C3xC2^2.F5 - GroupNames

G = C₃×C₂².F₅ order 240 = 2⁴·3·5

Direct product of C₃ and C₂².F₅

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C₃×C₂².F₅, C₁₅⋊₉M₄(2), Dic₅.₃C₁₂, C₅⋊C₈⋊₂C₆, C₂.₆(C₆×F₅), (C₂×C₆).₁F₅, (C₂×C₃₀).₄C₄, C₂².(C₃×F₅), C₅⋊₂(C₃×M₄(2)), C₆.₂₀(C₂×F₅), C₃₀.₂₀(C₂×C₄), C₁₀.₆(C₂×C₁₂), (C₂×C₁₀).₂C₁₂, (C₃×Dic₅).₇C₄, Dic₅.₈(C₂×C₆), (C₂×Dic₅).₅C₆, (C₆×Dic₅).₁₂C₂, (C₃×Dic₅).₂₈C₂², (C₃×C₅⋊C₈)⋊₆C₂, SmallGroup(240,116)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₁₀ — C₃×C₂².F₅

Chief series

C₁ — C₅ — C₁₀ — Dic₅ — C₃×Dic₅ — C₃×C₅⋊C₈ — C₃×C₂².F₅

Lower central

C₅ — C₁₀ — C₃×C₂².F₅

Upper central

C₁ — C₆ — C₂×C₆

Generators and relations for C₃×C₂².F₅
G = < a,b,c,d,e | a³=b²=c²=d⁵=1, e⁴=c, ab=ba, ac=ca, ad=da, ae=ea, ebe^-1=bc=cb, bd=db, cd=dc, ce=ec, ede^-1=d³ >

C₁

C₂

₂C₂

C₃

C₅

C₂²

₅C₄

C₆

₂C₆

C₁₀

₂C₁₀

C₁₅

₅C₈

₅C₂×C₄

₅C₈

C₂×C₆

₅C₁₂

Dic₅

C₂×C₁₀

C₃₀

₂C₃₀

₅M₄(2)

₅C₂₄

₅C₂×C₁₂

C₅⋊C₈

C₂×Dic₅

C₅⋊C₈

C₃×Dic₅

C₂×C₃₀

₅C₃×M₄(2)

C₂².F₅

C₃×C₅⋊C₈

C₆×Dic₅

C₃×C₂².F₅

Smallest permutation representation of C₃×C₂².F₅
►On 120 points

Generators in S₁₂₀

(1 100 60)(2 101 61)(3 102 62)(4 103 63)(5 104 64)(6 97 57)(7 98 58)(8 99 59)(9 86 51)(10 87 52)(11 88 53)(12 81 54)(13 82 55)(14 83 56)(15 84 49)(16 85 50)(17 94 119)(18 95 120)(19 96 113)(20 89 114)(21 90 115)(22 91 116)(23 92 117)(24 93 118)(25 45 66)(26 46 67)(27 47 68)(28 48 69)(29 41 70)(30 42 71)(31 43 72)(32 44 65)(33 108 74)(34 109 75)(35 110 76)(36 111 77)(37 112 78)(38 105 79)(39 106 80)(40 107 73)

(2 6)(4 8)(10 14)(12 16)(18 22)(20 24)(25 29)(27 31)(33 37)(35 39)(41 45)(43 47)(50 54)(52 56)(57 61)(59 63)(66 70)(68 72)(74 78)(76 80)(81 85)(83 87)(89 93)(91 95)(97 101)(99 103)(106 110)(108 112)(114 118)(116 120)

(1 5)(2 6)(3 7)(4 8)(9 13)(10 14)(11 15)(12 16)(17 21)(18 22)(19 23)(20 24)(25 29)(26 30)(27 31)(28 32)(33 37)(34 38)(35 39)(36 40)(41 45)(42 46)(43 47)(44 48)(49 53)(50 54)(51 55)(52 56)(57 61)(58 62)(59 63)(60 64)(65 69)(66 70)(67 71)(68 72)(73 77)(74 78)(75 79)(76 80)(81 85)(82 86)(83 87)(84 88)(89 93)(90 94)(91 95)(92 96)(97 101)(98 102)(99 103)(100 104)(105 109)(106 110)(107 111)(108 112)(113 117)(114 118)(115 119)(116 120)

