C5xD8:C2^2 - GroupNames

G = C₅×D₈⋊C₂² order 320 = 2⁶·5

Direct product of C₅ and D₈⋊C₂²

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

Aliases: C₅×D₈⋊C₂², C₂₀.₈₄C₂⁴, C₄₀.₅₁C₂³, C₄○D₈⋊₃C₁₀, D₈⋊₄(C₂×C₁₀), C₈⋊C₂²⋊₆C₁₀, Q₁₆⋊₄(C₂×C₁₀), C₄.₆₈(D₄×C₁₀), (C₂×C₄₀)⋊₃₀C₂², SD₁₆⋊₃(C₂×C₁₀), (C₂×C₂₀).₅₂₇D₄, C₂₀.₄₇₃(C₂×D₄), (C₅×D₈)⋊₂₀C₂², C₈.C₂²⋊₆C₁₀, C₄.₇(C₂³×C₁₀), C₈.₂(C₂²×C₁₀), C₂³.₂₀(C₅×D₄), (D₄×C₁₀)⋊₆₇C₂², (C₂×M₄(2))⋊₅C₁₀, M₄(2)⋊₅(C₂×C₁₀), (C₅×Q₁₆)⋊₁₈C₂², (Q₈×C₁₀)⋊₅₆C₂², D₄.₄(C₂²×C₁₀), (C₅×D₄).₃₇C₂³, C₂².₂₅(D₄×C₁₀), (C₂²×C₁₀).₃₈D₄, (C₅×Q₈).₃₈C₂³, Q₈.₄(C₂²×C₁₀), (C₁₀×M₄(2))⋊₁₅C₂, (C₂×C₂₀).₆₈₆C₂³, (C₅×SD₁₆)⋊₁₉C₂², C₁₀.₂₀₅(C₂²×D₄), (C₅×M₄(2))⋊₃₁C₂², (C₂²×C₂₀).₄₆₇C₂², (C₂×C₈)⋊₃(C₂×C₁₀), C₂.₂₉(D₄×C₂×C₁₀), C₄○D₄⋊₅(C₂×C₁₀), (C₅×C₄○D₈)⋊₁₀C₂, (C₂×C₄○D₄)⋊₁₂C₁₀, (C₁₀×C₄○D₄)⋊₂₈C₂, (C₂×D₄)⋊₁₆(C₂×C₁₀), (C₅×C₈⋊C₂²)⋊₁₃C₂, (C₂×Q₈)⋊₁₆(C₂×C₁₀), (C₂×C₄).₁₃₈(C₅×D₄), (C₂×C₁₀).₄₂₁(C₂×D₄), (C₅×C₄○D₄)⋊₂₅C₂², (C₅×C₈.C₂²)⋊₁₃C₂, (C₂×C₄).₄₇(C₂²×C₁₀), (C₂²×C₄).₇₈(C₂×C₁₀), SmallGroup(320,1577)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₄ — C₅×D₈⋊C₂²

Chief series

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

Lower central

C₁ — C₂ — C₄ — C₅×D₈⋊C₂²

Upper central

C₁ — C₂₀ — C₂²×C₂₀ — C₅×D₈⋊C₂²

Generators and relations for C₅×D₈⋊C₂²
G = < a,b,c,d,e | a⁵=b⁸=c²=d²=e²=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b^-1, dbd=b⁵, be=eb, dcd=ece=b⁴c, de=ed >

Subgroups: 402 in 262 conjugacy classes, 158 normal (22 characteristic)
C₁, C₂, C₂, C₄, C₄, C₄, C₂², C₂², C₂², C₅, C₈, C₂×C₄, C₂×C₄, C₂×C₄, D₄, D₄, Q₈, Q₈, C₂³, C₂³, C₁₀, C₁₀, C₂×C₈, M₄(2), D₈, SD₁₆, Q₁₆, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×D₄, C₂×Q₈, C₄○D₄, C₄○D₄, C₂₀, C₂₀, C₂₀, C₂×C₁₀, C₂×C₁₀, C₂×C₁₀, C₂×M₄(2), C₄○D₈, C₈⋊C₂², C₈.C₂², C₂×C₄○D₄, C₄₀, C₂×C₂₀, C₂×C₂₀, C₂×C₂₀, C₅×D₄, C₅×D₄, C₅×Q₈, C₅×Q₈, C₂²×C₁₀, C₂²×C₁₀, D₈⋊C₂², C₂×C₄₀, C₅×M₄(2), C₅×D₈, C₅×SD₁₆, C₅×Q₁₆, C₂²×C₂₀, C₂²×C₂₀, D₄×C₁₀, D₄×C₁₀, Q₈×C₁₀, C₅×C₄○D₄, C₅×C₄○D₄, C₁₀×M₄(2), C₅×C₄○D₈, C₅×C₈⋊C₂², C₅×C₈.C₂², C₁₀×C₄○D₄, C₅×D₈⋊C₂²
Quotients: C₁, C₂, C₂², C₅, D₄, C₂³, C₁₀, C₂×D₄, C₂⁴, C₂×C₁₀, C₂²×D₄, C₅×D₄, C₂²×C₁₀, D₈⋊C₂², D₄×C₁₀, C₂³×C₁₀, D₄×C₂×C₁₀, C₅×D₈⋊C₂²

