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G = C5×2+ 1+4order 160 = 25·5

Direct product of C5 and 2+ 1+4

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

Aliases: C5×2+ 1+4, C20.51C23, C10.19C24, C4○D43C10, D44(C2×C10), (C2×D4)⋊6C10, Q84(C2×C10), (D4×C10)⋊15C2, (C2×C20)⋊9C22, C232(C2×C10), (C5×D4)⋊13C22, (C2×C10).7C23, C4.9(C22×C10), C2.4(C23×C10), (C5×Q8)⋊12C22, (C22×C10)⋊2C22, C22.2(C22×C10), (C5×D4)(C5×D4), (C5×Q8)(C5×Q8), (C2×C4)⋊2(C2×C10), (C5×C4○D4)⋊8C2, SmallGroup(160,232)

Series: Derived Chief Lower central Upper central

C1C2 — C5×2+ 1+4
C1C2C10C2×C10C5×D4D4×C10 — C5×2+ 1+4
C1C2 — C5×2+ 1+4
C1C10 — C5×2+ 1+4

Generators and relations for C5×2+ 1+4
 G = < a,b,c,d,e | a5=b4=c2=e2=1, d2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=b2d >

Subgroups: 220 in 166 conjugacy classes, 136 normal (6 characteristic)
C1, C2, C2 [×9], C4 [×6], C22 [×9], C22 [×6], C5, C2×C4 [×9], D4 [×18], Q8 [×2], C23 [×6], C10, C10 [×9], C2×D4 [×9], C4○D4 [×6], C20 [×6], C2×C10 [×9], C2×C10 [×6], 2+ 1+4, C2×C20 [×9], C5×D4 [×18], C5×Q8 [×2], C22×C10 [×6], D4×C10 [×9], C5×C4○D4 [×6], C5×2+ 1+4
Quotients: C1, C2 [×15], C22 [×35], C5, C23 [×15], C10 [×15], C24, C2×C10 [×35], 2+ 1+4, C22×C10 [×15], C23×C10, C5×2+ 1+4

Smallest permutation representation of C5×2+ 1+4
On 40 points
Generators in S40
(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)
(1 31 11 27)(2 32 12 28)(3 33 13 29)(4 34 14 30)(5 35 15 26)(6 20 40 24)(7 16 36 25)(8 17 37 21)(9 18 38 22)(10 19 39 23)
(1 17)(2 18)(3 19)(4 20)(5 16)(6 34)(7 35)(8 31)(9 32)(10 33)(11 21)(12 22)(13 23)(14 24)(15 25)(26 36)(27 37)(28 38)(29 39)(30 40)
(1 27 11 31)(2 28 12 32)(3 29 13 33)(4 30 14 34)(5 26 15 35)(6 20 40 24)(7 16 36 25)(8 17 37 21)(9 18 38 22)(10 19 39 23)
(1 17)(2 18)(3 19)(4 20)(5 16)(6 30)(7 26)(8 27)(9 28)(10 29)(11 21)(12 22)(13 23)(14 24)(15 25)(31 37)(32 38)(33 39)(34 40)(35 36)

G:=sub<Sym(40)| (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), (1,31,11,27)(2,32,12,28)(3,33,13,29)(4,34,14,30)(5,35,15,26)(6,20,40,24)(7,16,36,25)(8,17,37,21)(9,18,38,22)(10,19,39,23), (1,17)(2,18)(3,19)(4,20)(5,16)(6,34)(7,35)(8,31)(9,32)(10,33)(11,21)(12,22)(13,23)(14,24)(15,25)(26,36)(27,37)(28,38)(29,39)(30,40), (1,27,11,31)(2,28,12,32)(3,29,13,33)(4,30,14,34)(5,26,15,35)(6,20,40,24)(7,16,36,25)(8,17,37,21)(9,18,38,22)(10,19,39,23), (1,17)(2,18)(3,19)(4,20)(5,16)(6,30)(7,26)(8,27)(9,28)(10,29)(11,21)(12,22)(13,23)(14,24)(15,25)(31,37)(32,38)(33,39)(34,40)(35,36)>;

