D24:5S3 - GroupNames

G = D₂₄⋊₅S₃ order 288 = 2⁵·3²

5^th semidirect product of D₂₄ and S₃ acting via S₃/C₃=C₂

Aliases: D₂₄⋊₅S₃, D₁₂.₃D₆, C₂₄.₂₅D₆, Dic₁₂⋊₅S₃, Dic₆.₃D₆, C₈.₁₇S₃², C₆.₃₃(S₃×D₄), (C₃×D₂₄)⋊₁₁C₂, C₃²⋊₇(C₄○D₈), D₁₂⋊S₃⋊₃C₂, C₃⋊₂(D₈⋊₃S₃), C₃⋊Dic₃.₄₃D₄, C₃⋊₂(D₂₄⋊C₂), (C₃×Dic₁₂)⋊₁₁C₂, Dic₆⋊S₃⋊₃C₂, C₁₂.₇₅(C₂²×S₃), (C₃×C₁₂).₅₅C₂³, (C₃×C₂₄).₂₅C₂², (C₃×D₁₂).₈C₂², C₂.₁₀(D₆⋊D₆), (C₃×Dic₆).₈C₂², C₃²⋊₄C₈.₂₃C₂², (C₈×C₃⋊S₃)⋊₂C₂, C₄.₇₀(C₂×S₃²), (C₂×C₃⋊S₃).₄₄D₄, (C₃×C₆).₃₉(C₂×D₄), (C₄×C₃⋊S₃).₆₉C₂², SmallGroup(288,458)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃×C₁₂ — D₂₄⋊₅S₃

Chief series

C₁ — C₃ — C₃² — C₃×C₆ — C₃×C₁₂ — C₃×D₁₂ — D₁₂⋊S₃ — D₂₄⋊₅S₃

Lower central

C₃² — C₃×C₆ — C₃×C₁₂ — D₂₄⋊₅S₃

Upper central

C₁ — C₂ — C₄ — C₈

Generators and relations for D₂₄⋊₅S₃
G = < a,b,c,d | a²⁴=b²=c³=d²=1, bab=a^-1, ac=ca, dad=a¹⁷, bc=cb, dbd=a⁴b, dcd=c^-1 >

Subgroups: 562 in 135 conjugacy classes, 38 normal (26 characteristic)
C₁, C₂, C₂, C₃, C₃, C₄, C₄, C₂², S₃, C₆, C₆, C₈, C₈, C₂×C₄, D₄, Q₈, C₃², Dic₃, C₁₂, C₁₂, D₆, C₂×C₆, C₂×C₈, D₈, SD₁₆, Q₁₆, C₄○D₄, C₃×S₃, C₃⋊S₃, C₃×C₆, C₃⋊C₈, C₂₄, C₂₄, Dic₆, C₄×S₃, D₁₂, D₁₂, C₂×Dic₃, C₃⋊D₄, C₃×D₄, C₃×Q₈, C₄○D₈, C₃×Dic₃, C₃⋊Dic₃, C₃×C₁₂, S₃×C₆, C₂×C₃⋊S₃, S₃×C₈, D₂₄, Dic₁₂, D₄.S₃, Q₈⋊₂S₃, C₃×D₈, C₃×Q₁₆, D₄⋊₂S₃, Q₈⋊₃S₃, C₃²⋊₄C₈, C₃×C₂₄, S₃×Dic₃, C₃⋊D₁₂, C₃×Dic₆, C₃×D₁₂, C₄×C₃⋊S₃, D₈⋊₃S₃, D₂₄⋊C₂, Dic₆⋊S₃, C₃×D₂₄, C₃×Dic₁₂, C₈×C₃⋊S₃, D₁₂⋊S₃, D₂₄⋊₅S₃
Quotients: C₁, C₂, C₂², S₃, D₄, C₂³, D₆, C₂×D₄, C₂²×S₃, C₄○D₈, S₃², S₃×D₄, C₂×S₃², D₈⋊₃S₃, D₂₄⋊C₂, D₆⋊D₆, D₂₄⋊₅S₃

Smallest permutation representation of D₂₄⋊₅S₃
►On 48 points

Generators in S₄₈

(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)

(1 47)(2 46)(3 45)(4 44)(5 43)(6 42)(7 41)(8 40)(9 39)(10 38)(11 37)(12 36)(13 35)(14 34)(15 33)(16 32)(17 31)(18 30)(19 29)(20 28)(21 27)(22 26)(23 25)(24 48)

(1 9 17)(2 10 18)(3 11 19)(4 12 20)(5 13 21)(6 14 22)(7 15 23)(8 16 24)(25 41 33)(26 42 34)(27 43 35)(28 44 36)(29 45 37)(30 46 38)(31 47 39)(32 48 40)

(2 18)(3 11)(5 21)(6 14)(8 24)(9 17)(12 20)(15 23)(25 29)(26 46)(27 39)(28 32)(30 42)(31 35)(33 45)(34 38)(36 48)(37 41)(40 44)(43 47)

