C6.(C4oD8) - GroupNames

G = C₆.(C₄○D₈) order 192 = 2⁶·3

22^nd non-split extension by C₆ of C₄○D₈ acting via C₄○D₈/SD₁₆=C₂

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C₄⋊C₄.₄₂D₆, C₄.Q₈⋊₁₀S₃, D₆⋊C₈.₁₄C₂, (C₂×C₈).₁₄₀D₆, C₆.₅₇(C₄○D₈), C₆.Q₁₆⋊₁₇C₂, C₄.D₁₂.₆C₂, C₁₂.₃₃(C₄○D₄), C₄.₇₅(C₄○D₁₂), C₆.SD₁₆⋊₁₈C₂, C₂.Dic₁₂⋊₃₂C₂, (C₂²×S₃).₂₇D₄, C₂².₂₂₀(S₃×D₄), (C₂×C₂₄).₂₈₇C₂², (C₂×C₁₂).₂₈₄C₂³, C₄.₂₇(Q₈⋊₃S₃), (C₂×Dic₃).₁₆₄D₄, C₂.₂₅(D₄.D₆), C₆.₄₄(C₈.C₂²), C₃⋊₄(C₂³.₂₀D₄), C₂.₂₄(Q₈.₇D₆), C₂.₁₄(D₆.D₄), C₄⋊Dic₃.₁₁₃C₂², (C₂×Dic₆).₈₅C₂², C₆.₄₄(C₂².D₄), (C₃×C₄.Q₈)⋊₁₈C₂, C₄⋊C₄⋊₇S₃.₆C₂, (C₂×C₆).₂₈₉(C₂×D₄), (C₂×C₃⋊C₈).₆₁C₂², (S₃×C₂×C₄).₃₆C₂², (C₃×C₄⋊C₄).₇₇C₂², (C₂×C₄).₃₈₇(C₂²×S₃), SmallGroup(192,427)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂×C₁₂ — C₆.(C₄○D₈)

Chief series

C₁ — C₃ — C₆ — C₂×C₆ — C₂×C₁₂ — S₃×C₂×C₄ — C₄⋊C₄⋊₇S₃ — C₆.(C₄○D₈)

Lower central

C₃ — C₆ — C₂×C₁₂ — C₆.(C₄○D₈)

Upper central

C₁ — C₂² — C₂×C₄ — C₄.Q₈

Generators and relations for C₆.(C₄○D₈)
G = < a,b,c,d | a⁶=b⁴=1, c⁴=b², d²=a³, bab^-1=cac^-1=dad^-1=a^-1, cbc^-1=a³b, bd=db, dcd^-1=b²c³ >

Subgroups: 272 in 96 conjugacy classes, 37 normal (all characteristic)
C₁, C₂, C₂, C₃, C₄, C₄, C₂², C₂², S₃, C₆, C₈, C₂×C₄, C₂×C₄, Q₈, C₂³, Dic₃, C₁₂, C₁₂, D₆, C₂×C₆, C₄², C₂²⋊C₄, C₄⋊C₄, C₄⋊C₄, C₂×C₈, C₂×C₈, C₂²×C₄, C₂×Q₈, C₃⋊C₈, C₂₄, Dic₆, C₄×S₃, C₂×Dic₃, C₂×Dic₃, C₂×C₁₂, C₂×C₁₂, C₂²×S₃, C₂²⋊C₈, Q₈⋊C₄, C₄.Q₈, C₂.D₈, C₄²⋊C₂, C₂²⋊Q₈, C₂×C₃⋊C₈, C₄×Dic₃, C₄⋊Dic₃, C₄⋊Dic₃, D₆⋊C₄, C₃×C₄⋊C₄, C₂×C₂₄, C₂×Dic₆, S₃×C₂×C₄, C₂³.₂₀D₄, C₆.Q₁₆, C₆.SD₁₆, C₂.Dic₁₂, D₆⋊C₈, C₃×C₄.Q₈, C₄⋊C₄⋊₇S₃, C₄.D₁₂, C₆.(C₄○D₈)
Quotients: C₁, C₂, C₂², S₃, D₄, C₂³, D₆, C₂×D₄, C₄○D₄, C₂²×S₃, C₂².D₄, C₄○D₈, C₈.C₂², C₄○D₁₂, S₃×D₄, Q₈⋊₃S₃, C₂³.₂₀D₄, D₆.D₄, D₄.D₆, Q₈.₇D₆, C₆.(C₄○D₈)

Smallest permutation representation of C₆.(C₄○D₈)
►On 96 points

Generators in S₉₆

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

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

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

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

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

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

33 conjugacy classes

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

33 irreducible representations

dim	1	1	1	1	1	1	1	1	2	2	2	2	2	2	2	2	4	4	4	4	4
type	+	+	+	+	+	+	+	+	+	+	+	+	+				-	+	+	-
image	C₁	C₂	C₂	C₂	C₂	C₂	C₂	C₂	S₃	D₄	D₄	D₆	D₆	C₄○D₄	C₄○D₈	C₄○D₁₂	C₈.C₂²	Q₈⋊₃S₃	S₃×D₄	D₄.D₆	Q₈.₇D₆
kernel	C₆.(C₄○D₈)	C₆.Q₁₆	C₆.SD₁₆	C₂.Dic₁₂	D₆⋊C₈	C₃×C₄.Q₈	C₄⋊C₄⋊₇S₃	C₄.D₁₂	C₄.Q₈	C₂×Dic₃	C₂²×S₃	C₄⋊C₄	C₂×C₈	C₁₂	C₆	C₄	C₆	C₄	C₂²	C₂	C₂
# reps	1	1	1	1	1	1	1	1	1	1	1	2	1	4	4	4	1	1	1	2	2

Matrix representation of C₆.(C₄○D₈) ►in GL₆(𝔽₇₃)

72	0	0	0	0	0
0	72	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	72

72	0	0	0	0	0
0	1	0	0	0	0
0	0	46	0	0	0
0	0	0	46	0	0
0	0	0	0	0	1
0	0	0	0	1	0

0	72	0	0	0	0
72	0	0	0	0	0
0	0	22	0	0	0
0	0	55	10	0	0
0	0	0	0	0	72
0	0	0	0	72	0

46	0	0	0	0	0
0	46	0	0	0	0
0	0	70	71	0	0
0	0	4	3	0	0
0	0	0	0	0	72
0	0	0	0	72	0

G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,72,72],[72,0,0,0,0,0,0,1,0,0,0,0,0,0,46,0,0,0,0,0,0,46,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[0,72,0,0,0,0,72,0,0,0,0,0,0,0,22,55,0,0,0,0,0,10,0,0,0,0,0,0,0,72,0,0,0,0,72,0],[46,0,0,0,0,0,0,46,0,0,0,0,0,0,70,4,0,0,0,0,71,3,0,0,0,0,0,0,0,72,0,0,0,0,72,0] >;

C₆.(C₄○D₈) in GAP, Magma, Sage, TeX

C_6.(C_4\circ D_8)

% in TeX

G:=Group("C6.(C4oD8)");

// GroupNames label

G:=SmallGroup(192,427);

// by ID

G=gap.SmallGroup(192,427);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,477,64,926,219,100,851,102,6278]);

// Polycyclic

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

// generators/relations

G = C6.(C4○D8) order 192 = 26·3

22nd non-split extension by C6 of C4○D8 acting via C4○D8/SD16=C2

G = C₆.(C₄○D₈) order 192 = 2⁶·3

22^nd non-split extension by C₆ of C₄○D₈ acting via C₄○D₈/SD₁₆=C₂