C5xC2^2:A4 - GroupNames

G = C₅×C₂²⋊A₄ order 240 = 2⁴·3·5

Direct product of C₅ and C₂²⋊A₄

direct product, metabelian, soluble, monomial, A-group

Aliases: C₅×C₂²⋊A₄, C₂⁴⋊₄C₁₅, (C₂×C₁₀)⋊A₄, C₂²⋊(C₅×A₄), (C₂³×C₁₀)⋊₂C₃, SmallGroup(240,204)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂⁴ — C₅×C₂²⋊A₄

Chief series

C₁ — C₂² — C₂⁴ — C₂³×C₁₀ — C₅×C₂²⋊A₄

Lower central

C₂⁴ — C₅×C₂²⋊A₄

Upper central

C₁ — C₅

Generators and relations for C₅×C₂²⋊A₄
G = < a,b,c,d,e,f | a⁵=b²=c²=d²=e²=f³=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, fbf^-1=bc=cb, bd=db, be=eb, cd=dc, ce=ec, fcf^-1=b, fdf^-1=de=ed, fef^-1=d >

Subgroups: 208 in 68 conjugacy classes, 16 normal (6 characteristic)
C₁, C₂, C₃, C₂², C₂², C₅, C₂³, C₁₀, A₄, C₁₅, C₂⁴, C₂×C₁₀, C₂×C₁₀, C₂²×C₁₀, C₂²⋊A₄, C₅×A₄, C₂³×C₁₀, C₅×C₂²⋊A₄
Quotients: C₁, C₃, C₅, A₄, C₁₅, C₂²⋊A₄, C₅×A₄, C₅×C₂²⋊A₄

Smallest permutation representation of C₅×C₂²⋊A₄
►On 60 points

Generators in S₆₀

(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)

(1 21)(2 22)(3 23)(4 24)(5 25)(6 20)(7 16)(8 17)(9 18)(10 19)(11 57)(12 58)(13 59)(14 60)(15 56)(26 34)(27 35)(28 31)(29 32)(30 33)

(6 20)(7 16)(8 17)(9 18)(10 19)(11 57)(12 58)(13 59)(14 60)(15 56)(36 41)(37 42)(38 43)(39 44)(40 45)(46 54)(47 55)(48 51)(49 52)(50 53)

(1 21)(2 22)(3 23)(4 24)(5 25)(6 14)(7 15)(8 11)(9 12)(10 13)(16 56)(17 57)(18 58)(19 59)(20 60)(26 34)(27 35)(28 31)(29 32)(30 33)(36 55)(37 51)(38 52)(39 53)(40 54)(41 47)(42 48)(43 49)(44 50)(45 46)

(1 35)(2 31)(3 32)(4 33)(5 34)(6 20)(7 16)(8 17)(9 18)(10 19)(11 57)(12 58)(13 59)(14 60)(15 56)(21 27)(22 28)(23 29)(24 30)(25 26)(36 47)(37 48)(38 49)(39 50)(40 46)(41 55)(42 51)(43 52)(44 53)(45 54)

(1 56 36)(2 57 37)(3 58 38)(4 59 39)(5 60 40)(6 54 34)(7 55 35)(8 51 31)(9 52 32)(10 53 33)(11 42 22)(12 43 23)(13 44 24)(14 45 25)(15 41 21)(16 47 27)(17 48 28)(18 49 29)(19 50 30)(20 46 26)

G:=sub<Sym(60)| (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), (1,21)(2,22)(3,23)(4,24)(5,25)(6,20)(7,16)(8,17)(9,18)(10,19)(11,57)(12,58)(13,59)(14,60)(15,56)(26,34)(27,35)(28,31)(29,32)(30,33), (6,20)(7,16)(8,17)(9,18)(10,19)(11,57)(12,58)(13,59)(14,60)(15,56)(36,41)(37,42)(38,43)(39,44)(40,45)(46,54)(47,55)(48,51)(49,52)(50,53), (1,21)(2,22)(3,23)(4,24)(5,25)(6,14)(7,15)(8,11)(9,12)(10,13)(16,56)(17,57)(18,58)(19,59)(20,60)(26,34)(27,35)(28,31)(29,32)(30,33)(36,55)(37,51)(38,52)(39,53)(40,54)(41,47)(42,48)(43,49)(44,50)(45,46), (1,35)(2,31)(3,32)(4,33)(5,34)(6,20)(7,16)(8,17)(9,18)(10,19)(11,57)(12,58)(13,59)(14,60)(15,56)(21,27)(22,28)(23,29)(24,30)(25,26)(36,47)(37,48)(38,49)(39,50)(40,46)(41,55)(42,51)(43,52)(44,53)(45,54), (1,56,36)(2,57,37)(3,58,38)(4,59,39)(5,60,40)(6,54,34)(7,55,35)(8,51,31)(9,52,32)(10,53,33)(11,42,22)(12,43,23)(13,44,24)(14,45,25)(15,41,21)(16,47,27)(17,48,28)(18,49,29)(19,50,30)(20,46,26)>;

