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G = C5×C24⋊C22  order 320 = 26·5

Direct product of C5 and C24⋊C22

direct product, metabelian, nilpotent (class 2), monomial, 2-elementary

Aliases: C5×C24⋊C22, C10.1702+ (1+4), C24⋊5(C2×C10), C22≀C2⋊8C10, (C4×C20)⋊46C22, C42⋊12(C2×C10), C4.4D4⋊15C10, (C23×C10)⋊5C22, (Q8×C10)⋊31C22, (C2×C10).381C24, (C2×C20).682C23, (D4×C10).223C22, C23.24(C22×C10), C22.55(C23×C10), C2.22(C5×2+ (1+4)), (C22×C10).107C23, (C2×Q8)⋊6(C2×C10), C22⋊C4⋊8(C2×C10), (C5×C22≀C2)⋊18C2, (C2×D4).36(C2×C10), (C5×C4.4D4)⋊35C2, (C5×C22⋊C4)⋊43C22, (C2×C4).41(C22×C10), SmallGroup(320,1563)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C22 — C5×C24⋊C22
Chief series
C1 — C2 — C22 — C2×C10 — C2×C20 — Q8×C10 — C5×C4.4D4 — C5×C24⋊C22
Lower central
C1 — C22 — C5×C24⋊C22
Upper central
C1 — C2×C10 — C5×C24⋊C22

Subgroups: 506 in 260 conjugacy classes, 142 normal (6 characteristic)
C1, C2 [×3], C2 [×6], C4 [×9], C22, C22 [×26], C5, C2×C4 [×9], D4 [×9], Q8 [×3], C23 [×6], C23 [×6], C10 [×3], C10 [×6], C42 [×3], C22⋊C4 [×18], C2×D4 [×9], C2×Q8 [×3], C24 [×2], C20 [×9], C2×C10, C2×C10 [×26], C22≀C2 [×6], C4.4D4 [×9], C2×C20 [×9], C5×D4 [×9], C5×Q8 [×3], C22×C10 [×6], C22×C10 [×6], C24⋊C22, C4×C20 [×3], C5×C22⋊C4 [×18], D4×C10 [×9], Q8×C10 [×3], C23×C10 [×2], C5×C22≀C2 [×6], C5×C4.4D4 [×9], C5×C24⋊C22

Quotients:
C1, C2 [×15], C22 [×35], C5, C23 [×15], C10 [×15], C24, C2×C10 [×35], 2+ (1+4) [×3], C22×C10 [×15], C24⋊C22, C23×C10, C5×2+ (1+4) [×3], C5×C24⋊C22

Generators and relations
 G = < a,b,c,d,e,f,g | a5=b2=c2=d2=e2=f2=g2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, bc=cb, bd=db, fbf=be=eb, gbg=bde, gcg=cd=dc, ce=ec, fcf=cde, de=ed, df=fd, dg=gd, ef=fe, eg=ge, fg=gf >

Smallest permutation representation
►On 80 points
Generators in S80
(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 56)(2 57)(3 58)(4 59)(5 60)(6 36)(7 37)(8 38)(9 39)(10 40)(11 43)(12 44)(13 45)(14 41)(15 42)(16 55)(17 51)(18 52)(19 53)(20 54)(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)(46 77)(47 78)(48 79)(49 80)(50 76)

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

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

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

(1 30)(2 26)(3 27)(4 28)(5 29)(6 11)(7 12)(8 13)(9 14)(10 15)(16 76)(17 77)(18 78)(19 79)(20 80)(21 33)(22 34)(23 35)(24 31)(25 32)(36 55)(37 51)(38 52)(39 53)(40 54)(41 48)(42 49)(43 50)(44 46)(45 47)

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

Matrix representation ►G ⊆ GL8(𝔽41)

10000000
01000000
00100000
00010000
000018000
000001800
000000180
000000018
,
4003900000
004010000
00100000
01100000
00000010
000000040
00001000
000004000
,
139000000
040000000
01010000
01100000
000004000
000040000
00000001
00000010
,
400000000
040000000
004000000
000400000
000040000
000004000
000000400
000000040
,
10000000
01000000
00100000
00010000
000040000
000004000
000000400
000000040
,
10000000
140000000
00100000
4000400000
000040000
000004000
00000010
00000001
,
10000000
140000000
4004000000
00010000
000040000
00000100
000000400
00000001

G:=sub<GL(8,GF(41))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,18],[40,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,39,40,1,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,0],[1,0,0,0,0,0,0,0,39,40,1,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0],[40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40],[1,1,0,40,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,1,40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1] >;
 

95 conjugacy classes

class 1 2A2B2C2D···2I4A···4I5A5B5C5D10A···10L10M···10AJ20A···20AJ
order12222···24···4555510···1010···1020···20
size11114···44···411111···14···44···4

95 irreducible representations

dim11111144
type++++
imageC1C2C2C5C10C102+ (1+4)C5×2+ (1+4)
kernelC5×C24⋊C22C5×C22≀C2C5×C4.4D4C24⋊C22C22≀C2C4.4D4C10C2
# reps16942436312

In GAP, Magma, Sage, TeX 

C_5\times C_2^4\rtimes C_2^2
 
% in TeX
 
G:=Group("C5xC2^4:C2^2");
 
// GroupNames label
 
G:=SmallGroup(320,1563);
 
// by ID
 
G=gap.SmallGroup(320,1563);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-5,-2,-2,560,1149,568,3446,2571,6947,1242]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^5=b^2=c^2=d^2=e^2=f^2=g^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,a*g=g*a,b*c=c*b,b*d=d*b,f*b*f=b*e=e*b,g*b*g=b*d*e,g*c*g=c*d=d*c,c*e=e*c,f*c*f=c*d*e,d*e=e*d,d*f=f*d,d*g=g*d,e*f=f*e,e*g=g*e,f*g=g*f>;
 
// generators/relations
 

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