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## G = C2×C23.D5order 160 = 25·5

### Direct product of C2 and C23.D5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C2×C23.D5
 Chief series C1 — C5 — C10 — C2×C10 — C2×Dic5 — C22×Dic5 — C2×C23.D5
 Lower central C5 — C10 — C2×C23.D5
 Upper central C1 — C23 — C24

Generators and relations for C2×C23.D5
G = < a,b,c,d,e,f | a2=b2=c2=d2=e5=1, f2=c, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, fbf-1=bd=db, be=eb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=e-1 >

Subgroups: 296 in 132 conjugacy classes, 65 normal (11 characteristic)
C1, C2, C2 [×6], C2 [×4], C4 [×4], C22, C22 [×10], C22 [×12], C5, C2×C4 [×8], C23, C23 [×6], C23 [×4], C10, C10 [×6], C10 [×4], C22⋊C4 [×4], C22×C4 [×2], C24, Dic5 [×4], C2×C10, C2×C10 [×10], C2×C10 [×12], C2×C22⋊C4, C2×Dic5 [×4], C2×Dic5 [×4], C22×C10, C22×C10 [×6], C22×C10 [×4], C23.D5 [×4], C22×Dic5 [×2], C23×C10, C2×C23.D5
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, D5, C22⋊C4 [×4], C22×C4, C2×D4 [×2], Dic5 [×4], D10 [×3], C2×C22⋊C4, C2×Dic5 [×6], C5⋊D4 [×4], C22×D5, C23.D5 [×4], C22×Dic5, C2×C5⋊D4 [×2], C2×C23.D5

Smallest permutation representation of C2×C23.D5
On 80 points
Generators in S80
(1 36)(2 37)(3 38)(4 39)(5 40)(6 31)(7 32)(8 33)(9 34)(10 35)(11 26)(12 27)(13 28)(14 29)(15 30)(16 21)(17 22)(18 23)(19 24)(20 25)(41 76)(42 77)(43 78)(44 79)(45 80)(46 71)(47 72)(48 73)(49 74)(50 75)(51 66)(52 67)(53 68)(54 69)(55 70)(56 61)(57 62)(58 63)(59 64)(60 65)
(41 46)(42 47)(43 48)(44 49)(45 50)(51 56)(52 57)(53 58)(54 59)(55 60)(61 66)(62 67)(63 68)(64 69)(65 70)(71 76)(72 77)(73 78)(74 79)(75 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 6)(2 7)(3 8)(4 9)(5 10)(11 16)(12 17)(13 18)(14 19)(15 20)(21 26)(22 27)(23 28)(24 29)(25 30)(31 36)(32 37)(33 38)(34 39)(35 40)(41 46)(42 47)(43 48)(44 49)(45 50)(51 56)(52 57)(53 58)(54 59)(55 60)(61 66)(62 67)(63 68)(64 69)(65 70)(71 76)(72 77)(73 78)(74 79)(75 80)
(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 53 11 43)(2 52 12 42)(3 51 13 41)(4 55 14 45)(5 54 15 44)(6 58 16 48)(7 57 17 47)(8 56 18 46)(9 60 19 50)(10 59 20 49)(21 73 31 63)(22 72 32 62)(23 71 33 61)(24 75 34 65)(25 74 35 64)(26 78 36 68)(27 77 37 67)(28 76 38 66)(29 80 39 70)(30 79 40 69)

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

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

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

52 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 4A ··· 4H 5A 5B 10A ··· 10AD order 1 2 ··· 2 2 2 2 2 4 ··· 4 5 5 10 ··· 10 size 1 1 ··· 1 2 2 2 2 10 ··· 10 2 2 2 ··· 2

52 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 type + + + + + + - + image C1 C2 C2 C2 C4 D4 D5 Dic5 D10 C5⋊D4 kernel C2×C23.D5 C23.D5 C22×Dic5 C23×C10 C22×C10 C2×C10 C24 C23 C23 C22 # reps 1 4 2 1 8 4 2 8 6 16

Matrix representation of C2×C23.D5 in GL4(𝔽41) generated by

 40 0 0 0 0 40 0 0 0 0 1 0 0 0 0 1
,
 40 0 0 0 0 40 0 0 0 0 1 0 0 0 20 40
,
 40 0 0 0 0 1 0 0 0 0 40 0 0 0 0 40
,
 1 0 0 0 0 1 0 0 0 0 40 0 0 0 0 40
,
 1 0 0 0 0 1 0 0 0 0 10 0 0 0 17 37
,
 32 0 0 0 0 1 0 0 0 0 5 20 0 0 11 36
G:=sub<GL(4,GF(41))| [40,0,0,0,0,40,0,0,0,0,1,0,0,0,0,1],[40,0,0,0,0,40,0,0,0,0,1,20,0,0,0,40],[40,0,0,0,0,1,0,0,0,0,40,0,0,0,0,40],[1,0,0,0,0,1,0,0,0,0,40,0,0,0,0,40],[1,0,0,0,0,1,0,0,0,0,10,17,0,0,0,37],[32,0,0,0,0,1,0,0,0,0,5,11,0,0,20,36] >;

C2×C23.D5 in GAP, Magma, Sage, TeX

C_2\times C_2^3.D_5
% in TeX

G:=Group("C2xC2^3.D5");
// GroupNames label

G:=SmallGroup(160,173);
// by ID

G=gap.SmallGroup(160,173);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-5,48,362,4613]);
// Polycyclic

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

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