C5xC3^2:2Q8 - GroupNames

G = C₅×C₃²⋊₂Q₈ order 360 = 2³·3²·5

Direct product of C₅ and C₃²⋊₂Q₈

direct product, metabelian, supersoluble, monomial

Aliases: C₅×C₃²⋊₂Q₈, C₁₅⋊₅Dic₆, C₃₀.₃₉D₆, (C₃×C₁₅)⋊₇Q₈, C₁₀.₁₇S₃², C₃²⋊₂(C₅×Q₈), C₆.₅(S₃×C₁₀), C₃⋊₁(C₅×Dic₆), Dic₃.(C₅×S₃), C₃⋊Dic₃.₂C₁₀, (C₅×Dic₃).₂S₃, (C₃×C₃₀).₃₁C₂², (C₃×Dic₃).₁C₁₀, (Dic₃×C₁₅).₃C₂, C₂.₅(C₅×S₃²), (C₃×C₆).₅(C₂×C₁₀), (C₅×C₃⋊Dic₃).₅C₂, SmallGroup(360,76)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃×C₆ — C₅×C₃²⋊₂Q₈

Chief series

C₁ — C₃ — C₃² — C₃×C₆ — C₃×C₃₀ — Dic₃×C₁₅ — C₅×C₃²⋊₂Q₈

Lower central

C₃² — C₃×C₆ — C₅×C₃²⋊₂Q₈

Upper central

C₁ — C₁₀

Generators and relations for C₅×C₃²⋊₂Q₈
G = < a,b,c,d,e | a⁵=b³=c³=d⁴=1, e²=d², ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd^-1=b^-1, be=eb, cd=dc, ece^-1=c^-1, ede^-1=d^-1 >

Subgroups: 132 in 54 conjugacy classes, 28 normal (12 characteristic)
C₁, C₂, C₃, C₃, C₄, C₅, C₆, C₆, Q₈, C₃², C₁₀, Dic₃, Dic₃, C₁₂, C₁₅, C₁₅, C₃×C₆, C₂₀, Dic₆, C₃₀, C₃₀, C₃×Dic₃, C₃⋊Dic₃, C₅×Q₈, C₃×C₁₅, C₅×Dic₃, C₅×Dic₃, C₆₀, C₃²⋊₂Q₈, C₃×C₃₀, C₅×Dic₆, Dic₃×C₁₅, C₅×C₃⋊Dic₃, C₅×C₃²⋊₂Q₈
Quotients: C₁, C₂, C₂², C₅, S₃, Q₈, C₁₀, D₆, C₂×C₁₀, Dic₆, C₅×S₃, S₃², C₅×Q₈, S₃×C₁₀, C₃²⋊₂Q₈, C₅×Dic₆, C₅×S₃², C₅×C₃²⋊₂Q₈

Smallest permutation representation of C₅×C₃²⋊₂Q₈
►On 120 points

Generators in S₁₂₀

(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)

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

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

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

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

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

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

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

75 conjugacy classes

class	1	2	3A	3B	3C	4A	4B	4C	5A	5B	5C	5D	6A	6B	6C	10A	10B	10C	10D	12A	12B	12C	12D	15A	···	15H	15I	15J	15K	15L	20A	···	20H	20I	20J	20K	20L	30A	···	30H	30I	30J	30K	30L	60A	···	60P
order	1	2	3	3	3	4	4	4	5	5	5	5	6	6	6	10	10	10	10	12	12	12	12	15	···	15	15	15	15	15	20	···	20	20	20	20	20	30	···	30	30	30	30	30	60	···	60
size	1	1	2	2	4	6	6	18	1	1	1	1	2	2	4	1	1	1	1	6	6	6	6	2	···	2	4	4	4	4	6	···	6	18	18	18	18	2	···	2	4	4	4	4	6	···	6

75 irreducible representations

dim	1	1	1	1	1	1	2	2	2	2	2	2	2	2	4	4	4	4
type	+	+	+				+	-	+	-					+	-
image	C₁	C₂	C₂	C₅	C₁₀	C₁₀	S₃	Q₈	D₆	Dic₆	C₅×S₃	C₅×Q₈	S₃×C₁₀	C₅×Dic₆	S₃²	C₃²⋊₂Q₈	C₅×S₃²	C₅×C₃²⋊₂Q₈
kernel	C₅×C₃²⋊₂Q₈	Dic₃×C₁₅	C₅×C₃⋊Dic₃	C₃²⋊₂Q₈	C₃×Dic₃	C₃⋊Dic₃	C₅×Dic₃	C₃×C₁₅	C₃₀	C₁₅	Dic₃	C₃²	C₆	C₃	C₁₀	C₅	C₂	C₁
# reps	1	2	1	4	8	4	2	1	2	4	8	4	8	16	1	1	4	4

Matrix representation of C₅×C₃²⋊₂Q₈ ►in GL₆(𝔽₆₁)

1	0	0	0	0	0
0	1	0	0	0	0
0	0	58	0	0	0
0	0	0	58	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	1	0	0	0
0	0	0	1	0	0
0	0	0	0	0	1
0	0	0	0	60	60

1	0	0	0	0	0
0	1	0	0	0	0
0	0	60	60	0	0
0	0	1	0	0	0
0	0	0	0	1	0
0	0	0	0	0	1

53	33	0	0	0	0
35	8	0	0	0	0
0	0	60	0	0	0
0	0	0	60	0	0
0	0	0	0	1	0
0	0	0	0	60	60

27	3	0	0	0	0
21	34	0	0	0	0
0	0	1	0	0	0
0	0	60	60	0	0
0	0	0	0	1	0
0	0	0	0	0	1

G:=sub<GL(6,GF(61))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,58,0,0,0,0,0,0,58,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,60,0,0,0,0,1,60],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,60,1,0,0,0,0,60,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[53,35,0,0,0,0,33,8,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,1,60,0,0,0,0,0,60],[27,21,0,0,0,0,3,34,0,0,0,0,0,0,1,60,0,0,0,0,0,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

C₅×C₃²⋊₂Q₈ in GAP, Magma, Sage, TeX

C_5\times C_3^2\rtimes_2Q_8

% in TeX

G:=Group("C5xC3^2:2Q8");

// GroupNames label

G:=SmallGroup(360,76);

// by ID

G=gap.SmallGroup(360,76);

# by ID

G:=PCGroup([6,-2,-2,-5,-2,-3,-3,120,265,127,1210,8645]);

// Polycyclic

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

// generators/relations

G = C5×C32⋊2Q8 order 360 = 23·32·5

Direct product of C5 and C32⋊2Q8

G = C₅×C₃²⋊₂Q₈ order 360 = 2³·3²·5

Direct product of C₅ and C₃²⋊₂Q₈