C10xC8:C2^2 - GroupNames

G = C₁₀xC₈:C₂² order 320 = 2⁶·5

Direct product of C₁₀ and C₈:C₂²

direct product, metabelian, nilpotent (class 3), monomial, 2-elementary

Aliases: C₁₀xC₈:C₂², C₄₀:₈C₂³, C₂₀.₈₂C₂⁴, C₈:(C₂²xC₁₀), D₈:₃(C₂xC₁₀), (C₁₀xD₈):₂₅C₂, (C₂xD₈):₁₁C₁₀, C₄.₆₆(D₄xC₁₀), (C₂xC₄₀):₂₉C₂², SD₁₆:₁(C₂xC₁₀), (C₂xSD₁₆):₄C₁₀, C₂₀.₃₂₉(C₂xD₄), (C₂xC₂₀).₅₂₅D₄, D₄:₂(C₂²xC₁₀), (C₅xD₈):₁₉C₂², (C₅xD₄):₁₃C₂³, C₄.₅(C₂³xC₁₀), Q₈:₂(C₂²xC₁₀), (C₅xQ₈):₁₂C₂³, C₂³.₅₀(C₅xD₄), (C₁₀xSD₁₆):₁₅C₂, (C₂²xD₄):₁₁C₁₀, (D₄xC₁₀):₆₆C₂², M₄(2):₃(C₂xC₁₀), (C₂xM₄(2)):₃C₁₀, (Q₈xC₁₀):₅₄C₂², C₂².₂₃(D₄xC₁₀), (C₁₀xM₄(2)):₁₃C₂, (C₂xC₂₀).₉₇₅C₂³, (C₅xSD₁₆):₁₇C₂², C₁₀.₂₀₃(C₂²xD₄), (C₂²xC₁₀).₁₇₂D₄, (C₅xM₄(2)):₂₉C₂², (C₂²xC₂₀).₄₆₅C₂², (C₂xC₈):₂(C₂xC₁₀), (D₄xC₂xC₁₀):₂₆C₂, C₂.₂₇(D₄xC₂xC₁₀), C₄oD₄:₄(C₂xC₁₀), (C₂xC₄oD₄):₁₁C₁₀, (C₁₀xC₄oD₄):₂₇C₂, (C₂xD₄):₁₅(C₂xC₁₀), (C₂xQ₈):₁₄(C₂xC₁₀), (C₂xC₄).₁₃₆(C₅xD₄), (C₂xC₁₀).₄₁₉(C₂xD₄), (C₅xC₄oD₄):₂₄C₂², (C₂xC₄).₄₅(C₂²xC₁₀), (C₂²xC₄).₇₆(C₂xC₁₀), SmallGroup(320,1575)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₄ — C₁₀xC₈:C₂²

Chief series

C₁ — C₂ — C₄ — C₂₀ — C₅xD₄ — C₅xD₈ — C₅xC₈:C₂² — C₁₀xC₈:C₂²

Lower central

C₁ — C₂ — C₄ — C₁₀xC₈:C₂²

Upper central

C₁ — C₂xC₁₀ — C₂²xC₂₀ — C₁₀xC₈:C₂²

Generators and relations for C₁₀xC₈:C₂²
G = < a,b,c,d | a¹⁰=b⁸=c²=d²=1, ab=ba, ac=ca, ad=da, cbc=b³, dbd=b⁵, cd=dc >

Subgroups: 530 in 298 conjugacy classes, 162 normal (30 characteristic)
C₁, C₂, C₂, C₂, C₄, C₄, C₄, C₂², C₂², C₂², C₅, C₈, C₂xC₄, C₂xC₄, C₂xC₄, D₄, D₄, Q₈, Q₈, C₂³, C₂³, C₁₀, C₁₀, C₁₀, C₂xC₈, M₄(2), D₈, SD₁₆, C₂²xC₄, C₂²xC₄, C₂xD₄, C₂xD₄, C₂xD₄, C₂xQ₈, C₄oD₄, C₄oD₄, C₂⁴, C₂₀, C₂₀, C₂₀, C₂xC₁₀, C₂xC₁₀, C₂xC₁₀, C₂xM₄(2), C₂xD₈, C₂xSD₁₆, C₈:C₂², C₂²xD₄, C₂xC₄oD₄, C₄₀, C₂xC₂₀, C₂xC₂₀, C₂xC₂₀, C₅xD₄, C₅xD₄, C₅xQ₈, C₅xQ₈, C₂²xC₁₀, C₂²xC₁₀, C₂xC₈:C₂², C₂xC₄₀, C₅xM₄(2), C₅xD₈, C₅xSD₁₆, C₂²xC₂₀, C₂²xC₂₀, D₄xC₁₀, D₄xC₁₀, D₄xC₁₀, Q₈xC₁₀, C₅xC₄oD₄, C₅xC₄oD₄, C₂³xC₁₀, C₁₀xM₄(2), C₁₀xD₈, C₁₀xSD₁₆, C₅xC₈:C₂², D₄xC₂xC₁₀, C₁₀xC₄oD₄, C₁₀xC₈:C₂²
Quotients: C₁, C₂, C₂², C₅, D₄, C₂³, C₁₀, C₂xD₄, C₂⁴, C₂xC₁₀, C₈:C₂², C₂²xD₄, C₅xD₄, C₂²xC₁₀, C₂xC₈:C₂², D₄xC₁₀, C₂³xC₁₀, C₅xC₈:C₂², D₄xC₂xC₁₀, C₁₀xC₈:C₂²

