C5xC5:Q16 - GroupNames

G = C₅xC₅:Q₁₆ order 400 = 2⁴·5²

Direct product of C₅ and C₅:Q₁₆

direct product, metabelian, supersoluble, monomial

Aliases: C₅xC₅:Q₁₆, C₅²:₆Q₁₆, C₂₀.₃₆D₁₀, Dic₁₀.₂C₁₀, Q₈.(C₅xD₅), C₅:₂(C₅xQ₁₆), C₄.₄(D₅xC₁₀), (C₅xQ₈).₆D₅, C₅:₂C₈.₁C₁₀, C₂₀.₄(C₂xC₁₀), C₁₀.₁₀(C₅xD₄), (C₅xC₁₀).₃₁D₄, (C₅xQ₈).₁C₁₀, (Q₈xC₅²).₁C₂, C₁₀.₃₂(C₅:D₄), (C₅xC₂₀).₁₁C₂², (C₅xDic₁₀).₃C₂, C₂.₇(C₅xC₅:D₄), (C₅xC₅:₂C₈).₄C₂, SmallGroup(400,90)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂₀ — C₅xC₅:Q₁₆

Chief series

C₁ — C₅ — C₁₀ — C₂₀ — C₅xC₂₀ — C₅xDic₁₀ — C₅xC₅:Q₁₆

Lower central

C₅ — C₁₀ — C₂₀ — C₅xC₅:Q₁₆

Upper central

C₁ — C₁₀ — C₂₀ — C₅xQ₈

Generators and relations for C₅xC₅:Q₁₆
G = < a,b,c,d | a⁵=b⁵=c⁸=1, d²=c⁴, ab=ba, ac=ca, ad=da, cbc^-1=b^-1, bd=db, dcd^-1=c^-1 >

Subgroups: 108 in 48 conjugacy classes, 22 normal (all characteristic)
Quotients: C₁, C₂, C₂², C₅, D₄, D₅, C₁₀, Q₁₆, D₁₀, C₂xC₁₀, C₅:D₄, C₅xD₄, C₅xD₅, C₅:Q₁₆, C₅xQ₁₆, D₅xC₁₀, C₅xC₅:D₄, C₅xC₅:Q₁₆

C₁

C₂

C₅

₂C₅

C₄

₂C₄

₁₀C₄

C₁₀

₂C₁₀

C₅²

Q₈

₅C₈

₅Q₈

C₂₀

₂C₂₀

₂Dic₅

₂C₂₀

₁₀C₂₀

C₅xC₁₀

₅Q₁₆

Dic₁₀

C₅:₂C₈

C₅xQ₈

₂C₅xQ₈

₅C₄₀

₅C₅xQ₈

C₅xC₂₀

₂C₅xDic₅

₂C₅xC₂₀

C₅:Q₁₆

₅C₅xQ₁₆

C₅xDic₁₀

C₅xC₅:₂C₈

Q₈xC₅²

C₅xC₅:Q₁₆

Smallest permutation representation of C₅xC₅:Q₁₆
►On 80 points

Generators in S₈₀

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

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

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

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

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

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

85 conjugacy classes

class	1	2	4A	4B	4C	5A	5B	5C	5D	5E	···	5N	8A	8B	10A	10B	10C	10D	10E	···	10N	20A	20B	20C	20D	20E	···	20AL	20AM	20AN	20AO	20AP	40A	···	40H
order	1	2	4	4	4	5	5	5	5	5	···	5	8	8	10	10	10	10	10	···	10	20	20	20	20	20	···	20	20	20	20	20	40	···	40
size	1	1	2	4	20	1	1	1	1	2	···	2	10	10	1	1	1	1	2	···	2	2	2	2	2	4	···	4	20	20	20	20	10	···	10

85 irreducible representations

dim	1	1	1	1	1	1	1	1	2	2	2	2	2	2	2	2	2	2	4	4
type	+	+	+	+					+	+	-	+							-
image	C₁	C₂	C₂	C₂	C₅	C₁₀	C₁₀	C₁₀	D₄	D₅	Q₁₆	D₁₀	C₅:D₄	C₅xD₄	C₅xD₅	C₅xQ₁₆	D₅xC₁₀	C₅xC₅:D₄	C₅:Q₁₆	C₅xC₅:Q₁₆
kernel	C₅xC₅:Q₁₆	C₅xC₅:₂C₈	C₅xDic₁₀	Q₈xC₅²	C₅:Q₁₆	C₅:₂C₈	Dic₁₀	C₅xQ₈	C₅xC₁₀	C₅xQ₈	C₅²	C₂₀	C₁₀	C₁₀	Q₈	C₅	C₄	C₂	C₅	C₁
# reps	1	1	1	1	4	4	4	4	1	2	2	2	4	4	8	8	8	16	2	8

Matrix representation of C₅xC₅:Q₁₆ ►in GL₄(F₄₁) generated by

10	0	0	0
0	10	0	0
0	0	1	0
0	0	0	1

10	0	0	0
0	37	0	0
0	0	1	0
0	0	0	1

0	40	0	0
40	0	0	0
0	0	0	12
0	0	17	17

40	0	0	0
0	40	0	0
0	0	34	7
0	0	28	7

G:=sub<GL(4,GF(41))| [10,0,0,0,0,10,0,0,0,0,1,0,0,0,0,1],[10,0,0,0,0,37,0,0,0,0,1,0,0,0,0,1],[0,40,0,0,40,0,0,0,0,0,0,17,0,0,12,17],[40,0,0,0,0,40,0,0,0,0,34,28,0,0,7,7] >;

C₅xC₅:Q₁₆ in GAP, Magma, Sage, TeX

C_5\times C_5\rtimes Q_{16}

% in TeX

G:=Group("C5xC5:Q16");

// GroupNames label

G:=SmallGroup(400,90);

// by ID

G=gap.SmallGroup(400,90);

# by ID

G:=PCGroup([6,-2,-2,-5,-2,-2,-5,240,265,247,1443,729,69,11525]);

// Polycyclic

G:=Group<a,b,c,d|a^5=b^5=c^8=1,d^2=c^4,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=c^-1>;

// generators/relations

Export

Subgroup lattice of C₅xC₅:Q₁₆ in TeX

G = C5xC5:Q16 order 400 = 24·52

Direct product of C5 and C5:Q16

G = C₅xC₅:Q₁₆ order 400 = 2⁴·5²

Direct product of C₅ and C₅:Q₁₆