C3xC3^2:5SD16 - GroupNames

G = C₃×C₃²⋊₅SD₁₆ order 432 = 2⁴·3³

Direct product of C₃ and C₃²⋊₅SD₁₆

direct product, metabelian, supersoluble, monomial

Aliases: C₃×C₃²⋊₅SD₁₆, C₃³⋊₁₁SD₁₆, C₁₂.₇₆S₃², C₁₂.₂₉(S₃×C₆), Dic₆⋊₁(C₃×S₃), (C₃×Dic₆)⋊₂C₆, C₆.₂₆(C₃×D₁₂), (C₃×C₆).₇₃D₁₂, C₁₂⋊S₃.₄C₆, (C₃×Dic₆)⋊₁₀S₃, (C₃×C₁₂).₁₇₄D₆, C₃²⋊₈(C₃×SD₁₆), (C₃²×C₆).₂₂D₄, (C₃²×Dic₆)⋊₂C₂, C₃²⋊₁₂(C₂₄⋊C₂), C₆.₄₄(C₃⋊D₁₂), (C₃²×C₁₂).₅C₂², C₃²⋊₁₀(Q₈⋊₂S₃), (C₃×C₃⋊C₈)⋊₃C₆, C₃⋊C₈⋊₃(C₃×S₃), C₄.₃(C₃×S₃²), (C₃×C₃⋊C₈)⋊₁₀S₃, C₃⋊₂(C₃×C₂₄⋊C₂), (C₃²×C₃⋊C₈)⋊₆C₂, C₆.₃(C₃×C₃⋊D₄), C₃⋊₁(C₃×Q₈⋊₂S₃), (C₃×C₆).₂₁(C₃×D₄), (C₃×C₁₂).₃₉(C₂×C₆), C₂.₆(C₃×C₃⋊D₁₂), (C₃×C₁₂⋊S₃).₁C₂, (C₃×C₆).₇₂(C₃⋊D₄), SmallGroup(432,422)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃×C₁₂ — C₃×C₃²⋊₅SD₁₆

Chief series

C₁ — C₃ — C₃² — C₃×C₆ — C₃×C₁₂ — C₃²×C₁₂ — C₃²×Dic₆ — C₃×C₃²⋊₅SD₁₆

Lower central

C₃² — C₃×C₆ — C₃×C₁₂ — C₃×C₃²⋊₅SD₁₆

Upper central

C₁ — C₆ — C₁₂

Generators and relations for C₃×C₃²⋊₅SD₁₆
G = < a,b,c,d,e | a³=b³=c³=d⁸=e²=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd^-1=ebe=b^-1, cd=dc, ece=c^-1, ede=d³ >

Subgroups: 520 in 122 conjugacy classes, 36 normal (all characteristic)
C₁, C₂, C₂, C₃, C₃, C₄, C₄, C₂², S₃, C₆, C₆, C₈, D₄, Q₈, C₃², C₃², Dic₃, C₁₂, C₁₂, D₆, C₂×C₆, SD₁₆, C₃×S₃, C₃⋊S₃, C₃×C₆, C₃×C₆, C₃⋊C₈, C₂₄, Dic₆, D₁₂, C₃×D₄, C₃×Q₈, C₃³, C₃×Dic₃, C₃×C₁₂, C₃×C₁₂, S₃×C₆, C₂×C₃⋊S₃, C₂₄⋊C₂, Q₈⋊₂S₃, C₃×SD₁₆, C₃×C₃⋊S₃, C₃²×C₆, C₃×C₃⋊C₈, C₃×C₃⋊C₈, C₃×C₂₄, C₃×Dic₆, C₃×Dic₆, C₃×D₁₂, C₁₂⋊S₃, Q₈×C₃², C₃²×Dic₃, C₃²×C₁₂, C₆×C₃⋊S₃, C₃²⋊₅SD₁₆, C₃×C₂₄⋊C₂, C₃×Q₈⋊₂S₃, C₃²×C₃⋊C₈, C₃²×Dic₆, C₃×C₁₂⋊S₃, C₃×C₃²⋊₅SD₁₆
Quotients: C₁, C₂, C₃, C₂², S₃, C₆, D₄, D₆, C₂×C₆, SD₁₆, C₃×S₃, D₁₂, C₃⋊D₄, C₃×D₄, S₃², S₃×C₆, C₂₄⋊C₂, Q₈⋊₂S₃, C₃×SD₁₆, C₃⋊D₁₂, C₃×D₁₂, C₃×C₃⋊D₄, C₃×S₃², C₃²⋊₅SD₁₆, C₃×C₂₄⋊C₂, C₃×Q₈⋊₂S₃, C₃×C₃⋊D₁₂, C₃×C₃²⋊₅SD₁₆

