Dic6.37D4 - GroupNames

G = Dic₆.₃₇D₄ order 192 = 2⁶·3

7^th non-split extension by Dic₆ of D₄ acting via D₄/C₂²=C₂

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Dic₆.₃₇D₄, (C₂xC₆):₂Q₁₆, C₄:C₄.₆₉D₆, C₄.₁₀₃(S₃xD₄), (C₂xC₁₂).₇₉D₄, (C₂xQ₈).₅₅D₆, C₆.₃₈(C₂xQ₁₆), C₂²:Q₈.₅S₃, C₆.₄₉C₂²wrC₂, C₁₂.₁₅₆(C₂xD₄), C₃:₃(C₂²:Q₁₆), (C₂²xC₆).₉₆D₄, C₆.SD₁₆:₃₈C₂, (C₂²xC₄).₁₄₆D₆, C₂²:₃(C₃:Q₁₆), (C₆xQ₈).₄₉C₂², C₂.₁₇(C₂³:₂D₆), (C₂xC₁₂).₃₆₉C₂³, C₁₂.₅₅D₄.₉C₂, C₂³.₇₀(C₃:D₄), C₂.₁₅(Q₈.₁₄D₆), C₆.₁₁₇(C₈.C₂²), (C₂²xDic₆).₁₃C₂, (C₂²xC₁₂).₁₇₃C₂², (C₂xDic₆).₂₇₁C₂², (C₂xC₃:Q₁₆):₉C₂, C₂.₉(C₂xC₃:Q₁₆), (C₂xC₆).₅₀₀(C₂xD₄), (C₃xC₂²:Q₈).₄C₂, (C₂xC₄).₅₇(C₃:D₄), (C₂xC₃:C₈).₁₁₇C₂², (C₃xC₄:C₄).₁₁₆C₂², (C₂xC₄).₄₆₉(C₂²xS₃), C₂².₁₇₅(C₂xC₃:D₄), SmallGroup(192,609)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂xC₁₂ — Dic₆.₃₇D₄

Chief series

C₁ — C₃ — C₆ — C₁₂ — C₂xC₁₂ — C₂xDic₆ — C₂²xDic₆ — Dic₆.₃₇D₄

Lower central

C₃ — C₆ — C₂xC₁₂ — Dic₆.₃₇D₄

Upper central

C₁ — C₂² — C₂²xC₄ — C₂²:Q₈

Generators and relations for Dic₆.₃₇D₄
G = < a,b,c,d | a¹²=c⁴=d²=1, b²=a⁶, bab^-1=a^-1, cac^-1=a⁷, ad=da, cbc^-1=a⁹b, bd=db, dcd=a⁶c^-1 >

Subgroups: 384 in 148 conjugacy classes, 47 normal (27 characteristic)
C₁, C₂, C₂, C₃, C₄, C₄, C₂², C₂², C₂², C₆, C₆, C₈, C₂xC₄, C₂xC₄, Q₈, C₂³, Dic₃, C₁₂, C₁₂, C₂xC₆, C₂xC₆, C₂xC₆, C₂²:C₄, C₄:C₄, C₄:C₄, C₂xC₈, Q₁₆, C₂²xC₄, C₂²xC₄, C₂xQ₈, C₂xQ₈, C₃:C₈, Dic₆, Dic₆, C₂xDic₃, C₂xC₁₂, C₂xC₁₂, C₃xQ₈, C₂²xC₆, C₂²:C₈, Q₈:C₄, C₂²:Q₈, C₂xQ₁₆, C₂²xQ₈, C₂xC₃:C₈, C₃:Q₁₆, C₃xC₂²:C₄, C₃xC₄:C₄, C₃xC₄:C₄, C₂xDic₆, C₂xDic₆, C₂²xDic₃, C₂²xC₁₂, C₆xQ₈, C₂²:Q₁₆, C₆.SD₁₆, C₁₂.₅₅D₄, C₂xC₃:Q₁₆, C₃xC₂²:Q₈, C₂²xDic₆, Dic₆.₃₇D₄
Quotients: C₁, C₂, C₂², S₃, D₄, C₂³, D₆, Q₁₆, C₂xD₄, C₃:D₄, C₂²xS₃, C₂²wrC₂, C₂xQ₁₆, C₈.C₂², C₃:Q₁₆, S₃xD₄, C₂xC₃:D₄, C₂²:Q₁₆, C₂³:₂D₆, C₂xC₃:Q₁₆, Q₈.₁₄D₆, Dic₆.₃₇D₄