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), (1,31,11,27)(2,32,12,28)(3,33,13,29)(4,34,14,30)(5,35,15,26)(6,20,40,24)(7,16,36,25)(8,17,37,21)(9,18,38,22)(10,19,39,23), (1,17)(2,18)(3,19)(4,20)(5,16)(6,34)(7,35)(8,31)(9,32)(10,33)(11,21)(12,22)(13,23)(14,24)(15,25)(26,36)(27,37)(28,38)(29,39)(30,40), (1,27,11,31)(2,28,12,32)(3,29,13,33)(4,30,14,34)(5,26,15,35)(6,20,40,24)(7,16,36,25)(8,17,37,21)(9,18,38,22)(10,19,39,23), (1,17)(2,18)(3,19)(4,20)(5,16)(6,30)(7,26)(8,27)(9,28)(10,29)(11,21)(12,22)(13,23)(14,24)(15,25)(31,37)(32,38)(33,39)(34,40)(35,36) );

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)], [(1,31,11,27),(2,32,12,28),(3,33,13,29),(4,34,14,30),(5,35,15,26),(6,20,40,24),(7,16,36,25),(8,17,37,21),(9,18,38,22),(10,19,39,23)], [(1,17),(2,18),(3,19),(4,20),(5,16),(6,34),(7,35),(8,31),(9,32),(10,33),(11,21),(12,22),(13,23),(14,24),(15,25),(26,36),(27,37),(28,38),(29,39),(30,40)], [(1,27,11,31),(2,28,12,32),(3,29,13,33),(4,30,14,34),(5,26,15,35),(6,20,40,24),(7,16,36,25),(8,17,37,21),(9,18,38,22),(10,19,39,23)], [(1,17),(2,18),(3,19),(4,20),(5,16),(6,30),(7,26),(8,27),(9,28),(10,29),(11,21),(12,22),(13,23),(14,24),(15,25),(31,37),(32,38),(33,39),(34,40),(35,36)])

C5×2+ 1+4 is a maximal subgroup of   2+ 1+4⋊D5  2+ 1+4.D5  2+ 1+4.2D5  2+ 1+42D5  D20.32C23  D20.33C23  D20.37C23
C5×2+ 1+4 is a maximal quotient of   C5×D42  C5×Q82

85 conjugacy classes

class 1 2A2B···2J4A···4F5A5B5C5D10A10B10C10D10E···10AN20A···20X
order122···24···455551010101010···1020···20
size112···22···2111111112···22···2

85 irreducible representations

dim11111144
type++++
imageC1C2C2C5C10C102+ 1+4C5×2+ 1+4
kernelC5×2+ 1+4D4×C10C5×C4○D42+ 1+4C2×D4C4○D4C5C1
# reps1964362414

Matrix representation of C5×2+ 1+4 in GL4(𝔽41) generated by

37000
03700
00370
00037
,
0010
401402
40000
040140
,
0100
1000
140139
00040
,
00400
401402
1000
140040
,
0100
1000
401402
40101
G:=sub<GL(4,GF(41))| [37,0,0,0,0,37,0,0,0,0,37,0,0,0,0,37],[0,40,40,0,0,1,0,40,1,40,0,1,0,2,0,40],[0,1,1,0,1,0,40,0,0,0,1,0,0,0,39,40],[0,40,1,1,0,1,0,40,40,40,0,0,0,2,0,40],[0,1,40,40,1,0,1,1,0,0,40,0,0,0,2,1] >;

C5×2+ 1+4 in GAP, Magma, Sage, TeX

C_5\times 2_+^{1+4}
% in TeX

G:=Group("C5xES+(2,2)");
// GroupNames label

G:=SmallGroup(160,232);
// by ID

G=gap.SmallGroup(160,232);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-5,-2,985,764,2115]);
// Polycyclic

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

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