(1 109 46 23 53)(2 24 110 54 47)(3 55 17 48 111)(4 41 56 112 18)(5 105 42 19 49)(6 20 106 50 43)(7 51 21 44 107)(8 45 52 108 22)(9 90 65 73 98)(10 74 91 99 66)(11 100 75 67 92)(12 68 101 93 76)(13 94 69 77 102)(14 78 95 103 70)(15 104 79 71 96)(16 72 97 89 80)(25 87 33 116 59)(26 117 88 60 34)(27 61 118 35 81)(28 36 62 82 119)(29 83 37 120 63)(30 113 84 64 38)(31 57 114 39 85)(32 40 58 86 115)

(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)(81 82 83 84 85 86 87 88)(89 90 91 92 93 94 95 96)(97 98 99 100 101 102 103 104)(105 106 107 108 109 110 111 112)(113 114 115 116 117 118 119 120)

G:=sub<Sym(120)| (1,100,60)(2,101,61)(3,102,62)(4,103,63)(5,104,64)(6,97,57)(7,98,58)(8,99,59)(9,86,51)(10,87,52)(11,88,53)(12,81,54)(13,82,55)(14,83,56)(15,84,49)(16,85,50)(17,94,119)(18,95,120)(19,96,113)(20,89,114)(21,90,115)(22,91,116)(23,92,117)(24,93,118)(25,45,66)(26,46,67)(27,47,68)(28,48,69)(29,41,70)(30,42,71)(31,43,72)(32,44,65)(33,108,74)(34,109,75)(35,110,76)(36,111,77)(37,112,78)(38,105,79)(39,106,80)(40,107,73), (2,6)(4,8)(10,14)(12,16)(18,22)(20,24)(25,29)(27,31)(33,37)(35,39)(41,45)(43,47)(50,54)(52,56)(57,61)(59,63)(66,70)(68,72)(74,78)(76,80)(81,85)(83,87)(89,93)(91,95)(97,101)(99,103)(106,110)(108,112)(114,118)(116,120), (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16)(17,21)(18,22)(19,23)(20,24)(25,29)(26,30)(27,31)(28,32)(33,37)(34,38)(35,39)(36,40)(41,45)(42,46)(43,47)(44,48)(49,53)(50,54)(51,55)(52,56)(57,61)(58,62)(59,63)(60,64)(65,69)(66,70)(67,71)(68,72)(73,77)(74,78)(75,79)(76,80)(81,85)(82,86)(83,87)(84,88)(89,93)(90,94)(91,95)(92,96)(97,101)(98,102)(99,103)(100,104)(105,109)(106,110)(107,111)(108,112)(113,117)(114,118)(115,119)(116,120), (1,109,46,23,53)(2,24,110,54,47)(3,55,17,48,111)(4,41,56,112,18)(5,105,42,19,49)(6,20,106,50,43)(7,51,21,44,107)(8,45,52,108,22)(9,90,65,73,98)(10,74,91,99,66)(11,100,75,67,92)(12,68,101,93,76)(13,94,69,77,102)(14,78,95,103,70)(15,104,79,71,96)(16,72,97,89,80)(25,87,33,116,59)(26,117,88,60,34)(27,61,118,35,81)(28,36,62,82,119)(29,83,37,120,63)(30,113,84,64,38)(31,57,114,39,85)(32,40,58,86,115), (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)(81,82,83,84,85,86,87,88)(89,90,91,92,93,94,95,96)(97,98,99,100,101,102,103,104)(105,106,107,108,109,110,111,112)(113,114,115,116,117,118,119,120)>;

