C3xC6^2:5C4 - GroupNames

G = C₃xC₆²:₅C₄ order 432 = 2⁴·3³

Direct product of C₃ and C₆²:₅C₄

direct product, metabelian, supersoluble, monomial

Aliases: C₃xC₆²:₅C₄, C₆³.₃C₂, C₆²:₁₅C₁₂, C₆²:₁₄Dic₃, C₆².₁₄₄D₆, (C₃xC₆²):₇C₄, C₆².₆₈(C₂xC₆), (C₂xC₆²).₂₇C₆, (C₂xC₆²).₂₇S₃, C₆.₂₈(C₆xDic₃), (C₃²xC₆).₇₇D₄, C₃³:₁₈(C₂²:C₄), C₆.₃₉(C₃²:₇D₄), (C₃xC₆²).₅₀C₂², C₃²:₁₀(C₆.D₄), (C₂xC₆).₇₁(S₃xC₆), (C₆xC₃:Dic₃):₆C₂, (C₂xC₃:Dic₃):₉C₆, (C₂xC₆):₆(C₃xDic₃), (C₃xC₆).₇₃(C₃xD₄), C₆.₄₂(C₃xC₃:D₄), C₂³.₃(C₃xC₃:S₃), C₂².₇(C₆xC₃:S₃), C₂.₅(C₆xC₃:Dic₃), (C₃xC₆).₆₅(C₂xC₁₂), (C₂xC₆):₃(C₃:Dic₃), C₃:₂(C₃xC₆.D₄), C₆.₂₄(C₂xC₃:Dic₃), C₂²:₃(C₃xC₃:Dic₃), C₂.₃(C₃xC₃²:₇D₄), C₃²:₁₁(C₃xC₂²:C₄), (C₂²xC₆).₃₃(C₃xS₃), (C₃²xC₆).₇₁(C₂xC₄), (C₃xC₆).₇₀(C₂xDic₃), (C₃xC₆).₁₁₂(C₃:D₄), (C₂²xC₆).₁₁(C₃:S₃), (C₂xC₆).₆₅(C₂xC₃:S₃), SmallGroup(432,495)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃xC₆ — C₃xC₆²:₅C₄

Chief series

C₁ — C₃ — C₃² — C₃xC₆ — C₆² — C₃xC₆² — C₆xC₃:Dic₃ — C₃xC₆²:₅C₄

Lower central

C₃² — C₃xC₆ — C₃xC₆²:₅C₄

Upper central

C₁ — C₂xC₆ — C₂²xC₆

Generators and relations for C₃xC₆²:₅C₄
G = < a,b,c,d | a³=b⁶=c⁶=d⁴=1, ab=ba, ac=ca, ad=da, bc=cb, dbd^-1=b^-1c³, dcd^-1=c^-1 >

Subgroups: 756 in 332 conjugacy classes, 102 normal (18 characteristic)
C₁, C₂, C₂, C₂, C₃, C₃, C₃, C₄, C₂², C₂², C₂², C₆, C₆, C₆, C₂xC₄, C₂³, C₃², C₃², C₃², Dic₃, C₁₂, C₂xC₆, C₂xC₆, C₂xC₆, C₂²:C₄, C₃xC₆, C₃xC₆, C₃xC₆, C₂xDic₃, C₂xC₁₂, C₂²xC₆, C₂²xC₆, C₂²xC₆, C₃³, C₃xDic₃, C₃:Dic₃, C₆², C₆², C₆², C₆.D₄, C₃xC₂²:C₄, C₃²xC₆, C₃²xC₆, C₃²xC₆, C₆xDic₃, C₂xC₃:Dic₃, C₂xC₆², C₂xC₆², C₂xC₆², C₃xC₃:Dic₃, C₃xC₆², C₃xC₆², C₃xC₆², C₃xC₆.D₄, C₆²:₅C₄, C₆xC₃:Dic₃, C₆³, C₃xC₆²:₅C₄
Quotients: C₁, C₂, C₃, C₄, C₂², S₃, C₆, C₂xC₄, D₄, Dic₃, C₁₂, D₆, C₂xC₆, C₂²:C₄, C₃xS₃, C₃:S₃, C₂xDic₃, C₃:D₄, C₂xC₁₂, C₃xD₄, C₃xDic₃, C₃:Dic₃, S₃xC₆, C₂xC₃:S₃, C₆.D₄, C₃xC₂²:C₄, C₃xC₃:S₃, C₆xDic₃, C₃xC₃:D₄, C₂xC₃:Dic₃, C₃²:₇D₄, C₃xC₃:Dic₃, C₆xC₃:S₃, C₃xC₆.D₄, C₆²:₅C₄, C₆xC₃:Dic₃, C₃xC₃²:₇D₄, C₃xC₆²:₅C₄

