C10xC8:C2^2 - GroupNames

G = C₁₀×C₈⋊C₂² order 320 = 2⁶·5

Direct product of C₁₀ and C₈⋊C₂²

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

Aliases: C₁₀×C₈⋊C₂², C₄₀⋊₈C₂³, C₂₀.₈₂C₂⁴, C₈⋊(C₂²×C₁₀), D₈⋊₃(C₂×C₁₀), (C₁₀×D₈)⋊₂₅C₂, (C₂×D₈)⋊₁₁C₁₀, C₄.₆₆(D₄×C₁₀), (C₂×C₄₀)⋊₂₉C₂², SD₁₆⋊₁(C₂×C₁₀), (C₂×SD₁₆)⋊₄C₁₀, C₂₀.₃₂₉(C₂×D₄), (C₂×C₂₀).₅₂₅D₄, D₄⋊₂(C₂²×C₁₀), (C₅×D₈)⋊₁₉C₂², (C₅×D₄)⋊₁₃C₂³, C₄.₅(C₂³×C₁₀), Q₈⋊₂(C₂²×C₁₀), (C₅×Q₈)⋊₁₂C₂³, C₂³.₅₀(C₅×D₄), (C₁₀×SD₁₆)⋊₁₅C₂, (C₂²×D₄)⋊₁₁C₁₀, (D₄×C₁₀)⋊₆₆C₂², M₄(2)⋊₃(C₂×C₁₀), (C₂×M₄(2))⋊₃C₁₀, (Q₈×C₁₀)⋊₅₄C₂², C₂².₂₃(D₄×C₁₀), (C₁₀×M₄(2))⋊₁₃C₂, (C₂×C₂₀).₉₇₅C₂³, (C₅×SD₁₆)⋊₁₇C₂², C₁₀.₂₀₃(C₂²×D₄), (C₂²×C₁₀).₁₇₂D₄, (C₅×M₄(2))⋊₂₉C₂², (C₂²×C₂₀).₄₆₅C₂², (C₂×C₈)⋊₂(C₂×C₁₀), (D₄×C₂×C₁₀)⋊₂₆C₂, C₂.₂₇(D₄×C₂×C₁₀), C₄○D₄⋊₄(C₂×C₁₀), (C₂×C₄○D₄)⋊₁₁C₁₀, (C₁₀×C₄○D₄)⋊₂₇C₂, (C₂×D₄)⋊₁₅(C₂×C₁₀), (C₂×Q₈)⋊₁₄(C₂×C₁₀), (C₂×C₄).₁₃₆(C₅×D₄), (C₂×C₁₀).₄₁₉(C₂×D₄), (C₅×C₄○D₄)⋊₂₄C₂², (C₂×C₄).₄₅(C₂²×C₁₀), (C₂²×C₄).₇₆(C₂×C₁₀), SmallGroup(320,1575)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₄ — C₁₀×C₈⋊C₂²

Chief series

C₁ — C₂ — C₄ — C₂₀ — C₅×D₄ — C₅×D₈ — C₅×C₈⋊C₂² — C₁₀×C₈⋊C₂²

Lower central

C₁ — C₂ — C₄ — C₁₀×C₈⋊C₂²

Upper central

C₁ — C₂×C₁₀ — C₂²×C₂₀ — C₁₀×C₈⋊C₂²

Generators and relations for C₁₀×C₈⋊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, C₂×C₄, C₂×C₄, C₂×C₄, D₄, D₄, Q₈, Q₈, C₂³, C₂³, C₁₀, C₁₀, C₁₀, C₂×C₈, M₄(2), D₈, SD₁₆, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×D₄, C₂×D₄, C₂×Q₈, C₄○D₄, C₄○D₄, C₂⁴, C₂₀, C₂₀, C₂₀, C₂×C₁₀, C₂×C₁₀, C₂×C₁₀, C₂×M₄(2), C₂×D₈, C₂×SD₁₆, C₈⋊C₂², C₂²×D₄, C₂×C₄○D₄, C₄₀, C₂×C₂₀, C₂×C₂₀, C₂×C₂₀, C₅×D₄, C₅×D₄, C₅×Q₈, C₅×Q₈, C₂²×C₁₀, C₂²×C₁₀, C₂×C₈⋊C₂², C₂×C₄₀, C₅×M₄(2), C₅×D₈, C₅×SD₁₆, C₂²×C₂₀, C₂²×C₂₀, D₄×C₁₀, D₄×C₁₀, D₄×C₁₀, Q₈×C₁₀, C₅×C₄○D₄, C₅×C₄○D₄, C₂³×C₁₀, C₁₀×M₄(2), C₁₀×D₈, C₁₀×SD₁₆, C₅×C₈⋊C₂², D₄×C₂×C₁₀, C₁₀×C₄○D₄, C₁₀×C₈⋊C₂²
Quotients: C₁, C₂, C₂², C₅, D₄, C₂³, C₁₀, C₂×D₄, C₂⁴, C₂×C₁₀, C₈⋊C₂², C₂²×D₄, C₅×D₄, C₂²×C₁₀, C₂×C₈⋊C₂², D₄×C₁₀, C₂³×C₁₀, C₅×C₈⋊C₂², D₄×C₂×C₁₀, C₁₀×C₈⋊C₂²

Smallest permutation representation of C₁₀×C₈⋊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	C₅×D₄	C₅×D₄	C₈⋊C₂²	C₅×C₈⋊C₂²
kernel	C₁₀×C₈⋊C₂²	C₁₀×M₄(2)	C₁₀×D₈	C₁₀×SD₁₆	C₅×C₈⋊C₂²	D₄×C₂×C₁₀	C₁₀×C₄○D₄	C₂×C₈⋊C₂²	C₂×M₄(2)	C₂×D₈	C₂×SD₁₆	C₈⋊C₂²	C₂²×D₄	C₂×C₄○D₄	C₂×C₂₀	C₂²×C₁₀	C₂×C₄	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 C₁₀×C₈⋊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] >;

C₁₀×C₈⋊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 = C10×C8⋊C22 order 320 = 26·5

Direct product of C10 and C8⋊C22

G = C₁₀×C₈⋊C₂² order 320 = 2⁶·5

Direct product of C₁₀ and C₈⋊C₂²