C8xD12 - GroupNames

G = C₈xD₁₂ order 192 = 2⁶·3

Direct product of C₈ and D₁₂

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

Aliases: C₈xD₁₂, C₂₄:₂₅D₄, C₄².₂₅₆D₆, C₃:₁(C₈xD₄), C₄:₂(S₃xC₈), (C₄xC₈):₃S₃, C₁₂:₄(C₂xC₈), D₆:₁(C₂xC₈), C₆.₄(C₄xD₄), (C₄xC₂₄):₂₃C₂, D₆:C₈:₄₁C₂, C₂.₁(C₄xD₁₂), D₆:C₄.₁₈C₄, C₁₂:C₈:₃₆C₂, C₆.₃(C₈oD₄), C₄.₇₄(C₂xD₁₂), (C₂xC₈).₃₃₉D₆, C₆.₄(C₂²xC₈), (C₄xD₁₂).₂₉C₂, (C₂xD₁₂).₁₈C₄, C₂.₂(C₈oD₁₂), C₁₂.₂₉₄(C₂xD₄), C₄:Dic₃.₂₃C₄, C₄.₁₂₅(C₄oD₁₂), C₁₂.₂₄₁(C₄oD₄), (C₄xC₁₂).₃₂₃C₂², (C₂xC₂₄).₃₄₂C₂², (C₂xC₁₂).₈₀₄C₂³, C₂.₆(S₃xC₂xC₈), (S₃xC₂xC₈):₁₁C₂, C₂².₃₆(S₃xC₂xC₄), (C₂xC₄).₁₀₃(C₄xS₃), (C₂xC₁₂).₂₂₀(C₂xC₄), (C₂xC₃:C₈).₂₉₀C₂², (S₃xC₂xC₄).₂₆₆C₂², (C₂xC₆).₅₉(C₂²xC₄), (C₂²xS₃).₃₁(C₂xC₄), (C₂xC₄).₇₄₆(C₂²xS₃), (C₂xDic₃).₄₆(C₂xC₄), SmallGroup(192,245)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₆ — C₈xD₁₂

Chief series

C₁ — C₃ — C₆ — C₁₂ — C₂xC₁₂ — S₃xC₂xC₄ — C₄xD₁₂ — C₈xD₁₂

Lower central

C₃ — C₆ — C₈xD₁₂

Upper central

C₁ — C₂xC₈ — C₄xC₈

Generators and relations for C₈xD₁₂
G = < a,b,c | a⁸=b¹²=c²=1, ab=ba, ac=ca, cbc=b^-1 >

Subgroups: 312 in 134 conjugacy classes, 63 normal (33 characteristic)
C₁, C₂, C₂, C₃, C₄, C₄, C₄, C₂², C₂², S₃, C₆, C₈, C₈, C₂xC₄, C₂xC₄, D₄, C₂³, Dic₃, C₁₂, C₁₂, C₁₂, D₆, D₆, C₂xC₆, C₄², C₂²:C₄, C₄:C₄, C₂xC₈, C₂xC₈, C₂²xC₄, C₂xD₄, C₃:C₈, C₂₄, C₂₄, C₄xS₃, D₁₂, C₂xDic₃, C₂xC₁₂, C₂²xS₃, C₄xC₈, C₂²:C₈, C₄:C₈, C₄xD₄, C₂²xC₈, S₃xC₈, C₂xC₃:C₈, C₄:Dic₃, D₆:C₄, C₄xC₁₂, C₂xC₂₄, S₃xC₂xC₄, C₂xD₁₂, C₈xD₄, C₁₂:C₈, D₆:C₈, C₄xC₂₄, C₄xD₁₂, S₃xC₂xC₈, C₈xD₁₂
Quotients: C₁, C₂, C₄, C₂², S₃, C₈, C₂xC₄, D₄, C₂³, D₆, C₂xC₈, C₂²xC₄, C₂xD₄, C₄oD₄, C₄xS₃, D₁₂, C₂²xS₃, C₄xD₄, C₂²xC₈, C₈oD₄, S₃xC₈, S₃xC₂xC₄, C₂xD₁₂, C₄oD₁₂, C₈xD₄, C₄xD₁₂, S₃xC₂xC₈, C₈oD₁₂, C₈xD₁₂

Smallest permutation representation of C₈xD₁₂
►On 96 points

Generators in S₉₆

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

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

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

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

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

72 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	2G	3	4A	4B	4C	4D	4E	4F	4G	4H	4I	4J	4K	4L	6A	6B	6C	8A	···	8H	8I	8J	8K	8L	8M	···	8T	12A	···	12L	24A	···	24P
order	1	2	2	2	2	2	2	2	3	4	4	4	4	4	4	4	4	4	4	4	4	6	6	6	8	···	8	8	8	8	8	8	···	8	12	···	12	24	···	24
size	1	1	1	1	6	6	6	6	2	1	1	1	1	2	2	2	2	6	6	6	6	2	2	2	1	···	1	2	2	2	2	6	···	6	2	···	2	2	···	2

72 irreducible representations

dim	1	1	1	1	1	1	1	1	1	1	2	2	2	2	2	2	2	2	2	2	2
type	+	+	+	+	+	+					+	+	+	+		+
image	C₁	C₂	C₂	C₂	C₂	C₂	C₄	C₄	C₄	C₈	S₃	D₄	D₆	D₆	C₄oD₄	D₁₂	C₄xS₃	C₈oD₄	S₃xC₈	C₄oD₁₂	C₈oD₁₂
kernel	C₈xD₁₂	C₁₂:C₈	D₆:C₈	C₄xC₂₄	C₄xD₁₂	S₃xC₂xC₈	C₄:Dic₃	D₆:C₄	C₂xD₁₂	D₁₂	C₄xC₈	C₂₄	C₄²	C₂xC₈	C₁₂	C₈	C₂xC₄	C₆	C₄	C₄	C₂
# reps	1	1	2	1	1	2	2	4	2	16	1	2	1	2	2	4	4	4	8	4	8

Matrix representation of C₈xD₁₂ ►in GL₃(F₇₃) generated by

10	0	0
0	63	0
0	0	63

72	0	0
0	59	7
0	66	66

72	0	0
0	1	0
0	1	72

G:=sub<GL(3,GF(73))| [10,0,0,0,63,0,0,0,63],[72,0,0,0,59,66,0,7,66],[72,0,0,0,1,1,0,0,72] >;

C₈xD₁₂ in GAP, Magma, Sage, TeX

C_8\times D_{12}

% in TeX

G:=Group("C8xD12");

// GroupNames label

G:=SmallGroup(192,245);

// by ID

G=gap.SmallGroup(192,245);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,120,58,136,6278]);

// Polycyclic

G:=Group<a,b,c|a^8=b^12=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;

// generators/relations

G = C8xD12 order 192 = 26·3

Direct product of C8 and D12

G = C₈xD₁₂ order 192 = 2⁶·3

Direct product of C₈ and D₁₂