Smallest permutation representation of C₃xC₆²:₅C₄
►On 72 points

Generators in S₇₂

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

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

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

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

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

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

126 conjugacy classes

class	1	2A	2B	2C	2D	2E	3A	3B	3C	···	3N	4A	4B	4C	4D	6A	···	6F	6G	···	6CP	12A	···	12H
order	1	2	2	2	2	2	3	3	3	···	3	4	4	4	4	6	···	6	6	···	6	12	···	12
size	1	1	1	1	2	2	1	1	2	···	2	18	18	18	18	1	···	1	2	···	2	18	···	18

126 irreducible representations

dim	1	1	1	1	1	1	1	1	2	2	2	2	2	2	2	2	2	2
type	+	+	+						+	+	-	+
image	C₁	C₂	C₂	C₃	C₄	C₆	C₆	C₁₂	S₃	D₄	Dic₃	D₆	C₃xS₃	C₃:D₄	C₃xD₄	C₃xDic₃	S₃xC₆	C₃xC₃:D₄
kernel	C₃xC₆²:₅C₄	C₆xC₃:Dic₃	C₆³	C₆²:₅C₄	C₃xC₆²	C₂xC₃:Dic₃	C₂xC₆²	C₆²	C₂xC₆²	C₃²xC₆	C₆²	C₆²	C₂²xC₆	C₃xC₆	C₃xC₆	C₂xC₆	C₂xC₆	C₆
# reps	1	2	1	2	4	4	2	8	4	2	8	4	8	16	4	16	8	32

Matrix representation of C₃xC₆²:₅C₄ ►in GL₆(F₁₃)

9	0	0	0	0	0
0	9	0	0	0	0
0	0	3	0	0	0
0	0	0	3	0	0
0	0	0	0	9	0
0	0	0	0	0	9

9	0	0	0	0	0
0	3	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	12

1	0	0	0	0	0
0	1	0	0	0	0
0	0	9	0	0	0
0	0	0	3	0	0
0	0	0	0	10	0
0	0	0	0	0	4

0	5	0	0	0	0
8	0	0	0	0	0
0	0	0	1	0	0
0	0	12	0	0	0
0	0	0	0	0	1
0	0	0	0	1	0

G:=sub<GL(6,GF(13))| [9,0,0,0,0,0,0,9,0,0,0,0,0,0,3,0,0,0,0,0,0,3,0,0,0,0,0,0,9,0,0,0,0,0,0,9],[9,0,0,0,0,0,0,3,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,9,0,0,0,0,0,0,3,0,0,0,0,0,0,10,0,0,0,0,0,0,4],[0,8,0,0,0,0,5,0,0,0,0,0,0,0,0,12,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;

C₃xC₆²:₅C₄ in GAP, Magma, Sage, TeX

C_3\times C_6^2\rtimes_5C_4

% in TeX

G:=Group("C3xC6^2:5C4");

// GroupNames label

G:=SmallGroup(432,495);

// by ID

G=gap.SmallGroup(432,495);

# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-3,-3,84,365,4037,14118]);

// Polycyclic

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

// generators/relations

G = C3xC62:5C4 order 432 = 24·33

Direct product of C3 and C62:5C4

G = C₃xC₆²:₅C₄ order 432 = 2⁴·3³

Direct product of C₃ and C₆²:₅C₄