C10.122ES+(2,2) - GroupNames

G = C₁₀.₁₂₂2₊¹⁺⁴ order 320 = 2⁶·5

31^st non-split extension by C₁₀ of 2₊¹⁺⁴ acting via 2₊¹⁺⁴/C₄○D₄=C₂

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C₁₀.₁₂₂2₊¹⁺⁴, C₄⋊C₄⋊₃₀D₁₀, C₂²⋊C₄⋊₁₇D₁₀, (D₄×Dic₅)⋊₂₈C₂, D₁₀⋊₂Q₈⋊₃₂C₂, C₂³⋊D₁₀.₂C₂, (C₂×D₄).₁₆₆D₁₀, (C₂×C₂₀).₇₄C₂³, C₄⋊Dic₅⋊₃₉C₂², C₂².D₄⋊₇D₅, Dic₅⋊₄D₄⋊₂₁C₂, D₁₀.₃₅(C₄○D₄), (C₂×C₁₀).₂₀₂C₂⁴, (C₄×Dic₅)⋊₅₅C₂², (C₂²×C₄).₂₅₈D₁₀, D₁₀.₁₂D₄⋊₃₁C₂, C₂.₄₃(D₄⋊₈D₁₀), C₂³.D₅⋊₃₁C₂², Dic₅.₅D₄⋊₃₃C₂, C₅⋊₆(C₂².₄₅C₂⁴), (C₂×Dic₁₀)⋊₃₀C₂², (D₄×C₁₀).₁₄₀C₂², C₂².D₂₀⋊₂₁C₂, C₁₀.D₄⋊₂₄C₂², (C₂²×D₅).₈₆C₂³, (C₂³×D₅).₅₉C₂², C₂².₂₂₃(C₂³×D₅), C₂³.₂₀₂(C₂²×D₅), Dic₅.₁₄D₄⋊₃₂C₂, D₁₀⋊C₄.₃₃C₂², C₂².₁₉(D₄⋊₂D₅), C₂³.₂₁D₁₀⋊₁₂C₂, (C₂²×C₁₀).₂₂₂C₂³, (C₂²×C₂₀).₁₁₄C₂², (C₂×Dic₅).₂₅₅C₂³, (C₂²×Dic₅)⋊₂₆C₂², C₂.₆₄(D₅×C₄○D₄), C₄⋊C₄⋊₇D₅⋊₃₂C₂, C₄⋊C₄⋊D₅⋊₂₈C₂, (C₅×C₄⋊C₄)⋊₂₈C₂², (D₅×C₂²⋊C₄)⋊₁₄C₂, C₁₀.₁₇₆(C₂×C₄○D₄), C₂.₅₃(C₂×D₄⋊₂D₅), (C₂×D₁₀⋊C₄)⋊₂₄C₂, (C₂×C₄×D₅).₁₂₀C₂², (C₂×C₁₀).₄₇(C₄○D₄), (C₅×C₂²⋊C₄)⋊₂₄C₂², (C₂×C₄).₂₉₇(C₂²×D₅), (C₂×C₅⋊D₄).₄₇C₂², (C₅×C₂².D₄)⋊₁₀C₂, SmallGroup(320,1330)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂×C₁₀ — C₁₀.₁₂₂2₊¹⁺⁴

Chief series

C₁ — C₅ — C₁₀ — C₂×C₁₀ — C₂²×D₅ — C₂³×D₅ — C₂×D₁₀⋊C₄ — C₁₀.₁₂₂2₊¹⁺⁴

Lower central

C₅ — C₂×C₁₀ — C₁₀.₁₂₂2₊¹⁺⁴

Upper central

C₁ — C₂² — C₂².D₄

Generators and relations for C₁₀.₁₂₂2₊¹⁺⁴
G = < a,b,c,d,e | a¹⁰=b⁴=c²=1, d²=b², e²=a⁵, bab^-1=eae^-1=a^-1, ac=ca, ad=da, cbc=b^-1, dbd^-1=a⁵b, be=eb, dcd^-1=a⁵c, ce=ec, ede^-1=a⁵b²d >

Subgroups: 950 in 248 conjugacy classes, 97 normal (91 characteristic)
C₁, C₂, C₂, C₄, C₂², C₂², C₂², C₅, C₂×C₄, C₂×C₄, D₄, Q₈, C₂³, C₂³, D₅, C₁₀, C₁₀, C₄², C₂²⋊C₄, C₂²⋊C₄, C₄⋊C₄, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×D₄, C₂×Q₈, C₂⁴, Dic₅, C₂₀, D₁₀, D₁₀, C₂×C₁₀, C₂×C₁₀, C₂×C₁₀, C₂×C₂²⋊C₄, C₄²⋊C₂, C₄×D₄, C₂²≀C₂, C₂²⋊Q₈, C₂².D₄, C₂².D₄, C₄.₄D₄, C₄²⋊₂C₂, Dic₁₀, C₄×D₅, C₂×Dic₅, C₂×Dic₅, C₅⋊D₄, C₂×C₂₀, C₂×C₂₀, C₅×D₄, C₂²×D₅, C₂²×D₅, C₂²×C₁₀, C₂².₄₅C₂⁴, C₄×Dic₅, C₁₀.D₄, C₄⋊Dic₅, D₁₀⋊C₄, C₂³.D₅, C₅×C₂²⋊C₄, C₅×C₄⋊C₄, C₂×Dic₁₀, C₂×C₄×D₅, C₂²×Dic₅, C₂×C₅⋊D₄, C₂²×C₂₀, D₄×C₁₀, C₂³×D₅, Dic₅.₁₄D₄, D₅×C₂²⋊C₄, Dic₅⋊₄D₄, D₁₀.₁₂D₄, Dic₅.₅D₄, C₂².D₂₀, C₄⋊C₄⋊₇D₅, D₁₀⋊₂Q₈, C₄⋊C₄⋊D₅, C₂³.₂₁D₁₀, C₂×D₁₀⋊C₄, D₄×Dic₅, C₂³⋊D₁₀, C₅×C₂².D₄, C₁₀.₁₂₂2₊¹⁺⁴
Quotients: C₁, C₂, C₂², C₂³, D₅, C₄○D₄, C₂⁴, D₁₀, C₂×C₄○D₄, 2₊¹⁺⁴, C₂²×D₅, C₂².₄₅C₂⁴, D₄⋊₂D₅, C₂³×D₅, C₂×D₄⋊₂D₅, D₅×C₄○D₄, D₄⋊₈D₁₀, C₁₀.₁₂₂2₊¹⁺⁴