Smallest permutation representation of C₃×C₃²⋊₅SD₁₆
►On 48 points

Generators in S₄₈

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

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

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

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

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

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

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

72 conjugacy classes

class	1	2A	2B	3A	3B	3C	···	3H	3I	3J	3K	4A	4B	6A	6B	6C	···	6H	6I	6J	6K	6L	6M	8A	8B	12A	···	12H	12I	···	12Q	12R	···	12Y	24A	···	24P
order	1	2	2	3	3	3	···	3	3	3	3	4	4	6	6	6	···	6	6	6	6	6	6	8	8	12	···	12	12	···	12	12	···	12	24	···	24
size	1	1	36	1	1	2	···	2	4	4	4	2	12	1	1	2	···	2	4	4	4	36	36	6	6	2	···	2	4	···	4	12	···	12	6	···	6

72 irreducible representations

dim	1	1	1	1	1	1	1	1	2	2	2	2	2	2	2	2	2	2	2	2	2	2	2	2	4	4	4	4	4	4	4	4
type	+	+	+	+					+	+	+	+				+									+	+	+		+
image	C₁	C₂	C₂	C₂	C₃	C₆	C₆	C₆	S₃	S₃	D₄	D₆	SD₁₆	C₃×S₃	C₃×S₃	D₁₂	C₃⋊D₄	C₃×D₄	S₃×C₆	C₂₄⋊C₂	C₃×SD₁₆	C₃×D₁₂	C₃×C₃⋊D₄	C₃×C₂₄⋊C₂	S₃²	Q₈⋊₂S₃	C₃⋊D₁₂	C₃×S₃²	C₃²⋊₅SD₁₆	C₃×Q₈⋊₂S₃	C₃×C₃⋊D₁₂	C₃×C₃²⋊₅SD₁₆
kernel	C₃×C₃²⋊₅SD₁₆	C₃²×C₃⋊C₈	C₃²×Dic₆	C₃×C₁₂⋊S₃	C₃²⋊₅SD₁₆	C₃×C₃⋊C₈	C₃×Dic₆	C₁₂⋊S₃	C₃×C₃⋊C₈	C₃×Dic₆	C₃²×C₆	C₃×C₁₂	C₃³	C₃⋊C₈	Dic₆	C₃×C₆	C₃×C₆	C₃×C₆	C₁₂	C₃²	C₃²	C₆	C₆	C₃	C₁₂	C₃²	C₆	C₄	C₃	C₃	C₂	C₁
# reps	1	1	1	1	2	2	2	2	1	1	1	2	2	2	2	2	2	2	4	4	4	4	4	8	1	1	1	2	2	2	2	4

Matrix representation of C₃×C₃²⋊₅SD₁₆ ►in GL₆(𝔽₇₃)

1	0	0	0	0	0
0	1	0	0	0	0
0	0	8	0	0	0
0	0	0	8	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	0	1
0	0	0	0	72	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	1	0
0	0	0	0	0	1

51	47	0	0	0	0
0	10	0	0	0	0
0	0	72	0	0	0
0	0	0	72	0	0
0	0	0	0	1	0
0	0	0	0	72	72

21	57	0	0	0	0
64	52	0	0	0	0
0	0	0	72	0	0
0	0	72	0	0	0
0	0	0	0	1	0
0	0	0	0	72	72

G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,8,0,0,0,0,0,0,8,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,72,0,0,0,0,1,72],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,72,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[51,0,0,0,0,0,47,10,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,72,0,0,0,0,0,72],[21,64,0,0,0,0,57,52,0,0,0,0,0,0,0,72,0,0,0,0,72,0,0,0,0,0,0,0,1,72,0,0,0,0,0,72] >;

C₃×C₃²⋊₅SD₁₆ in GAP, Magma, Sage, TeX

C_3\times C_3^2\rtimes_5{\rm SD}_{16}

% in TeX

G:=Group("C3xC3^2:5SD16");

// GroupNames label

G:=SmallGroup(432,422);

// by ID

G=gap.SmallGroup(432,422);

# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-3,-3,168,197,92,1011,80,2028,14118]);

// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^3=c^3=d^8=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,d*b*d^-1=e*b*e=b^-1,c*d=d*c,e*c*e=c^-1,e*d*e=d^3>;

// generators/relations

G = C3×C32⋊5SD16 order 432 = 24·33

Direct product of C3 and C32⋊5SD16

G = C₃×C₃²⋊₅SD₁₆ order 432 = 2⁴·3³

Direct product of C₃ and C₃²⋊₅SD₁₆