Smallest permutation representation of C₅×D₈⋊C₂²
►On 80 points

Generators in S₈₀

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

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

(2 6)(4 8)(9 13)(11 15)(17 21)(19 23)(25 29)(27 31)(33 37)(35 39)(41 45)(43 47)(50 54)(52 56)(58 62)(60 64)(66 70)(68 72)(74 78)(76 80)

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

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

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

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

110 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	2G	2H	4A	4B	4C	4D	4E	4F	4G	4H	4I	5A	5B	5C	5D	8A	8B	8C	8D	10A	10B	10C	10D	10E	···	10P	10Q	···	10AF	20A	···	20H	20I	···	20T	20U	···	20AJ	40A	···	40P
order	1	2	2	2	2	2	2	2	2	4	4	4	4	4	4	4	4	4	5	5	5	5	8	8	8	8	10	10	10	10	10	···	10	10	···	10	20	···	20	20	···	20	20	···	20	40	···	40
size	1	1	2	2	2	4	4	4	4	1	1	2	2	2	4	4	4	4	1	1	1	1	4	4	4	4	1	1	1	1	2	···	2	4	···	4	1	···	1	2	···	2	4	···	4	4	···	4

110 irreducible representations

dim	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	C₅	C₁₀	C₁₀	C₁₀	C₁₀	C₁₀	D₄	D₄	C₅×D₄	C₅×D₄	D₈⋊C₂²	C₅×D₈⋊C₂²
kernel	C₅×D₈⋊C₂²	C₁₀×M₄(2)	C₅×C₄○D₈	C₅×C₈⋊C₂²	C₅×C₈.C₂²	C₁₀×C₄○D₄	D₈⋊C₂²	C₂×M₄(2)	C₄○D₈	C₈⋊C₂²	C₈.C₂²	C₂×C₄○D₄	C₂×C₂₀	C₂²×C₁₀	C₂×C₄	C₂³	C₅	C₁
# reps	1	1	4	4	4	2	4	4	16	16	16	8	3	1	12	4	2	8

Matrix representation of C₅×D₈⋊C₂² ►in GL₆(𝔽₄₁)

18	0	0	0	0	0
0	18	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

40	1	0	0	0	0
39	1	0	0	0	0
0	0	22	0	1	16
0	0	22	0	0	17
0	0	40	1	0	0
0	0	39	0	0	19

40	0	0	0	0	0
39	1	0	0	0	0
0	0	0	22	1	16
0	0	0	22	0	16
0	0	1	40	0	0
0	0	0	39	0	19

40	0	0	0	0	0
0	40	0	0	0	0
0	0	1	0	0	22
0	0	0	1	0	22
0	0	0	0	40	0
0	0	0	0	0	40

40	0	0	0	0	0
0	40	0	0	0	0
0	0	0	32	7	0
0	0	9	0	7	34
0	0	0	0	9	32
0	0	0	0	18	32

G:=sub<GL(6,GF(41))| [18,0,0,0,0,0,0,18,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[40,39,0,0,0,0,1,1,0,0,0,0,0,0,22,22,40,39,0,0,0,0,1,0,0,0,1,0,0,0,0,0,16,17,0,19],[40,39,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,22,22,40,39,0,0,1,0,0,0,0,0,16,16,0,19],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,22,22,0,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,9,0,0,0,0,32,0,0,0,0,0,7,7,9,18,0,0,0,34,32,32] >;

C₅×D₈⋊C₂² in GAP, Magma, Sage, TeX

C_5\times D_8\rtimes C_2^2

% in TeX

G:=Group("C5xD8:C2^2");

// GroupNames label

G:=SmallGroup(320,1577);

// by ID

G=gap.SmallGroup(320,1577);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-5,-2,-2,1149,3446,584,10085,5052,124]);

// Polycyclic

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

// generators/relations

G = C5×D8⋊C22 order 320 = 26·5

Direct product of C5 and D8⋊C22

G = C₅×D₈⋊C₂² order 320 = 2⁶·5

Direct product of C₅ and D₈⋊C₂²