Dic6.32D4 - GroupNames

G = Dic₆.₃₂D₄ order 192 = 2⁶·3

2^nd non-split extension by Dic₆ of D₄ acting via D₄/C₂²=C₂

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Dic₆.₃₂D₄, C₂²⋊₃Dic₁₂, C₂³.₄₆D₁₂, (C₂×C₆)⋊₁Q₁₆, (C₂×C₈).₅D₆, C₆.₆(C₂×Q₁₆), (C₂×C₄).₃₆D₁₂, C₄.₁₂₄(S₃×D₄), (C₂×C₁₂).₄₇D₄, C₂²⋊C₈.₄S₃, C₆.₁₂C₂²≀C₂, (C₂×Dic₁₂)⋊₃C₂, C₁₂.₃₃₆(C₂×D₄), (C₂×C₂₄).₅C₂², C₂.Dic₁₂⋊₇C₂, C₃⋊₁(C₂²⋊Q₁₆), C₂.₈(C₂×Dic₁₂), (C₂²×C₆).₅₉D₄, (C₂²×C₄).₁₀₅D₆, C₂.₁₅(D₆⋊D₄), (C₂×C₁₂).₇₄₉C₂³, C₁₂.₄₈D₄.₄C₂, C₂.₁₅(C₈.D₆), C₂².₁₁₂(C₂×D₁₂), C₆.₁₂(C₈.C₂²), C₄⋊Dic₃.₁₆C₂², (C₂²×Dic₆).₃C₂, (C₂²×C₁₂).₅₅C₂², (C₂×Dic₆).₁₃C₂², (C₂×C₆).₁₃₂(C₂×D₄), (C₃×C₂²⋊C₈).₆C₂, (C₂×C₄).₆₉₄(C₂²×S₃), SmallGroup(192,298)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂×C₁₂ — Dic₆.₃₂D₄

Chief series

C₁ — C₃ — C₆ — C₁₂ — C₂×C₁₂ — C₂×Dic₆ — C₂²×Dic₆ — Dic₆.₃₂D₄

Lower central

C₃ — C₆ — C₂×C₁₂ — Dic₆.₃₂D₄

Upper central

C₁ — C₂² — C₂²×C₄ — C₂²⋊C₈

Generators and relations for Dic₆.₃₂D₄
G = < a,b,c,d | a¹²=d²=1, b²=c⁴=a⁶, bab^-1=a^-1, ac=ca, ad=da, cbc^-1=a³b, bd=db, dcd=a⁹c³ >

Subgroups: 416 in 148 conjugacy classes, 47 normal (25 characteristic)
C₁, C₂, C₂, C₃, C₄, C₄, C₂², C₂², C₂², C₆, C₆, C₈, C₂×C₄, C₂×C₄, Q₈, C₂³, Dic₃, C₁₂, C₁₂, C₂×C₆, C₂×C₆, C₂×C₆, C₂²⋊C₄, C₄⋊C₄, C₂×C₈, Q₁₆, C₂²×C₄, C₂²×C₄, C₂×Q₈, C₂₄, Dic₆, Dic₆, C₂×Dic₃, C₂×C₁₂, C₂×C₁₂, C₂²×C₆, C₂²⋊C₈, Q₈⋊C₄, C₂²⋊Q₈, C₂×Q₁₆, C₂²×Q₈, Dic₁₂, Dic₃⋊C₄, C₄⋊Dic₃, C₆.D₄, C₂×C₂₄, C₂×Dic₆, C₂×Dic₆, C₂×Dic₆, C₂²×Dic₃, C₂²×C₁₂, C₂²⋊Q₁₆, C₂.Dic₁₂, C₃×C₂²⋊C₈, C₂×Dic₁₂, C₁₂.₄₈D₄, C₂²×Dic₆, Dic₆.₃₂D₄
Quotients: C₁, C₂, C₂², S₃, D₄, C₂³, D₆, Q₁₆, C₂×D₄, D₁₂, C₂²×S₃, C₂²≀C₂, C₂×Q₁₆, C₈.C₂², Dic₁₂, C₂×D₁₂, S₃×D₄, C₂²⋊Q₁₆, D₆⋊D₄, C₂×Dic₁₂, C₈.D₆, Dic₆.₃₂D₄