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), (1,21)(2,22)(3,23)(4,24)(5,25)(6,20)(7,16)(8,17)(9,18)(10,19)(11,57)(12,58)(13,59)(14,60)(15,56)(26,34)(27,35)(28,31)(29,32)(30,33), (6,20)(7,16)(8,17)(9,18)(10,19)(11,57)(12,58)(13,59)(14,60)(15,56)(36,41)(37,42)(38,43)(39,44)(40,45)(46,54)(47,55)(48,51)(49,52)(50,53), (1,21)(2,22)(3,23)(4,24)(5,25)(6,14)(7,15)(8,11)(9,12)(10,13)(16,56)(17,57)(18,58)(19,59)(20,60)(26,34)(27,35)(28,31)(29,32)(30,33)(36,55)(37,51)(38,52)(39,53)(40,54)(41,47)(42,48)(43,49)(44,50)(45,46), (1,35)(2,31)(3,32)(4,33)(5,34)(6,20)(7,16)(8,17)(9,18)(10,19)(11,57)(12,58)(13,59)(14,60)(15,56)(21,27)(22,28)(23,29)(24,30)(25,26)(36,47)(37,48)(38,49)(39,50)(40,46)(41,55)(42,51)(43,52)(44,53)(45,54), (1,56,36)(2,57,37)(3,58,38)(4,59,39)(5,60,40)(6,54,34)(7,55,35)(8,51,31)(9,52,32)(10,53,33)(11,42,22)(12,43,23)(13,44,24)(14,45,25)(15,41,21)(16,47,27)(17,48,28)(18,49,29)(19,50,30)(20,46,26) );

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)], [(1,21),(2,22),(3,23),(4,24),(5,25),(6,20),(7,16),(8,17),(9,18),(10,19),(11,57),(12,58),(13,59),(14,60),(15,56),(26,34),(27,35),(28,31),(29,32),(30,33)], [(6,20),(7,16),(8,17),(9,18),(10,19),(11,57),(12,58),(13,59),(14,60),(15,56),(36,41),(37,42),(38,43),(39,44),(40,45),(46,54),(47,55),(48,51),(49,52),(50,53)], [(1,21),(2,22),(3,23),(4,24),(5,25),(6,14),(7,15),(8,11),(9,12),(10,13),(16,56),(17,57),(18,58),(19,59),(20,60),(26,34),(27,35),(28,31),(29,32),(30,33),(36,55),(37,51),(38,52),(39,53),(40,54),(41,47),(42,48),(43,49),(44,50),(45,46)], [(1,35),(2,31),(3,32),(4,33),(5,34),(6,20),(7,16),(8,17),(9,18),(10,19),(11,57),(12,58),(13,59),(14,60),(15,56),(21,27),(22,28),(23,29),(24,30),(25,26),(36,47),(37,48),(38,49),(39,50),(40,46),(41,55),(42,51),(43,52),(44,53),(45,54)], [(1,56,36),(2,57,37),(3,58,38),(4,59,39),(5,60,40),(6,54,34),(7,55,35),(8,51,31),(9,52,32),(10,53,33),(11,42,22),(12,43,23),(13,44,24),(14,45,25),(15,41,21),(16,47,27),(17,48,28),(18,49,29),(19,50,30),(20,46,26)]])

C₅×C₂²⋊A₄ is a maximal subgroup of (C₂²×D₅)⋊A₄ C₂⁴⋊₄D₁₅

40 conjugacy classes

class	1	2A	···	2E	3A	3B	5A	5B	5C	5D	10A	···	10T	15A	···	15H
order	1	2	···	2	3	3	5	5	5	5	10	···	10	15	···	15
size	1	3	···	3	16	16	1	1	1	1	3	···	3	16	···	16

40 irreducible representations

dim	1	1	1	1	3	3
type	+				+
image	C₁	C₃	C₅	C₁₅	A₄	C₅×A₄
kernel	C₅×C₂²⋊A₄	C₂³×C₁₀	C₂²⋊A₄	C₂⁴	C₂×C₁₀	C₂²
# reps	1	2	4	8	5	20

Matrix representation of C₅×C₂²⋊A₄ ►in GL₆(𝔽₃₁)

4	0	0	0	0	0
0	4	0	0	0	0
0	0	4	0	0	0
0	0	0	8	0	0
0	0	0	0	8	0
0	0	0	0	0	8

1	0	0	0	0	0
0	1	0	0	0	0
0	0	1	0	0	0
0	0	0	30	0	0
0	0	0	0	1	0
0	0	0	0	0	30

1	0	0	0	0	0
0	1	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	0	0	0	30	0
0	0	0	0	0	30

1	0	0	0	0	0
0	30	0	0	0	0
0	0	30	0	0	0
0	0	0	30	0	0
0	0	0	0	1	0
0	0	0	0	0	30

30	0	0	0	0	0
0	30	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	0	0	0	30	0
0	0	0	0	0	30

0	1	0	0	0	0
0	0	1	0	0	0
1	0	0	0	0	0
0	0	0	0	1	0
0	0	0	0	0	1
0	0	0	1	0	0

G:=sub<GL(6,GF(31))| [4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,8,0,0,0,0,0,0,8,0,0,0,0,0,0,8],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,30,0,0,0,0,0,0,1,0,0,0,0,0,0,30],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,30,0,0,0,0,0,0,30],[1,0,0,0,0,0,0,30,0,0,0,0,0,0,30,0,0,0,0,0,0,30,0,0,0,0,0,0,1,0,0,0,0,0,0,30],[30,0,0,0,0,0,0,30,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,30,0,0,0,0,0,0,30],[0,0,1,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1,0] >;

C₅×C₂²⋊A₄ in GAP, Magma, Sage, TeX

C_5\times C_2^2\rtimes A_4

% in TeX

G:=Group("C5xC2^2:A4");

// GroupNames label

G:=SmallGroup(240,204);

// by ID

G=gap.SmallGroup(240,204);

# by ID

G:=PCGroup([6,-3,-5,-2,2,-2,2,542,1083,3604,6485]);

// Polycyclic

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

// generators/relations

G = C5×C22⋊A4 order 240 = 24·3·5

Direct product of C5 and C22⋊A4

G = C₅×C₂²⋊A₄ order 240 = 2⁴·3·5

Direct product of C₅ and C₂²⋊A₄