32,0,0,0,0,17,1,0,0,0,0,40,24],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,14,14,0,0,0,1,2,2,0,0,0,0,40,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,5,13,23,0,0,1,35,9,28,0,0,0,0,40,7,0,0,0,0,34,7],[32,0,0,0,0,0,0,9,0,0,0,0,0,0,0,22,4,25,0,0,0,9,30,6,0,0,40,39,32,32,0,0,1,34,0,0] >;

C2×C23.F5 in GAP, Magma, Sage, TeX

C_2\times C_2^3.F_5
% in TeX

G:=Group("C2xC2^3.F5");
// GroupNames label

G:=SmallGroup(320,1137);
// by ID

G=gap.SmallGroup(320,1137);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,422,297,136,1684,6278,1595]);
// Polycyclic

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

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