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

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

(13 31)(14 32)(15 33)(16 34)(17 35)(18 36)(19 25)(20 26)(21 27)(22 28)(23 29)(24 30)(37 90)(38 91)(39 92)(40 93)(41 94)(42 95)(43 96)(44 85)(45 86)(46 87)(47 88)(48 89)

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

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

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

39 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	24A	···	24H
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	24	···	24
size	1	1	1	1	2	2	2	2	2	4	12	12	12	12	24	24	2	2	2	4	4	4	4	4	4	2	2	2	2	4	4	4	···	4

39 irreducible representations

dim	1	1	1	1	1	1	2	2	2	2	2	2	2	2	2	2	4	4	4
type	+	+	+	+	+	+	+	+	+	+	+	+	-	+	+	-	-	+	-
image	C₁	C₂	C₂	C₂	C₂	C₂	S₃	D₄	D₄	D₄	D₆	D₆	Q₁₆	D₁₂	D₁₂	Dic₁₂	C₈.C₂²	S₃×D₄	C₈.D₆
kernel	Dic₆.₃₂D₄	C₂.Dic₁₂	C₃×C₂²⋊C₈	C₂×Dic₁₂	C₁₂.₄₈D₄	C₂²×Dic₆	C₂²⋊C₈	Dic₆	C₂×C₁₂	C₂²×C₆	C₂×C₈	C₂²×C₄	C₂×C₆	C₂×C₄	C₂³	C₂²	C₆	C₄	C₂
# reps	1	2	1	2	1	1	1	4	1	1	2	1	4	2	2	8	1	2	2

Matrix representation of Dic₆.₃₂D₄ ►in GL₆(𝔽₇₃)

0	1	0	0	0	0
72	0	0	0	0	0
0	0	72	0	0	0
0	0	0	72	0	0
0	0	0	0	1	72
0	0	0	0	1	0

56	41	0	0	0	0
41	17	0	0	0	0
0	0	1	0	0	0
0	0	41	72	0	0
0	0	0	0	1	0
0	0	0	0	1	72

16	57	0	0	0	0
16	16	0	0	0	0
0	0	41	71	0	0
0	0	37	32	0	0
0	0	0	0	7	59
0	0	0	0	14	66

1	0	0	0	0	0
0	1	0	0	0	0
0	0	1	0	0	0
0	0	41	72	0	0
0	0	0	0	1	0
0	0	0	0	0	1

G:=sub<GL(6,GF(73))| [0,72,0,0,0,0,1,0,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,1,0,0,0,0,72,0],[56,41,0,0,0,0,41,17,0,0,0,0,0,0,1,41,0,0,0,0,0,72,0,0,0,0,0,0,1,1,0,0,0,0,0,72],[16,16,0,0,0,0,57,16,0,0,0,0,0,0,41,37,0,0,0,0,71,32,0,0,0,0,0,0,7,14,0,0,0,0,59,66],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,41,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

Dic₆.₃₂D₄ in GAP, Magma, Sage, TeX

{\rm Dic}_6._{32}D_4

% in TeX

G:=Group("Dic6.32D4");

// GroupNames label

G:=SmallGroup(192,298);

// by ID

G=gap.SmallGroup(192,298);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,254,219,226,1123,136,6278]);

// Polycyclic

G:=Group<a,b,c,d|a^12=d^2=1,b^2=c^4=a^6,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=a^3*b,b*d=d*b,d*c*d=a^9*c^3>;

// generators/relations

G = Dic6.32D4 order 192 = 26·3

2nd non-split extension by Dic6 of D4 acting via D4/C22=C2

G = Dic₆.₃₂D₄ order 192 = 2⁶·3

2^nd non-split extension by Dic₆ of D₄ acting via D₄/C₂²=C₂