M4(2).25D10 - GroupNames

G = M₄(2).₂₅D₁₀ order 320 = 2⁶·5

8^th non-split extension by M₄(2) of D₁₀ acting via D₁₀/D₅=C₂

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: M₄(2).₂₅D₁₀, C₈.₁₀(C₄×D₅), C₄₀.₆₉(C₂×C₄), C₈.C₄⋊₃D₅, C₈⋊D₅.₃C₄, (C₂×C₈).₆₈D₁₀, C₄.₂₁₁(D₄×D₅), C₄₀.₆C₄⋊₇C₂, (C₄×D₅).₁₀₆D₄, C₂₀.₃₇₀(C₂×D₄), C₂².₄(Q₈×D₅), (C₂×Dic₅).₄Q₈, D₁₀.₁₈(C₄⋊C₄), (C₂²×D₅).₃Q₈, C₂₀.₅₃D₄⋊₈C₂, (C₂×C₄₀).₄₀C₂², (D₅×M₄(2)).₉C₂, C₅⋊₂(M₄(2).C₄), Dic₅.₂₀(C₄⋊C₄), C₂₀.₁₁₂(C₂²×C₄), (C₂×C₂₀).₃₀₉C₂³, C₄.Dic₅.₁₃C₂², (C₅×M₄(2)).₁₉C₂², C₄.₈₃(C₂×C₄×D₅), C₂.₁₈(D₅×C₄⋊C₄), C₁₀.₄₀(C₂×C₄⋊C₄), C₅⋊₂C₈.₃(C₂×C₄), (C₄×D₅).₈(C₂×C₄), (C₂×C₁₀).₂(C₂×Q₈), (C₅×C₈.C₄)⋊₃C₂, (C₂×C₈⋊D₅).₁C₂, (C₂×C₄×D₅).₄₇C₂², (C₂×C₅⋊₂C₈).₇₇C₂², (C₂×C₄).₄₁₂(C₂²×D₅), SmallGroup(320,520)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂₀ — M₄(2).₂₅D₁₀

Chief series

C₁ — C₅ — C₁₀ — C₂₀ — C₂×C₂₀ — C₂×C₄×D₅ — C₂×C₈⋊D₅ — M₄(2).₂₅D₁₀

Lower central

C₅ — C₁₀ — C₂₀ — M₄(2).₂₅D₁₀

Upper central

C₁ — C₄ — C₂×C₄ — C₈.C₄

Generators and relations for M₄(2).₂₅D₁₀
G = < a,b,c,d | a⁸=b²=1, c¹⁰=a²b, d²=a⁶b, bab=a⁵, cac^-1=a^-1b, dad^-1=a³b, bc=cb, bd=db, dcd^-1=a⁴c⁹ >

Subgroups: 318 in 102 conjugacy classes, 51 normal (27 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₂², C₅, C₈, C₈, C₂×C₄, C₂×C₄, C₂³, D₅, C₁₀, C₁₀, C₂×C₈, C₂×C₈, M₄(2), M₄(2), C₂²×C₄, Dic₅, C₂₀, D₁₀, D₁₀, C₂×C₁₀, C₈.C₄, C₈.C₄, C₂×M₄(2), C₅⋊₂C₈, C₅⋊₂C₈, C₄₀, C₄₀, C₄×D₅, C₂×Dic₅, C₂×C₂₀, C₂²×D₅, M₄(2).C₄, C₈×D₅, C₈⋊D₅, C₈⋊D₅, C₂×C₅⋊₂C₈, C₄.Dic₅, C₂×C₄₀, C₅×M₄(2), C₂×C₄×D₅, C₄₀.₆C₄, C₂₀.₅₃D₄, C₅×C₈.C₄, C₂×C₈⋊D₅, D₅×M₄(2), M₄(2).₂₅D₁₀
Quotients: C₁, C₂, C₄, C₂², C₂×C₄, D₄, Q₈, C₂³, D₅, C₄⋊C₄, C₂²×C₄, C₂×D₄, C₂×Q₈, D₁₀, C₂×C₄⋊C₄, C₄×D₅, C₂²×D₅, M₄(2).C₄, C₂×C₄×D₅, D₄×D₅, Q₈×D₅, D₅×C₄⋊C₄, M₄(2).₂₅D₁₀

