C36.38D6 - GroupNames

G = C₃₆.₃₈D₆ order 432 = 2⁴·3³

9^th non-split extension by C₃₆ of D₆ acting via D₆/S₃=C₂

metabelian, supersoluble, monomial, A-group

Aliases: C₃₆.₃₈D₆, C₁₂.₃₈D₁₈, C₉⋊S₃⋊C₈, C₉⋊C₈⋊₆S₃, C₃⋊C₈⋊₆D₉, C₉⋊₁(S₃×C₈), C₃⋊₁(C₈×D₉), C₁₂.₆₀S₃², C₆.₁(C₄×D₉), C₁₈.₁(C₄×S₃), C₄.₂₃(S₃×D₉), C₉⋊Dic₃.₁C₄, C₃².₃(S₃×C₈), (C₃×C₁₂).₁₅₆D₆, (C₃×C₃₆).₃₇C₂², C₆.₁(C₆.D₆), C₂.₁(C₁₈.D₆), C₃.₁(C₁₂.₂₉D₆), (C₃×C₉⋊C₈)⋊₇C₂, (C₉×C₃⋊C₈)⋊₅C₂, (C₃×C₉)⋊₂(C₂×C₈), (C₃×C₃⋊C₈).₈S₃, (C₂×C₉⋊S₃).₁C₄, (C₄×C₉⋊S₃).₃C₂, (C₃×C₁₈).₂(C₂×C₄), (C₃×C₆).₃₄(C₄×S₃), SmallGroup(432,59)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃×C₉ — C₃₆.₃₈D₆

Chief series

C₁ — C₃ — C₃² — C₃×C₉ — C₃×C₁₈ — C₃×C₃₆ — C₉×C₃⋊C₈ — C₃₆.₃₈D₆

Lower central

C₃×C₉ — C₃₆.₃₈D₆

Upper central

C₁ — C₄

Generators and relations for C₃₆.₃₈D₆
G = < a,b,c | a³⁶=1, b⁶=a²⁷, c²=a¹⁸, bab^-1=cac^-1=a¹⁷, cbc^-1=a¹⁸b⁵ >

Subgroups: 500 in 82 conjugacy classes, 29 normal (27 characteristic)
C₁, C₂, C₂, C₃, C₃, C₄, C₄, C₂², S₃, C₆, C₆, C₈, C₂×C₄, C₉, C₉, C₃², Dic₃, C₁₂, C₁₂, D₆, C₂×C₈, D₉, C₁₈, C₁₈, C₃⋊S₃, C₃×C₆, C₃⋊C₈, C₃⋊C₈, C₂₄, C₄×S₃, C₃×C₉, Dic₉, C₃₆, C₃₆, D₁₈, C₃⋊Dic₃, C₃×C₁₂, C₂×C₃⋊S₃, S₃×C₈, C₉⋊S₃, C₃×C₁₈, C₉⋊C₈, C₇₂, C₄×D₉, C₃×C₃⋊C₈, C₃×C₃⋊C₈, C₄×C₃⋊S₃, C₉⋊Dic₃, C₃×C₃₆, C₂×C₉⋊S₃, C₈×D₉, C₁₂.₂₉D₆, C₃×C₉⋊C₈, C₉×C₃⋊C₈, C₄×C₉⋊S₃, C₃₆.₃₈D₆
Quotients: C₁, C₂, C₄, C₂², S₃, C₈, C₂×C₄, D₆, C₂×C₈, D₉, C₄×S₃, D₁₈, S₃², S₃×C₈, C₄×D₉, C₆.D₆, S₃×D₉, C₈×D₉, C₁₂.₂₉D₆, C₁₈.D₆, C₃₆.₃₈D₆

Smallest permutation representation of C₃₆.₃₈D₆
►On 72 points

Generators in S₇₂

(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)

(1 49 22 46 7 43 28 40 13 37 34 70 19 67 4 64 25 61 10 58 31 55 16 52)(2 66 23 63 8 60 29 57 14 54 35 51 20 48 5 45 26 42 11 39 32 72 17 69)(3 47 24 44 9 41 30 38 15 71 36 68 21 65 6 62 27 59 12 56 33 53 18 50)

(1 4 19 22)(2 21 20 3)(5 36 23 18)(6 17 24 35)(7 34 25 16)(8 15 26 33)(9 32 27 14)(10 13 28 31)(11 30 29 12)(37 52 55 70)(38 69 56 51)(39 50 57 68)(40 67 58 49)(41 48 59 66)(42 65 60 47)(43 46 61 64)(44 63 62 45)(53 72 71 54)

G:=sub<Sym(72)| (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), (1,49,22,46,7,43,28,40,13,37,34,70,19,67,4,64,25,61,10,58,31,55,16,52)(2,66,23,63,8,60,29,57,14,54,35,51,20,48,5,45,26,42,11,39,32,72,17,69)(3,47,24,44,9,41,30,38,15,71,36,68,21,65,6,62,27,59,12,56,33,53,18,50), (1,4,19,22)(2,21,20,3)(5,36,23,18)(6,17,24,35)(7,34,25,16)(8,15,26,33)(9,32,27,14)(10,13,28,31)(11,30,29,12)(37,52,55,70)(38,69,56,51)(39,50,57,68)(40,67,58,49)(41,48,59,66)(42,65,60,47)(43,46,61,64)(44,63,62,45)(53,72,71,54)>;