G:=Group( (1,100,60)(2,101,61)(3,102,62)(4,103,63)(5,104,64)(6,97,57)(7,98,58)(8,99,59)(9,86,51)(10,87,52)(11,88,53)(12,81,54)(13,82,55)(14,83,56)(15,84,49)(16,85,50)(17,94,119)(18,95,120)(19,96,113)(20,89,114)(21,90,115)(22,91,116)(23,92,117)(24,93,118)(25,45,66)(26,46,67)(27,47,68)(28,48,69)(29,41,70)(30,42,71)(31,43,72)(32,44,65)(33,108,74)(34,109,75)(35,110,76)(36,111,77)(37,112,78)(38,105,79)(39,106,80)(40,107,73), (2,6)(4,8)(10,14)(12,16)(18,22)(20,24)(25,29)(27,31)(33,37)(35,39)(41,45)(43,47)(50,54)(52,56)(57,61)(59,63)(66,70)(68,72)(74,78)(76,80)(81,85)(83,87)(89,93)(91,95)(97,101)(99,103)(106,110)(108,112)(114,118)(116,120), (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16)(17,21)(18,22)(19,23)(20,24)(25,29)(26,30)(27,31)(28,32)(33,37)(34,38)(35,39)(36,40)(41,45)(42,46)(43,47)(44,48)(49,53)(50,54)(51,55)(52,56)(57,61)(58,62)(59,63)(60,64)(65,69)(66,70)(67,71)(68,72)(73,77)(74,78)(75,79)(76,80)(81,85)(82,86)(83,87)(84,88)(89,93)(90,94)(91,95)(92,96)(97,101)(98,102)(99,103)(100,104)(105,109)(106,110)(107,111)(108,112)(113,117)(114,118)(115,119)(116,120), (1,109,46,23,53)(2,24,110,54,47)(3,55,17,48,111)(4,41,56,112,18)(5,105,42,19,49)(6,20,106,50,43)(7,51,21,44,107)(8,45,52,108,22)(9,90,65,73,98)(10,74,91,99,66)(11,100,75,67,92)(12,68,101,93,76)(13,94,69,77,102)(14,78,95,103,70)(15,104,79,71,96)(16,72,97,89,80)(25,87,33,116,59)(26,117,88,60,34)(27,61,118,35,81)(28,36,62,82,119)(29,83,37,120,63)(30,113,84,64,38)(31,57,114,39,85)(32,40,58,86,115), (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)(81,82,83,84,85,86,87,88)(89,90,91,92,93,94,95,96)(97,98,99,100,101,102,103,104)(105,106,107,108,109,110,111,112)(113,114,115,116,117,118,119,120) );

G=PermutationGroup([[(1,100,60),(2,101,61),(3,102,62),(4,103,63),(5,104,64),(6,97,57),(7,98,58),(8,99,59),(9,86,51),(10,87,52),(11,88,53),(12,81,54),(13,82,55),(14,83,56),(15,84,49),(16,85,50),(17,94,119),(18,95,120),(19,96,113),(20,89,114),(21,90,115),(22,91,116),(23,92,117),(24,93,118),(25,45,66),(26,46,67),(27,47,68),(28,48,69),(29,41,70),(30,42,71),(31,43,72),(32,44,65),(33,108,74),(34,109,75),(35,110,76),(36,111,77),(37,112,78),(38,105,79),(39,106,80),(40,107,73)], [(2,6),(4,8),(10,14),(12,16),(18,22),(20,24),(25,29),(27,31),(33,37),(35,39),(41,45),(43,47),(50,54),(52,56),(57,61),(59,63),(66,70),(68,72),(74,78),(76,80),(81,85),(83,87),(89,93),(91,95),(97,101),(99,103),(106,110),(108,112),(114,118),(116,120)], [(1,5),(2,6),(3,7),(4,8),(9,13),(10,14),(11,15),(12,16),(17,21),(18,22),(19,23),(20,24),(25,29),(26,30),(27,31),(28,32),(33,37),(34,38),(35,39),(36,40),(41,45),(42,46),(43,47),(44,48),(49,53),(50,54),(51,55),(52,56),(57,61),(58,62),(59,63),(60,64),(65,69),(66,70),(67,71),(68,72),(73,77),(74,78),(75,79),(76,80),(81,85),(82,86),(83,87),(84,88),(89,93),(90,94),(91,95),(92,96),(97,101),(98,102),(99,103),(100,104),(105,109),(106,110),(107,111),(108,112),(113,117),(114,118),(115,119),(116,120)], [(1,109,46,23,53),(2,24,110,54,47),(3,55,17,48,111),(4,41,56,112,18),(5,105,42,19,49),(6,20,106,50,43),(7,51,21,44,107),(8,45,52,108,22),(9,90,65,73,98),(10,74,91,99,66),(11,100,75,67,92),(12,68,101,93,76),(13,94,69,77,102),(14,78,95,103,70),(15,104,79,71,96),(16,72,97,89,80),(25,87,33,116,59),(26,117,88,60,34),(27,61,118,35,81),(28,36,62,82,119),(29,83,37,120,63),(30,113,84,64,38),(31,57,114,39,85),(32,40,58,86,115)], [(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),(81,82,83,84,85,86,87,88),(89,90,91,92,93,94,95,96),(97,98,99,100,101,102,103,104),(105,106,107,108,109,110,111,112),(113,114,115,116,117,118,119,120)]])