Smallest permutation representation of Dic₆.₃₇D₄
►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 75 7 81)(2 74 8 80)(3 73 9 79)(4 84 10 78)(5 83 11 77)(6 82 12 76)(13 34 19 28)(14 33 20 27)(15 32 21 26)(16 31 22 25)(17 30 23 36)(18 29 24 35)(37 55 43 49)(38 54 44 60)(39 53 45 59)(40 52 46 58)(41 51 47 57)(42 50 48 56)(61 91 67 85)(62 90 68 96)(63 89 69 95)(64 88 70 94)(65 87 71 93)(66 86 72 92)

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

(13 51)(14 52)(15 53)(16 54)(17 55)(18 56)(19 57)(20 58)(21 59)(22 60)(23 49)(24 50)(25 38)(26 39)(27 40)(28 41)(29 42)(30 43)(31 44)(32 45)(33 46)(34 47)(35 48)(36 37)

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

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

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

33 conjugacy classes

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

33 irreducible representations

dim	1	1	1	1	1	1	2	2	2	2	2	2	2	2	2	2	4	4	4	4
type	+	+	+	+	+	+	+	+	+	+	+	+	+	-			-	+	-	-
image	C₁	C₂	C₂	C₂	C₂	C₂	S₃	D₄	D₄	D₄	D₆	D₆	D₆	Q₁₆	C₃:D₄	C₃:D₄	C₈.C₂²	S₃xD₄	C₃:Q₁₆	Q₈.₁₄D₆
kernel	Dic₆.₃₇D₄	C₆.SD₁₆	C₁₂.₅₅D₄	C₂xC₃:Q₁₆	C₃xC₂²:Q₈	C₂²xDic₆	C₂²:Q₈	Dic₆	C₂xC₁₂	C₂²xC₆	C₄:C₄	C₂²xC₄	C₂xQ₈	C₂xC₆	C₂xC₄	C₂³	C₆	C₄	C₂²	C₂
# reps	1	2	1	2	1	1	1	4	1	1	1	1	1	4	2	2	1	2	2	2

Matrix representation of Dic₆.₃₇D₄ ►in GL₆(F₇₃)

72	1	0	0	0	0
72	0	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	72	0

11	60	0	0	0	0
71	62	0	0	0	0
0	0	72	0	0	0
0	0	0	72	0	0
0	0	0	0	60	66
0	0	0	0	66	13

72	0	0	0	0	0
0	72	0	0	0	0
0	0	56	70	0	0
0	0	48	17	0	0
0	0	0	0	50	45
0	0	0	0	45	23

1	0	0	0	0	0
0	1	0	0	0	0
0	0	1	0	0	0
0	0	13	72	0	0
0	0	0	0	1	0
0	0	0	0	0	1

G:=sub<GL(6,GF(73))| [72,72,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,72,0,0,0,0,1,0],[11,71,0,0,0,0,60,62,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,60,66,0,0,0,0,66,13],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,56,48,0,0,0,0,70,17,0,0,0,0,0,0,50,45,0,0,0,0,45,23],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,13,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

Dic₆.₃₇D₄ in GAP, Magma, Sage, TeX

{\rm Dic}_6._{37}D_4

% in TeX

G:=Group("Dic6.37D4");

// GroupNames label

G:=SmallGroup(192,609);

// by ID

G=gap.SmallGroup(192,609);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,254,219,184,1123,297,136,6278]);

// Polycyclic

G:=Group<a,b,c,d|a^12=c^4=d^2=1,b^2=a^6,b*a*b^-1=a^-1,c*a*c^-1=a^7,a*d=d*a,c*b*c^-1=a^9*b,b*d=d*b,d*c*d=a^6*c^-1>;

// generators/relations

G = Dic6.37D4 order 192 = 26·3

7th non-split extension by Dic6 of D4 acting via D4/C22=C2

G = Dic₆.₃₇D₄ order 192 = 2⁶·3

7^th non-split extension by Dic₆ of D₄ acting via D₄/C₂²=C₂