D72:C3 - GroupNames

G = D₇₂⋊C₃ order 432 = 2⁴·3³

The semidirect product of D₇₂ and C₃ acting faithfully

Aliases: D₇₂⋊C₃, C₇₂⋊₁C₆, D₃₆⋊₄C₆, C₃².D₂₄, 3_-¹⁺²⋊₁D₈, C₉⋊₁(C₃×D₈), C₈⋊₁(C₉⋊C₆), D₃₆⋊C₃⋊₄C₂, C₂₄.₆(C₃×S₃), (C₃×C₂₄).₃S₃, C₁₈.₃(C₃×D₄), C₃.₃(C₃×D₂₄), C₆.₉(C₃×D₁₂), C₁₂.₇₂(S₃×C₆), C₃₆.₁₀(C₂×C₆), (C₃×C₁₂).₄₅D₆, (C₃×C₆).₁₈D₁₂, C₂.₅(D₃₆⋊C₃), (C₈×3_-¹⁺²)⋊₁C₂, (C₂×3_-¹⁺²).₃D₄, (C₄×3_-¹⁺²).₁₀C₂², C₄.₁₀(C₂×C₉⋊C₆), SmallGroup(432,123)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃₆ — D₇₂⋊C₃

Chief series

C₁ — C₃ — C₉ — C₁₈ — C₃₆ — C₄×3_-¹⁺² — D₃₆⋊C₃ — D₇₂⋊C₃

Lower central

C₉ — C₁₈ — C₃₆ — D₇₂⋊C₃

Upper central

C₁ — C₂ — C₄ — C₈

Generators and relations for D₇₂⋊C₃
G = < a,b,c | a⁷²=b²=c³=1, bab=a^-1, cac^-1=a⁴⁹, cbc^-1=a⁴⁸b >

Subgroups: 430 in 70 conjugacy classes, 26 normal (22 characteristic)
C₁, C₂, C₂, C₃, C₃, C₄, C₂², S₃, C₆, C₆, C₈, D₄, C₉, C₉, C₃², C₁₂, C₁₂, D₆, C₂×C₆, D₈, D₉, C₁₈, C₁₈, C₃×S₃, C₃×C₆, C₂₄, C₂₄, D₁₂, C₃×D₄, 3_-¹⁺², C₃₆, C₃₆, D₁₈, C₃×C₁₂, S₃×C₆, D₂₄, C₃×D₈, C₉⋊C₆, C₂×3_-¹⁺², C₇₂, C₇₂, D₃₆, C₃×C₂₄, C₃×D₁₂, C₄×3_-¹⁺², C₂×C₉⋊C₆, D₇₂, C₃×D₂₄, C₈×3_-¹⁺², D₃₆⋊C₃, D₇₂⋊C₃
Quotients: C₁, C₂, C₃, C₂², S₃, C₆, D₄, D₆, C₂×C₆, D₈, C₃×S₃, D₁₂, C₃×D₄, S₃×C₆, D₂₄, C₃×D₈, C₉⋊C₆, C₃×D₁₂, C₂×C₉⋊C₆, C₃×D₂₄, D₃₆⋊C₃, D₇₂⋊C₃

Smallest permutation representation of D₇₂⋊C₃
►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 63)(2 62)(3 61)(4 60)(5 59)(6 58)(7 57)(8 56)(9 55)(10 54)(11 53)(12 52)(13 51)(14 50)(15 49)(16 48)(17 47)(18 46)(19 45)(20 44)(21 43)(22 42)(23 41)(24 40)(25 39)(26 38)(27 37)(28 36)(29 35)(30 34)(31 33)(64 72)(65 71)(66 70)(67 69)

(2 26 50)(3 51 27)(5 29 53)(6 54 30)(8 32 56)(9 57 33)(11 35 59)(12 60 36)(14 38 62)(15 63 39)(17 41 65)(18 66 42)(20 44 68)(21 69 45)(23 47 71)(24 72 48)

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

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

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

53 conjugacy classes

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

53 irreducible representations

dim	1	1	1	1	1	1	2	2	2	2	2	2	2	2	2	2	2	2	6	6	6	6
type	+	+	+				+	+	+	+			+			+			+	+	+	+
image	C₁	C₂	C₂	C₃	C₆	C₆	S₃	D₄	D₆	D₈	C₃×S₃	C₃×D₄	D₁₂	S₃×C₆	C₃×D₈	D₂₄	C₃×D₁₂	C₃×D₂₄	C₉⋊C₆	C₂×C₉⋊C₆	D₃₆⋊C₃	D₇₂⋊C₃
kernel	D₇₂⋊C₃	C₈×3_-¹⁺²	D₃₆⋊C₃	D₇₂	C₇₂	D₃₆	C₃×C₂₄	C₂×3_-¹⁺²	C₃×C₁₂	3_-¹⁺²	C₂₄	C₁₈	C₃×C₆	C₁₂	C₉	C₃²	C₆	C₃	C₈	C₄	C₂	C₁
# reps	1	1	2	2	2	4	1	1	1	2	2	2	2	2	4	4	4	8	1	1	2	4

Matrix representation of D₇₂⋊C₃ ►in GL₆(𝔽₇₃)

0	0	55	50	0	0
0	0	23	5	0	0
0	0	0	0	55	50
0	0	0	0	23	5
68	18	0	0	0	0
55	50	0	0	0	0

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

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

G:=sub<GL(6,GF(73))| [0,0,0,0,68,55,0,0,0,0,18,50,55,23,0,0,0,0,50,5,0,0,0,0,0,0,55,23,0,0,0,0,50,5,0,0],[0,0,72,0,0,0,0,0,72,1,0,0,72,0,0,0,0,0,72,1,0,0,0,0,0,0,0,0,1,72,0,0,0,0,0,72],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,1,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,1,72] >;

D₇₂⋊C₃ in GAP, Magma, Sage, TeX

D_{72}\rtimes C_3

% in TeX

G:=Group("D72:C3");

// GroupNames label

G:=SmallGroup(432,123);

// by ID

G=gap.SmallGroup(432,123);

# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-3,-3,197,260,1011,80,10085,2035,292,14118]);

// Polycyclic

G:=Group<a,b,c|a^72=b^2=c^3=1,b*a*b=a^-1,c*a*c^-1=a^49,c*b*c^-1=a^48*b>;

// generators/relations

G = D72⋊C3 order 432 = 24·33

The semidirect product of D72 and C3 acting faithfully

G = D₇₂⋊C₃ order 432 = 2⁴·3³

The semidirect product of D₇₂ and C₃ acting faithfully