Smallest permutation representation of M₄(2).₂₅D₁₀
►On 80 points

Generators in S₈₀

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

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

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

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

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

50 conjugacy classes

class	1	2A	2B	2C	2D	4A	4B	4C	4D	4E	5A	5B	8A	···	8F	8G	···	8L	10A	10B	10C	10D	20A	20B	20C	20D	20E	20F	40A	···	40H	40I	···	40P
order	1	2	2	2	2	4	4	4	4	4	5	5	8	···	8	8	···	8	10	10	10	10	20	20	20	20	20	20	40	···	40	40	···	40
size	1	1	2	10	10	1	1	2	10	10	2	2	4	···	4	20	···	20	2	2	4	4	2	2	2	2	4	4	4	···	4	8	···	8

50 irreducible representations

dim	1	1	1	1	1	1	1	2	2	2	2	2	2	2	4	4	4	4
type	+	+	+	+	+	+		+	-	-	+	+	+			+	-
image	C₁	C₂	C₂	C₂	C₂	C₂	C₄	D₄	Q₈	Q₈	D₅	D₁₀	D₁₀	C₄×D₅	M₄(2).C₄	D₄×D₅	Q₈×D₅	M₄(2).₂₅D₁₀
kernel	M₄(2).₂₅D₁₀	C₄₀.₆C₄	C₂₀.₅₃D₄	C₅×C₈.C₄	C₂×C₈⋊D₅	D₅×M₄(2)	C₈⋊D₅	C₄×D₅	C₂×Dic₅	C₂²×D₅	C₈.C₄	C₂×C₈	M₄(2)	C₈	C₅	C₄	C₂²	C₁
# reps	1	1	2	1	1	2	8	2	1	1	2	2	4	8	2	2	2	8

Matrix representation of M₄(2).₂₅D₁₀ ►in GL₆(𝔽₄₁)

32	0	0	0	0	0
0	32	0	0	0	0
0	0	0	30	0	15
0	0	9	0	12	0
0	0	0	9	0	40
0	0	40	0	17	0

40	0	0	0	0	0
0	40	0	0	0	0
0	0	1	0	0	0
0	0	0	40	0	0
0	0	0	0	1	0
0	0	0	0	0	40

6	6	0	0	0	0
35	1	0	0	0	0
0	0	24	0	34	0
0	0	0	32	0	33
0	0	40	0	17	0
0	0	0	1	0	9

6	6	0	0	0	0
1	35	0	0	0	0
0	0	24	0	11	0
0	0	0	32	0	10
0	0	40	0	17	0
0	0	0	1	0	9

G:=sub<GL(6,GF(41))| [32,0,0,0,0,0,0,32,0,0,0,0,0,0,0,9,0,40,0,0,30,0,9,0,0,0,0,12,0,17,0,0,15,0,40,0],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,40],[6,35,0,0,0,0,6,1,0,0,0,0,0,0,24,0,40,0,0,0,0,32,0,1,0,0,34,0,17,0,0,0,0,33,0,9],[6,1,0,0,0,0,6,35,0,0,0,0,0,0,24,0,40,0,0,0,0,32,0,1,0,0,11,0,17,0,0,0,0,10,0,9] >;

M₄(2).₂₅D₁₀ in GAP, Magma, Sage, TeX

M_4(2)._{25}D_{10}

% in TeX

G:=Group("M4(2).25D10");

// GroupNames label

G:=SmallGroup(320,520);

// by ID

G=gap.SmallGroup(320,520);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,477,120,219,58,136,438,102,12550]);

// Polycyclic

G:=Group<a,b,c,d|a^8=b^2=1,c^10=a^2*b,d^2=a^6*b,b*a*b=a^5,c*a*c^-1=a^-1*b,d*a*d^-1=a^3*b,b*c=c*b,b*d=d*b,d*c*d^-1=a^4*c^9>;

// generators/relations

G = M4(2).25D10 order 320 = 26·5

8th non-split extension by M4(2) of D10 acting via D10/D5=C2

G = M₄(2).₂₅D₁₀ order 320 = 2⁶·5

8^th non-split extension by M₄(2) of D₁₀ acting via D₁₀/D₅=C₂