G:=sub<Sym(48)| (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), (1,47)(2,46)(3,45)(4,44)(5,43)(6,42)(7,41)(8,40)(9,39)(10,38)(11,37)(12,36)(13,35)(14,34)(15,33)(16,32)(17,31)(18,30)(19,29)(20,28)(21,27)(22,26)(23,25)(24,48), (1,9,17)(2,10,18)(3,11,19)(4,12,20)(5,13,21)(6,14,22)(7,15,23)(8,16,24)(25,41,33)(26,42,34)(27,43,35)(28,44,36)(29,45,37)(30,46,38)(31,47,39)(32,48,40), (2,18)(3,11)(5,21)(6,14)(8,24)(9,17)(12,20)(15,23)(25,29)(26,46)(27,39)(28,32)(30,42)(31,35)(33,45)(34,38)(36,48)(37,41)(40,44)(43,47)>;

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), (1,47)(2,46)(3,45)(4,44)(5,43)(6,42)(7,41)(8,40)(9,39)(10,38)(11,37)(12,36)(13,35)(14,34)(15,33)(16,32)(17,31)(18,30)(19,29)(20,28)(21,27)(22,26)(23,25)(24,48), (1,9,17)(2,10,18)(3,11,19)(4,12,20)(5,13,21)(6,14,22)(7,15,23)(8,16,24)(25,41,33)(26,42,34)(27,43,35)(28,44,36)(29,45,37)(30,46,38)(31,47,39)(32,48,40), (2,18)(3,11)(5,21)(6,14)(8,24)(9,17)(12,20)(15,23)(25,29)(26,46)(27,39)(28,32)(30,42)(31,35)(33,45)(34,38)(36,48)(37,41)(40,44)(43,47) );

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)], [(1,47),(2,46),(3,45),(4,44),(5,43),(6,42),(7,41),(8,40),(9,39),(10,38),(11,37),(12,36),(13,35),(14,34),(15,33),(16,32),(17,31),(18,30),(19,29),(20,28),(21,27),(22,26),(23,25),(24,48)], [(1,9,17),(2,10,18),(3,11,19),(4,12,20),(5,13,21),(6,14,22),(7,15,23),(8,16,24),(25,41,33),(26,42,34),(27,43,35),(28,44,36),(29,45,37),(30,46,38),(31,47,39),(32,48,40)], [(2,18),(3,11),(5,21),(6,14),(8,24),(9,17),(12,20),(15,23),(25,29),(26,46),(27,39),(28,32),(30,42),(31,35),(33,45),(34,38),(36,48),(37,41),(40,44),(43,47)]])

36 conjugacy classes

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

36 irreducible representations

dim	1	1	1	1	1	1	2	2	2	2	2	2	2	2	4	4	4	4	4	4	4
type	+	+	+	+	+	+	+	+	+	+	+	+	+		+	+	+	-	+
image	C₁	C₂	C₂	C₂	C₂	C₂	S₃	S₃	D₄	D₄	D₆	D₆	D₆	C₄○D₈	S₃²	S₃×D₄	C₂×S₃²	D₈⋊₃S₃	D₂₄⋊C₂	D₆⋊D₆	D₂₄⋊₅S₃
kernel	D₂₄⋊₅S₃	Dic₆⋊S₃	C₃×D₂₄	C₃×Dic₁₂	C₈×C₃⋊S₃	D₁₂⋊S₃	D₂₄	Dic₁₂	C₃⋊Dic₃	C₂×C₃⋊S₃	C₂₄	Dic₆	D₁₂	C₃²	C₈	C₆	C₄	C₃	C₃	C₂	C₁
# reps	1	2	1	1	1	2	1	1	1	1	2	2	2	4	1	2	1	2	2	2	4

Matrix representation of D₂₄⋊₅S₃ ►in GL₆(𝔽₇₃)

22	0	0	0	0	0
18	10	0	0	0	0
0	0	0	1	0	0
0	0	72	1	0	0
0	0	0	0	1	0
0	0	0	0	0	1

51	39	0	0	0	0
55	22	0	0	0	0
0	0	1	72	0	0
0	0	0	72	0	0
0	0	0	0	1	0
0	0	0	0	0	1

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

1	0	0	0	0	0
3	72	0	0	0	0
0	0	0	1	0	0
0	0	1	0	0	0
0	0	0	0	0	1
0	0	0	0	1	0

G:=sub<GL(6,GF(73))| [22,18,0,0,0,0,0,10,0,0,0,0,0,0,0,72,0,0,0,0,1,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[51,55,0,0,0,0,39,22,0,0,0,0,0,0,1,0,0,0,0,0,72,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,72,0,0,0,0,1,0],[1,3,0,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;

D₂₄⋊₅S₃ in GAP, Magma, Sage, TeX

D_{24}\rtimes_5S_3

% in TeX

G:=Group("D24:5S3");

// GroupNames label

G:=SmallGroup(288,458);

// by ID

G=gap.SmallGroup(288,458);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,120,254,135,142,675,346,80,1356,9414]);

// Polycyclic

G:=Group<a,b,c,d|a^24=b^2=c^3=d^2=1,b*a*b=a^-1,a*c=c*a,d*a*d=a^17,b*c=c*b,d*b*d=a^4*b,d*c*d=c^-1>;

// generators/relations

G = D24⋊5S3 order 288 = 25·32

5th semidirect product of D24 and S3 acting via S3/C3=C2

G = D₂₄⋊₅S₃ order 288 = 2⁵·3²

5^th semidirect product of D₂₄ and S₃ acting via S₃/C₃=C₂