C5xC2^3.C8 - GroupNames

G = C₅×C₂³.C₈ order 320 = 2⁶·5

Direct product of C₅ and C₂³.C₈

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

Aliases: C₅×C₂³.C₈, C₂³.C₄₀, C₄₀.₁₀₄D₄, M₅(2)⋊₃C₁₀, C₂₀.₅₇M₄(2), (C₂×C₄).C₄₀, (C₂×C₈).₃C₂₀, C₈.₂₄(C₅×D₄), (C₂×C₄₀).₃₀C₄, (C₂×C₂₀).₁₁C₈, (C₂²×C₄).₄C₂₀, C₂².₄(C₂×C₄₀), (C₂²×C₁₀).₁C₈, C₄.₇(C₅×M₄(2)), (C₅×M₅(2))⋊₁₁C₂, (C₂²×C₂₀).₃₇C₄, C₁₀.₄₁(C₂²⋊C₈), (C₂×C₄₀).₃₀₉C₂², C₂₀.₁₅₉(C₂²⋊C₄), (C₁₀×M₄(2)).₂₂C₂, (C₂×M₄(2)).₁₀C₁₀, C₂.₇(C₅×C₂²⋊C₈), (C₂×C₄).₆₇(C₂×C₂₀), (C₂×C₈).₄₆(C₂×C₁₀), (C₂×C₁₀).₅₀(C₂×C₈), C₄.₂₉(C₅×C₂²⋊C₄), (C₂×C₂₀).₅₀₁(C₂×C₄), SmallGroup(320,154)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂² — C₅×C₂³.C₈

Chief series

C₁ — C₂ — C₄ — C₈ — C₂×C₈ — C₂×C₄₀ — C₅×M₅(2) — C₅×C₂³.C₈

Lower central

C₁ — C₂ — C₂² — C₅×C₂³.C₈

Upper central

C₁ — C₂₀ — C₂×C₄₀ — C₅×C₂³.C₈

Generators and relations for C₅×C₂³.C₈
G = < a,b,c,d,e | a⁵=b²=c²=d²=1, e⁸=d, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, ebe^-1=bcd, ece^-1=cd=dc, de=ed >

Subgroups: 90 in 58 conjugacy classes, 34 normal (26 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₂², C₅, C₈, C₈, C₂×C₄, C₂×C₄, C₂³, C₁₀, C₁₀, C₁₆, C₂×C₈, M₄(2), C₂²×C₄, C₂₀, C₂₀, C₂×C₁₀, C₂×C₁₀, M₅(2), C₂×M₄(2), C₄₀, C₄₀, C₂×C₂₀, C₂×C₂₀, C₂²×C₁₀, C₂³.C₈, C₈₀, C₂×C₄₀, C₅×M₄(2), C₂²×C₂₀, C₅×M₅(2), C₁₀×M₄(2), C₅×C₂³.C₈
Quotients: C₁, C₂, C₄, C₂², C₅, C₈, C₂×C₄, D₄, C₁₀, C₂²⋊C₄, C₂×C₈, M₄(2), C₂₀, C₂×C₁₀, C₂²⋊C₈, C₄₀, C₂×C₂₀, C₅×D₄, C₂³.C₈, C₅×C₂²⋊C₄, C₂×C₄₀, C₅×M₄(2), C₅×C₂²⋊C₈, C₅×C₂³.C₈

Smallest permutation representation of C₅×C₂³.C₈
►On 80 points

Generators in S₈₀

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

(2 10)(3 11)(6 14)(7 15)(18 26)(19 27)(22 30)(23 31)(34 42)(35 43)(38 46)(39 47)(49 57)(52 60)(53 61)(56 64)(65 73)(68 76)(69 77)(72 80)

(2 10)(4 12)(6 14)(8 16)(18 26)(20 28)(22 30)(24 32)(34 42)(36 44)(38 46)(40 48)(50 58)(52 60)(54 62)(56 64)(66 74)(68 76)(70 78)(72 80)

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

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

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

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

110 conjugacy classes

class	1	2A	2B	2C	4A	4B	4C	4D	5A	5B	5C	5D	8A	8B	8C	8D	8E	8F	10A	10B	10C	10D	10E	10F	10G	10H	10I	10J	10K	10L	16A	···	16H	20A	···	20H	20I	20J	20K	20L	20M	20N	20O	20P	40A	···	40P	40Q	···	40X	80A	···	80AF
order	1	2	2	2	4	4	4	4	5	5	5	5	8	8	8	8	8	8	10	10	10	10	10	10	10	10	10	10	10	10	16	···	16	20	···	20	20	20	20	20	20	20	20	20	40	···	40	40	···	40	80	···	80
size	1	1	2	4	1	1	2	4	1	1	1	1	2	2	2	2	4	4	1	1	1	1	2	2	2	2	4	4	4	4	4	···	4	1	···	1	2	2	2	2	4	4	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	C₄	C₄	C₅	C₈	C₈	C₁₀	C₁₀	C₂₀	C₂₀	C₄₀	C₄₀	D₄	M₄(2)	C₅×D₄	C₅×M₄(2)	C₂³.C₈	C₅×C₂³.C₈
kernel	C₅×C₂³.C₈	C₅×M₅(2)	C₁₀×M₄(2)	C₂×C₄₀	C₂²×C₂₀	C₂³.C₈	C₂×C₂₀	C₂²×C₁₀	M₅(2)	C₂×M₄(2)	C₂×C₈	C₂²×C₄	C₂×C₄	C₂³	C₄₀	C₂₀	C₈	C₄	C₅	C₁
# reps	1	2	1	2	2	4	4	4	8	4	8	8	16	16	2	2	8	8	2	8

Matrix representation of C₅×C₂³.C₈ ►in GL₆(𝔽₂₄₁)

98	0	0	0	0	0
0	98	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	0	0	0	1	0
0	0	0	0	0	1

0	240	0	0	0	0
240	0	0	0	0	0
0	0	1	0	0	0
0	0	0	240	0	0
0	0	0	0	1	0
0	0	0	0	0	240

240	0	0	0	0	0
0	240	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	0	0	0	240	0
0	0	0	0	0	240

1	0	0	0	0	0
0	1	0	0	0	0
0	0	240	0	0	0
0	0	0	240	0	0
0	0	0	0	240	0
0	0	0	0	0	240

0	240	0	0	0	0
1	0	0	0	0	0
0	0	0	0	1	0
0	0	0	0	0	1
0	0	0	1	0	0
0	0	177	0	0	0

G:=sub<GL(6,GF(241))| [98,0,0,0,0,0,0,98,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,240,0,0,0,0,240,0,0,0,0,0,0,0,1,0,0,0,0,0,0,240,0,0,0,0,0,0,1,0,0,0,0,0,0,240],[240,0,0,0,0,0,0,240,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,240,0,0,0,0,0,0,240],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,240],[0,1,0,0,0,0,240,0,0,0,0,0,0,0,0,0,0,177,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1,0,0] >;

C₅×C₂³.C₈ in GAP, Magma, Sage, TeX

C_5\times C_2^3.C_8

% in TeX

G:=Group("C5xC2^3.C8");

// GroupNames label

G:=SmallGroup(320,154);

// by ID

G=gap.SmallGroup(320,154);

# by ID

G:=PCGroup([7,-2,-2,-5,-2,-2,-2,-2,280,309,5043,3511,102,124]);

// Polycyclic

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

// generators/relations

G = C5×C23.C8 order 320 = 26·5

Direct product of C5 and C23.C8

G = C₅×C₂³.C₈ order 320 = 2⁶·5

Direct product of C₅ and C₂³.C₈