C12.16D8 - GroupNames

G = C₁₂.₁₆D₈ order 192 = 2⁶·3

16^th non-split extension by C₁₂ of D₈ acting via D₈/C₄=C₂²

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C₁₂.₁₆D₈, C₁₂.₁₅SD₁₆, C₄².₂₁₇D₆, C₆.₅₆(C₂xD₈), C₄.₅(D₄:S₃), (C₂xD₄).₅₃D₆, C₄:₁D₄.₄S₃, C₃:₃(C₄.₄D₈), C₁₂:₂Q₈:₁₉C₂, (C₂xC₁₂).₁₄₆D₄, C₄.₃(D₄.S₃), C₆.₅₇(C₂xSD₁₆), C₁₂.₇₄(C₄oD₄), D₄:Dic₃:₂₀C₂, (C₆xD₄).₆₉C₂², C₄.₂₂(D₄:₂S₃), (C₂xC₁₂).₃₈₈C₂³, (C₄xC₁₂).₁₁₈C₂², C₆.₄₃(C₄.₄D₄), C₄:Dic₃.₁₅₄C₂², C₂.₁₀(C₂³.₁₂D₆), (C₄xC₃:C₈):₁₄C₂, C₂.₁₁(C₂xD₄:S₃), (C₃xC₄:₁D₄).₂C₂, (C₂xC₆).₅₁₉(C₂xD₄), C₂.₁₁(C₂xD₄.S₃), (C₂xC₃:C₈).₂₅₆C₂², (C₂xC₄).₁₃₀(C₃:D₄), (C₂xC₄).₄₈₆(C₂²xS₃), C₂².₁₉₂(C₂xC₃:D₄), SmallGroup(192,629)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂xC₁₂ — C₁₂.₁₆D₈

Chief series

C₁ — C₃ — C₆ — C₂xC₆ — C₂xC₁₂ — C₂xC₃:C₈ — C₄xC₃:C₈ — C₁₂.₁₆D₈

Lower central

C₃ — C₆ — C₂xC₁₂ — C₁₂.₁₆D₈

Upper central

C₁ — C₂² — C₄² — C₄:₁D₄

Generators and relations for C₁₂.₁₆D₈
G = < a,b,c | a¹²=b⁸=1, c²=a⁶, bab^-1=a⁵, cac^-1=a^-1, cbc^-1=a⁶b^-1 >

Subgroups: 336 in 118 conjugacy classes, 47 normal (23 characteristic)
C₁, C₂, C₂, C₃, C₄, C₄, C₂², C₂², C₆, C₆, C₈, C₂xC₄, C₂xC₄, D₄, Q₈, C₂³, Dic₃, C₁₂, C₂xC₆, C₂xC₆, C₄², C₄:C₄, C₂xC₈, C₂xD₄, C₂xD₄, C₂xQ₈, C₃:C₈, Dic₆, C₂xDic₃, C₂xC₁₂, C₃xD₄, C₂²xC₆, C₄xC₈, D₄:C₄, C₄:₁D₄, C₄:Q₈, C₂xC₃:C₈, C₄:Dic₃, C₄:Dic₃, C₄xC₁₂, C₂xDic₆, C₆xD₄, C₆xD₄, C₄.₄D₈, C₄xC₃:C₈, D₄:Dic₃, C₁₂:₂Q₈, C₃xC₄:₁D₄, C₁₂.₁₆D₈
Quotients: C₁, C₂, C₂², S₃, D₄, C₂³, D₆, D₈, SD₁₆, C₂xD₄, C₄oD₄, C₃:D₄, C₂²xS₃, C₄.₄D₄, C₂xD₈, C₂xSD₁₆, D₄:S₃, D₄.S₃, D₄:₂S₃, C₂xC₃:D₄, C₄.₄D₈, C₂xD₄:S₃, C₂xD₄.S₃, C₂³.₁₂D₆, C₁₂.₁₆D₈

Smallest permutation representation of C₁₂.₁₆D₈
►On 96 points

Generators in S₉₆

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

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

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

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

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

36 conjugacy classes

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

36 irreducible representations

dim	1	1	1	1	1	2	2	2	2	2	2	2	2	4	4	4
type	+	+	+	+	+	+	+	+	+	+				+	-	-
image	C₁	C₂	C₂	C₂	C₂	S₃	D₄	D₆	D₆	D₈	SD₁₆	C₄oD₄	C₃:D₄	D₄:S₃	D₄.S₃	D₄:₂S₃
kernel	C₁₂.₁₆D₈	C₄xC₃:C₈	D₄:Dic₃	C₁₂:₂Q₈	C₃xC₄:₁D₄	C₄:₁D₄	C₂xC₁₂	C₄²	C₂xD₄	C₁₂	C₁₂	C₁₂	C₂xC₄	C₄	C₄	C₄
# reps	1	1	4	1	1	1	2	1	2	4	4	4	4	2	2	2

Matrix representation of C₁₂.₁₆D₈ ►in GL₆(F₇₃)

1	2	0	0	0	0
72	72	0	0	0	0
0	0	72	0	0	0
0	0	0	72	0	0
0	0	0	0	8	0
0	0	0	0	0	64

46	0	0	0	0	0
0	46	0	0	0	0
0	0	16	16	0	0
0	0	57	16	0	0
0	0	0	0	0	2
0	0	0	0	36	0

27	54	0	0	0	0
0	46	0	0	0	0
0	0	57	57	0	0
0	0	57	16	0	0
0	0	0	0	0	2
0	0	0	0	37	0

G:=sub<GL(6,GF(73))| [1,72,0,0,0,0,2,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,8,0,0,0,0,0,0,64],[46,0,0,0,0,0,0,46,0,0,0,0,0,0,16,57,0,0,0,0,16,16,0,0,0,0,0,0,0,36,0,0,0,0,2,0],[27,0,0,0,0,0,54,46,0,0,0,0,0,0,57,57,0,0,0,0,57,16,0,0,0,0,0,0,0,37,0,0,0,0,2,0] >;

C₁₂.₁₆D₈ in GAP, Magma, Sage, TeX

C_{12}._{16}D_8

% in TeX

G:=Group("C12.16D8");

// GroupNames label

G:=SmallGroup(192,629);

// by ID

G=gap.SmallGroup(192,629);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,64,590,135,438,102,6278]);

// Polycyclic

G:=Group<a,b,c|a^12=b^8=1,c^2=a^6,b*a*b^-1=a^5,c*a*c^-1=a^-1,c*b*c^-1=a^6*b^-1>;

// generators/relations

G = C12.16D8 order 192 = 26·3

16th non-split extension by C12 of D8 acting via D8/C4=C22

G = C₁₂.₁₆D₈ order 192 = 2⁶·3

16^th non-split extension by C₁₂ of D₈ acting via D₈/C₄=C₂²