C₃×C₂².F₅ is a maximal subgroup of Dic₅.D₁₂ Dic₅.₄D₁₂ D₁₅⋊C₈⋊C₂ D₁₅⋊₂M₄(2)

42 conjugacy classes

class	1	2A	2B	3A	3B	4A	4B	4C	5	6A	6B	6C	6D	8A	8B	8C	8D	10A	10B	10C	12A	12B	12C	12D	12E	12F	15A	15B	24A	···	24H	30A	···	30F
order	1	2	2	3	3	4	4	4	5	6	6	6	6	8	8	8	8	10	10	10	12	12	12	12	12	12	15	15	24	···	24	30	···	30
size	1	1	2	1	1	5	5	10	4	1	1	2	2	10	10	10	10	4	4	4	5	5	5	5	10	10	4	4	10	···	10	4	···	4

42 irreducible representations

dim	1	1	1	1	1	1	1	1	1	1	2	2	4	4	4	4	4	4
type	+	+	+										+	+		-
image	C₁	C₂	C₂	C₃	C₄	C₄	C₆	C₆	C₁₂	C₁₂	M₄(2)	C₃×M₄(2)	F₅	C₂×F₅	C₃×F₅	C₂².F₅	C₆×F₅	C₃×C₂².F₅
kernel	C₃×C₂².F₅	C₃×C₅⋊C₈	C₆×Dic₅	C₂².F₅	C₃×Dic₅	C₂×C₃₀	C₅⋊C₈	C₂×Dic₅	Dic₅	C₂×C₁₀	C₁₅	C₅	C₂×C₆	C₆	C₂²	C₃	C₂	C₁
# reps	1	2	1	2	2	2	4	2	4	4	2	4	1	1	2	2	2	4

Matrix representation of C₃×C₂².F₅ ►in GL₆(𝔽₂₄₁)

225	0	0	0	0	0
0	225	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

1	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	1	0
0	0	0	0	0	1

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	1	0
0	0	0	0	0	1

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

0	1	0	0	0	0
64	0	0	0	0	0
0	0	181	52	64	72
0	0	4	124	177	12
0	0	117	64	229	76
0	0	169	128	60	189

G:=sub<GL(6,GF(241))| [225,0,0,0,0,0,0,225,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,240,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[240,0,0,0,0,0,0,240,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,240,240,240,240,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0],[0,64,0,0,0,0,1,0,0,0,0,0,0,0,181,4,117,169,0,0,52,124,64,128,0,0,64,177,229,60,0,0,72,12,76,189] >;

C₃×C₂².F₅ in GAP, Magma, Sage, TeX

C_3\times C_2^2.F_5

% in TeX

G:=Group("C3xC2^2.F5");

// GroupNames label

G:=SmallGroup(240,116);

// by ID

G=gap.SmallGroup(240,116);

# by ID

G:=PCGroup([6,-2,-2,-3,-2,-2,-5,72,313,69,3461,599]);

// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^2=c^2=d^5=1,e^4=c,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,e*b*e^-1=b*c=c*b,b*d=d*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^3>;

// generators/relations

Export

Subgroup lattice of C₃×C₂².F₅ in TeX

0	1	0	0	0	0
64	0	0	0	0	0
0	0	181	52	64	72
0	0	4	124	177	12
0	0	117	64	229	76
0	0	169	128	60	189

0	1	0	0	0	0
64	0	0	0	0	0
0	0	181	52	64	72
0	0	4	124	177	12
0	0	117	64	229	76
0	0	169	128	60	189

G = C3×C22.F5 order 240 = 24·3·5

Direct product of C3 and C22.F5

G = C₃×C₂².F₅ order 240 = 2⁴·3·5

Direct product of C₃ and C₂².F₅

0	1	0	0	0	0
64	0	0	0	0	0
0	0	181	52	64	72
0	0	4	124	177	12
0	0	117	64	229	76
0	0	169	128	60	189