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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, C2, C4, C22, C22, C5, C8, C2×C4, D4, C23, C23, C23, D5, C10, C10, C10, C2×C8, M4(2), C22×C4, C2×D4, C24, C24, Dic5, D10, C2×C10, C2×C10, C4.D4, C2×M4(2), C22×D4, C5⋊C8, C2×Dic5, C2×Dic5, C5⋊D4, C22×D5, C22×D5, C22×C10, C22×C10, C22×C10, C2×C4.D4, C2×C5⋊C8, C22.F5, C22.F5, C22×Dic5, C2×C5⋊D4, C2×C5⋊D4, C23×D5, C23×C10, C23.F5, C2×C22.F5, C22×C5⋊D4, C2×C23.F5
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, F5, C4.D4, C2×C22⋊C4, C2×F5, C2×C4.D4, C22⋊F5, C22×F5, C23.F5, C2×C22⋊F5, C2×C23.F5

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