Smallest permutation representation of C₁₀xC₈:C₂²
►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 28 65 79 59 37 50 11)(2 29 66 80 60 38 41 12)(3 30 67 71 51 39 42 13)(4 21 68 72 52 40 43 14)(5 22 69 73 53 31 44 15)(6 23 70 74 54 32 45 16)(7 24 61 75 55 33 46 17)(8 25 62 76 56 34 47 18)(9 26 63 77 57 35 48 19)(10 27 64 78 58 36 49 20)

(11 37)(12 38)(13 39)(14 40)(15 31)(16 32)(17 33)(18 34)(19 35)(20 36)(21 72)(22 73)(23 74)(24 75)(25 76)(26 77)(27 78)(28 79)(29 80)(30 71)(41 66)(42 67)(43 68)(44 69)(45 70)(46 61)(47 62)(48 63)(49 64)(50 65)

(1 59)(2 60)(3 51)(4 52)(5 53)(6 54)(7 55)(8 56)(9 57)(10 58)(41 66)(42 67)(43 68)(44 69)(45 70)(46 61)(47 62)(48 63)(49 64)(50 65)

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

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

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

110 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	···	2K	4A	4B	4C	4D	4E	4F	5A	5B	5C	5D	8A	8B	8C	8D	10A	···	10L	10M	···	10T	10U	···	10AR	20A	···	20P	20Q	···	20X	40A	···	40P
order	1	2	2	2	2	2	2	···	2	4	4	4	4	4	4	5	5	5	5	8	8	8	8	10	···	10	10	···	10	10	···	10	20	···	20	20	···	20	40	···	40
size	1	1	1	1	2	2	4	···	4	2	2	2	2	4	4	1	1	1	1	4	4	4	4	1	···	1	2	···	2	4	···	4	2	···	2	4	···	4	4	···	4

110 irreducible representations

dim	1	1	1	1	1	1	1	1	1	1	1	1	1	1	2	2	2	2	4	4
type	+	+	+	+	+	+	+								+	+			+
image	C₁	C₂	C₂	C₂	C₂	C₂	C₂	C₅	C₁₀	C₁₀	C₁₀	C₁₀	C₁₀	C₁₀	D₄	D₄	C₅xD₄	C₅xD₄	C₈:C₂²	C₅xC₈:C₂²
kernel	C₁₀xC₈:C₂²	C₁₀xM₄(2)	C₁₀xD₈	C₁₀xSD₁₆	C₅xC₈:C₂²	D₄xC₂xC₁₀	C₁₀xC₄oD₄	C₂xC₈:C₂²	C₂xM₄(2)	C₂xD₈	C₂xSD₁₆	C₈:C₂²	C₂²xD₄	C₂xC₄oD₄	C₂xC₂₀	C₂²xC₁₀	C₂xC₄	C₂³	C₁₀	C₂
# reps	1	1	2	2	8	1	1	4	4	8	8	32	4	4	3	1	12	4	2	8

Matrix representation of C₁₀xC₈:C₂² ►in GL₆(F₄₁)

40	0	0	0	0	0
0	40	0	0	0	0
0	0	16	0	0	0
0	0	0	16	0	0
0	0	0	0	16	0
0	0	0	0	0	16

40	2	0	0	0	0
40	1	0	0	0	0
0	0	0	0	1	0
0	0	6	0	0	40
0	0	0	1	0	0
0	0	1	0	6	0

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

1	0	0	0	0	0
0	1	0	0	0	0
0	0	40	0	0	0
0	0	0	40	0	0
0	0	0	0	1	0
0	0	29	0	0	1

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[40,40,0,0,0,0,2,1,0,0,0,0,0,0,0,6,0,1,0,0,0,0,1,0,0,0,1,0,0,6,0,0,0,40,0,0],[1,1,0,0,0,0,0,40,0,0,0,0,0,0,1,0,6,6,0,0,0,40,0,0,0,0,0,0,0,40,0,0,0,0,40,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,29,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

C₁₀xC₈:C₂² in GAP, Magma, Sage, TeX

C_{10}\times C_8\rtimes C_2^2

% in TeX

G:=Group("C10xC8:C2^2");

// GroupNames label

G:=SmallGroup(320,1575);

// by ID

G=gap.SmallGroup(320,1575);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-5,-2,-2,1149,3446,10085,5052,124]);

// Polycyclic

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

// generators/relations

G = C10xC8:C22 order 320 = 26·5

Direct product of C10 and C8:C22

G = C₁₀xC₈:C₂² order 320 = 2⁶·5

Direct product of C₁₀ and C₈:C₂²