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), (1,49,22,46,7,43,28,40,13,37,34,70,19,67,4,64,25,61,10,58,31,55,16,52)(2,66,23,63,8,60,29,57,14,54,35,51,20,48,5,45,26,42,11,39,32,72,17,69)(3,47,24,44,9,41,30,38,15,71,36,68,21,65,6,62,27,59,12,56,33,53,18,50), (1,4,19,22)(2,21,20,3)(5,36,23,18)(6,17,24,35)(7,34,25,16)(8,15,26,33)(9,32,27,14)(10,13,28,31)(11,30,29,12)(37,52,55,70)(38,69,56,51)(39,50,57,68)(40,67,58,49)(41,48,59,66)(42,65,60,47)(43,46,61,64)(44,63,62,45)(53,72,71,54) );

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)], [(1,49,22,46,7,43,28,40,13,37,34,70,19,67,4,64,25,61,10,58,31,55,16,52),(2,66,23,63,8,60,29,57,14,54,35,51,20,48,5,45,26,42,11,39,32,72,17,69),(3,47,24,44,9,41,30,38,15,71,36,68,21,65,6,62,27,59,12,56,33,53,18,50)], [(1,4,19,22),(2,21,20,3),(5,36,23,18),(6,17,24,35),(7,34,25,16),(8,15,26,33),(9,32,27,14),(10,13,28,31),(11,30,29,12),(37,52,55,70),(38,69,56,51),(39,50,57,68),(40,67,58,49),(41,48,59,66),(42,65,60,47),(43,46,61,64),(44,63,62,45),(53,72,71,54)]])

72 conjugacy classes

class	1	2A	2B	2C	3A	3B	3C	4A	4B	4C	4D	6A	6B	6C	8A	8B	8C	8D	8E	8F	8G	8H	9A	9B	9C	9D	9E	9F	12A	12B	12C	12D	12E	12F	18A	18B	18C	18D	18E	18F	24A	24B	24C	24D	24E	24F	24G	24H	36A	···	36F	36G	···	36L	72A	···	72L
order	1	2	2	2	3	3	3	4	4	4	4	6	6	6	8	8	8	8	8	8	8	8	9	9	9	9	9	9	12	12	12	12	12	12	18	18	18	18	18	18	24	24	24	24	24	24	24	24	36	···	36	36	···	36	72	···	72
size	1	1	27	27	2	2	4	1	1	27	27	2	2	4	3	3	3	3	9	9	9	9	2	2	2	4	4	4	2	2	2	2	4	4	2	2	2	4	4	4	6	6	6	6	18	18	18	18	2	···	2	4	···	4	6	···	6

72 irreducible representations

dim	1	1	1	1	1	1	1	2	2	2	2	2	2	2	2	2	2	2	2	4	4	4	4	4	4
type	+	+	+	+				+	+	+	+	+			+					+	+	+		+
image	C₁	C₂	C₂	C₂	C₄	C₄	C₈	S₃	S₃	D₆	D₆	D₉	C₄×S₃	C₄×S₃	D₁₈	S₃×C₈	S₃×C₈	C₄×D₉	C₈×D₉	S₃²	C₆.D₆	S₃×D₉	C₁₂.₂₉D₆	C₁₈.D₆	C₃₆.₃₈D₆
kernel	C₃₆.₃₈D₆	C₃×C₉⋊C₈	C₉×C₃⋊C₈	C₄×C₉⋊S₃	C₉⋊Dic₃	C₂×C₉⋊S₃	C₉⋊S₃	C₉⋊C₈	C₃×C₃⋊C₈	C₃₆	C₃×C₁₂	C₃⋊C₈	C₁₈	C₃×C₆	C₁₂	C₉	C₃²	C₆	C₃	C₁₂	C₆	C₄	C₃	C₂	C₁
# reps	1	1	1	1	2	2	8	1	1	1	1	3	2	2	3	4	4	6	12	1	1	3	2	3	6

Matrix representation of C₃₆.₃₈D₆ ►in GL₄(𝔽₇₃) generated by

1	0	0	0
0	1	0	0
0	0	39	47
0	0	26	65

0	72	0	0
1	1	0	0
0	0	63	0
0	0	10	10

0	1	0	0
1	0	0	0
0	0	27	0
0	0	46	46

G:=sub<GL(4,GF(73))| [1,0,0,0,0,1,0,0,0,0,39,26,0,0,47,65],[0,1,0,0,72,1,0,0,0,0,63,10,0,0,0,10],[0,1,0,0,1,0,0,0,0,0,27,46,0,0,0,46] >;

C₃₆.₃₈D₆ in GAP, Magma, Sage, TeX

C_{36}._{38}D_6

% in TeX

G:=Group("C36.38D6");

// GroupNames label

G:=SmallGroup(432,59);

// by ID

G=gap.SmallGroup(432,59);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,28,36,58,3091,662,4037,7069]);

// Polycyclic

G:=Group<a,b,c|a^36=1,b^6=a^27,c^2=a^18,b*a*b^-1=c*a*c^-1=a^17,c*b*c^-1=a^18*b^5>;

// generators/relations

G = C36.38D6 order 432 = 24·33

9th non-split extension by C36 of D6 acting via D6/S3=C2

G = C₃₆.₃₈D₆ order 432 = 2⁴·3³

9^th non-split extension by C₃₆ of D₆ acting via D₆/S₃=C₂