C3xC4oD16 - GroupNames

G = C₃xC₄oD₁₆ order 192 = 2⁶·3

Direct product of C₃ and C₄oD₁₆

direct product, metabelian, nilpotent (class 4), monomial, 2-elementary

Aliases: C₃xC₄oD₁₆, D₁₆:₃C₆, Q₃₂:₃C₆, SD₃₂:₃C₆, C₁₂.₇₁D₈, C₂₄.₇₆D₄, C₄₈.₂₃C₂², C₂₄.₆₆C₂³, (C₂xC₁₆):₆C₆, C₄oD₈:₁C₆, (C₂xC₄₈):₁₃C₂, (C₃xD₁₆):₇C₂, C₁₆.₆(C₂xC₆), (C₃xQ₃₂):₇C₂, D₈.₂(C₂xC₆), C₄.₂₀(C₃xD₈), (C₂xC₆).₁₂D₈, C₄.₁₀(C₆xD₄), C₆.₈₇(C₂xD₈), C₂.₁₅(C₆xD₈), C₈.₁₃(C₃xD₄), (C₃xSD₃₂):₇C₂, C₈.₆(C₂²xC₆), Q₁₆.₂(C₂xC₆), C₂².₁(C₃xD₈), C₁₂.₃₁₇(C₂xD₄), (C₂xC₁₂).₄₂₉D₄, (C₃xD₈).₁₂C₂², (C₂xC₂₄).₄₁₂C₂², (C₃xQ₁₆).₁₄C₂², (C₃xC₄oD₈):₈C₂, (C₂xC₈).₉₁(C₂xC₆), (C₂xC₄).₈₅(C₃xD₄), SmallGroup(192,941)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₈ — C₃xC₄oD₁₆

Chief series

C₁ — C₂ — C₄ — C₈ — C₂₄ — C₃xD₈ — C₃xD₁₆ — C₃xC₄oD₁₆

Lower central

C₁ — C₂ — C₄ — C₈ — C₃xC₄oD₁₆

Upper central

C₁ — C₁₂ — C₂xC₁₂ — C₂xC₂₄ — C₃xC₄oD₁₆

Generators and relations for C₃xC₄oD₁₆
G = < a,b,c,d | a³=b⁴=d²=1, c⁸=b², ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=b²c⁷ >

Subgroups: 178 in 84 conjugacy classes, 46 normal (30 characteristic)
C₁, C₂, C₂, C₃, C₄, C₄, C₂², C₂², C₆, C₆, C₈, C₂xC₄, C₂xC₄, D₄, Q₈, C₁₂, C₁₂, C₂xC₆, C₂xC₆, C₁₆, C₂xC₈, D₈, SD₁₆, Q₁₆, C₄oD₄, C₂₄, C₂xC₁₂, C₂xC₁₂, C₃xD₄, C₃xQ₈, C₂xC₁₆, D₁₆, SD₃₂, Q₃₂, C₄oD₈, C₄₈, C₂xC₂₄, C₃xD₈, C₃xSD₁₆, C₃xQ₁₆, C₃xC₄oD₄, C₄oD₁₆, C₂xC₄₈, C₃xD₁₆, C₃xSD₃₂, C₃xQ₃₂, C₃xC₄oD₈, C₃xC₄oD₁₆
Quotients: C₁, C₂, C₃, C₂², C₆, D₄, C₂³, C₂xC₆, D₈, C₂xD₄, C₃xD₄, C₂²xC₆, C₂xD₈, C₃xD₈, C₆xD₄, C₄oD₁₆, C₆xD₈, C₃xC₄oD₁₆

Smallest permutation representation of C₃xC₄oD₁₆
►On 96 points

Generators in S₉₆

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

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

(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)

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

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

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

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

66 conjugacy classes

class	1	2A	2B	2C	2D	3A	3B	4A	4B	4C	4D	4E	6A	6B	6C	6D	6E	6F	6G	6H	8A	8B	8C	8D	12A	12B	12C	12D	12E	12F	12G	12H	12I	12J	16A	···	16H	24A	···	24H	48A	···	48P
order	1	2	2	2	2	3	3	4	4	4	4	4	6	6	6	6	6	6	6	6	8	8	8	8	12	12	12	12	12	12	12	12	12	12	16	···	16	24	···	24	48	···	48
size	1	1	2	8	8	1	1	1	1	2	8	8	1	1	2	2	8	8	8	8	2	2	2	2	1	1	1	1	2	2	8	8	8	8	2	···	2	2	···	2	2	···	2

66 irreducible representations

dim	1	1	1	1	1	1	1	1	1	1	1	1	2	2	2	2	2	2	2	2	2	2
type	+	+	+	+	+	+							+	+	+	+
image	C₁	C₂	C₂	C₂	C₂	C₂	C₃	C₆	C₆	C₆	C₆	C₆	D₄	D₄	D₈	D₈	C₃xD₄	C₃xD₄	C₃xD₈	C₃xD₈	C₄oD₁₆	C₃xC₄oD₁₆
kernel	C₃xC₄oD₁₆	C₂xC₄₈	C₃xD₁₆	C₃xSD₃₂	C₃xQ₃₂	C₃xC₄oD₈	C₄oD₁₆	C₂xC₁₆	D₁₆	SD₃₂	Q₃₂	C₄oD₈	C₂₄	C₂xC₁₂	C₁₂	C₂xC₆	C₈	C₂xC₄	C₄	C₂²	C₃	C₁
# reps	1	1	1	2	1	2	2	2	2	4	2	4	1	1	2	2	2	2	4	4	8	16

Matrix representation of C₃xC₄oD₁₆ ►in GL₂(F₉₇) generated by

35	0
0	35

75	0
0	75

2	26
71	2

71	2
2	26

G:=sub<GL(2,GF(97))| [35,0,0,35],[75,0,0,75],[2,71,26,2],[71,2,2,26] >;

C₃xC₄oD₁₆ in GAP, Magma, Sage, TeX

C_3\times C_4\circ D_{16}

% in TeX

G:=Group("C3xC4oD16");

// GroupNames label

G:=SmallGroup(192,941);

// by ID

G=gap.SmallGroup(192,941);

# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-2,365,520,2524,1271,242,6053,3036,124]);

// Polycyclic

G:=Group<a,b,c,d|a^3=b^4=d^2=1,c^8=b^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=b^2*c^7>;

// generators/relations

G = C3xC4oD16 order 192 = 26·3

Direct product of C3 and C4oD16

G = C₃xC₄oD₁₆ order 192 = 2⁶·3

Direct product of C₃ and C₄oD₁₆