Smallest permutation representation of C₁₀.₁₂₂2₊¹⁺⁴
►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 73 27 16)(2 72 28 15)(3 71 29 14)(4 80 30 13)(5 79 21 12)(6 78 22 11)(7 77 23 20)(8 76 24 19)(9 75 25 18)(10 74 26 17)(31 62 48 55)(32 61 49 54)(33 70 50 53)(34 69 41 52)(35 68 42 51)(36 67 43 60)(37 66 44 59)(38 65 45 58)(39 64 46 57)(40 63 47 56)

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

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

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

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

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

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

53 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	2G	2H	2I	4A	4B	4C	4D	4E	4F	4G	···	4L	4M	4N	4O	5A	5B	10A	···	10F	10G	10H	10I	10J	10K	10L	20A	···	20H	20I	···	20N
order	1	2	2	2	2	2	2	2	2	2	4	4	4	4	4	4	4	···	4	4	4	4	5	5	10	···	10	10	10	10	10	10	10	20	···	20	20	···	20
size	1	1	1	1	2	2	4	10	10	20	2	2	4	4	4	4	10	···	10	20	20	20	2	2	2	···	2	4	4	4	4	8	8	4	···	4	8	···	8

53 irreducible representations

dim	1	1	1	1	1	1	1	1	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	C₂	C₂	C₂	C₂	C₂	C₂	C₂	C₂	C₂	D₅	C₄○D₄	C₄○D₄	D₁₀	D₁₀	D₁₀	D₁₀	2₊¹⁺⁴	D₄⋊₂D₅	D₅×C₄○D₄	D₄⋊₈D₁₀
kernel	C₁₀.₁₂₂2₊¹⁺⁴	Dic₅.₁₄D₄	D₅×C₂²⋊C₄	Dic₅⋊₄D₄	D₁₀.₁₂D₄	Dic₅.₅D₄	C₂².D₂₀	C₄⋊C₄⋊₇D₅	D₁₀⋊₂Q₈	C₄⋊C₄⋊D₅	C₂³.₂₁D₁₀	C₂×D₁₀⋊C₄	D₄×Dic₅	C₂³⋊D₁₀	C₅×C₂².D₄	C₂².D₄	D₁₀	C₂×C₁₀	C₂²⋊C₄	C₄⋊C₄	C₂²×C₄	C₂×D₄	C₁₀	C₂²	C₂	C₂
# reps	1	1	1	1	1	1	1	1	1	2	1	1	1	1	1	2	4	4	6	4	2	2	1	4	4	4

Matrix representation of C₁₀.₁₂₂2₊¹⁺⁴ ►in GL₆(𝔽₄₁)

40	0	0	0	0	0
0	40	0	0	0	0
0	0	35	35	0	0
0	0	6	40	0	0
0	0	0	0	40	0
0	0	0	0	0	40

0	9	0	0	0	0
32	0	0	0	0	0
0	0	40	0	0	0
0	0	35	1	0	0
0	0	0	0	1	2
0	0	0	0	40	40

0	32	0	0	0	0
9	0	0	0	0	0
0	0	40	0	0	0
0	0	0	40	0	0
0	0	0	0	1	2
0	0	0	0	0	40

0	1	0	0	0	0
1	0	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	0	0	0	9	0
0	0	0	0	32	32

0	1	0	0	0	0
40	0	0	0	0	0
0	0	1	0	0	0
0	0	6	40	0	0
0	0	0	0	9	0
0	0	0	0	0	9

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,35,6,0,0,0,0,35,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[0,32,0,0,0,0,9,0,0,0,0,0,0,0,40,35,0,0,0,0,0,1,0,0,0,0,0,0,1,40,0,0,0,0,2,40],[0,9,0,0,0,0,32,0,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,2,40],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,9,32,0,0,0,0,0,32],[0,40,0,0,0,0,1,0,0,0,0,0,0,0,1,6,0,0,0,0,0,40,0,0,0,0,0,0,9,0,0,0,0,0,0,9] >;

C₁₀.₁₂₂2₊¹⁺⁴ in GAP, Magma, Sage, TeX

C_{10}._{122}2_+^{1+4}

% in TeX

G:=Group("C10.122ES+(2,2)");

// GroupNames label

G:=SmallGroup(320,1330);

// by ID

G=gap.SmallGroup(320,1330);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,758,100,346,297,136,12550]);

// Polycyclic

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

// generators/relations

G = C10.1222+ 1+4 order 320 = 26·5

31st non-split extension by C10 of 2+ 1+4 acting via 2+ 1+4/C4○D4=C2

G = C₁₀.₁₂₂2₊¹⁺⁴ order 320 = 2⁶·5

31^st non-split extension by C₁₀ of 2₊¹⁺⁴ acting via 2₊¹⁺⁴/C₄○D₄=C₂