C2xC3^2:2C16 - GroupNames

G = C₂xC₃²:₂C₁₆ order 288 = 2⁵·3²

Direct product of C₂ and C₃²:₂C₁₆

direct product, metabelian, soluble, monomial, A-group

Aliases: C₂xC₃²:₂C₁₆, C₆².₃C₈, (C₃xC₆):₂C₁₆, (C₆xC₁₂).₂C₄, (C₃xC₁₂).₅C₈, C₃²:₆(C₂xC₁₆), C₃²:₄C₈.₉C₄, C₄.₃(C₃²:₂C₈), C₃²:₄C₈.₃₄C₂², C₂².₂(C₃²:₂C₈), (C₃xC₆).₂₂(C₂xC₈), C₄.₁₉(C₂xC₃²:C₄), (C₃xC₁₂).₁₆(C₂xC₄), (C₂xC₄).₉(C₃²:C₄), C₂.₁(C₂xC₃²:₂C₈), (C₂xC₃²:₄C₈).₁₉C₂, SmallGroup(288,420)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃² — C₂xC₃²:₂C₁₆

Chief series

C₁ — C₃² — C₃xC₆ — C₃xC₁₂ — C₃²:₄C₈ — C₃²:₂C₁₆ — C₂xC₃²:₂C₁₆

Lower central

C₃² — C₂xC₃²:₂C₁₆

Upper central

C₁ — C₂xC₄

Generators and relations for C₂xC₃²:₂C₁₆
G = < a,b,c,d | a²=b³=c³=d¹⁶=1, ab=ba, ac=ca, ad=da, dcd^-1=bc=cb, dbd^-1=b^-1c >

Subgroups: 144 in 50 conjugacy classes, 22 normal (16 characteristic)
Quotients: C₁, C₂, C₄, C₂², C₈, C₂xC₄, C₁₆, C₂xC₈, C₂xC₁₆, C₃²:C₄, C₃²:₂C₈, C₂xC₃²:C₄, C₃²:₂C₁₆, C₂xC₃²:₂C₈, C₂xC₃²:₂C₁₆

C₁

C₂

₂C₃

C₄

C₂²

₂C₆

C₃²

C₂xC₄

₉C₈

₂C₁₂

₂C₂xC₆

₂C₁₂

C₃xC₆

₉C₂xC₈

₉C₁₆

₂C₂xC₁₂

₆C₃:C₈

C₃xC₁₂

C₆²

₉C₂xC₁₆

₆C₂xC₃:C₈

C₃²:₄C₈

C₆xC₁₂

C₃²:₄C₈

C₂xC₃²:₄C₈

C₃²:₂C₁₆

C₂xC₃²:₂C₁₆

Smallest permutation representation of C₂xC₃²:₂C₁₆
►On 96 points

Generators in S₉₆

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

(2 48 95)(4 81 34)(6 36 83)(8 85 38)(10 40 87)(12 89 42)(14 44 91)(16 93 46)(17 72 50)(19 52 74)(21 76 54)(23 56 78)(25 80 58)(27 60 66)(29 68 62)(31 64 70)

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

(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)

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

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

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

48 conjugacy classes

class	1	2A	2B	2C	3A	3B	4A	4B	4C	4D	6A	···	6F	8A	···	8H	12A	···	12H	16A	···	16P
order	1	2	2	2	3	3	4	4	4	4	6	···	6	8	···	8	12	···	12	16	···	16
size	1	1	1	1	4	4	1	1	1	1	4	···	4	9	···	9	4	···	4	9	···	9

48 irreducible representations

dim	1	1	1	1	1	1	1	1	4	4	4	4	4
type	+	+	+						+	-	+	-
image	C₁	C₂	C₂	C₄	C₄	C₈	C₈	C₁₆	C₃²:C₄	C₃²:₂C₈	C₂xC₃²:C₄	C₃²:₂C₈	C₃²:₂C₁₆
kernel	C₂xC₃²:₂C₁₆	C₃²:₂C₁₆	C₂xC₃²:₄C₈	C₃²:₄C₈	C₆xC₁₂	C₃xC₁₂	C₆²	C₃xC₆	C₂xC₄	C₄	C₄	C₂²	C₂
# reps	1	2	1	2	2	4	4	16	2	2	2	2	8

Matrix representation of C₂xC₃²:₂C₁₆ ►in GL₅(F₉₇)

1	0	0	0	0
0	96	0	0	0
0	0	96	0	0
0	0	0	96	0
0	0	0	0	96

1	0	0	0	0
0	1	0	0	0
0	0	1	0	0
0	0	0	0	1
0	13	29	96	96

1	0	0	0	0
0	0	96	0	0
0	1	96	0	0
0	70	41	0	1
0	83	70	96	96

85	0	0	0	0
0	0	0	96	1
0	13	29	95	96
0	34	44	41	27
0	51	91	41	27

G:=sub<GL(5,GF(97))| [1,0,0,0,0,0,96,0,0,0,0,0,96,0,0,0,0,0,96,0,0,0,0,0,96],[1,0,0,0,0,0,1,0,0,13,0,0,1,0,29,0,0,0,0,96,0,0,0,1,96],[1,0,0,0,0,0,0,1,70,83,0,96,96,41,70,0,0,0,0,96,0,0,0,1,96],[85,0,0,0,0,0,0,13,34,51,0,0,29,44,91,0,96,95,41,41,0,1,96,27,27] >;

C₂xC₃²:₂C₁₆ in GAP, Magma, Sage, TeX

C_2\times C_3^2\rtimes_2C_{16}

% in TeX

G:=Group("C2xC3^2:2C16");

// GroupNames label

G:=SmallGroup(288,420);

// by ID

G=gap.SmallGroup(288,420);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,28,58,80,9413,691,12550,2372]);

// Polycyclic

G:=Group<a,b,c,d|a^2=b^3=c^3=d^16=1,a*b=b*a,a*c=c*a,a*d=d*a,d*c*d^-1=b*c=c*b,d*b*d^-1=b^-1*c>;

// generators/relations

Export

Subgroup lattice of C₂xC₃²:₂C₁₆ in TeX

G = C2xC32:2C16 order 288 = 25·32

Direct product of C2 and C32:2C16

G = C₂xC₃²:₂C₁₆ order 288 = 2⁵·3²

Direct product of C₂ and C₃²:₂C₁₆