C3^2xC4.D4 - GroupNames

G = C₃²×C₄.D₄ order 288 = 2⁵·3²

Direct product of C₃² and C₄.D₄

direct product, metabelian, nilpotent (class 3), monomial

Aliases: C₃²×C₄.D₄, C₂³.(C₃×C₁₂), (C₆×D₄).₁₉C₆, C₁₂.₇₅(C₃×D₄), (C₂×C₄).₁C₆², (C₂×C₆²).₁C₄, C₄.₉(D₄×C₃²), (C₃×C₁₂).₁₇₆D₄, (C₃×M₄(2))⋊₉C₆, M₄(2)⋊₃(C₃×C₆), (C₂²×C₆).₄C₁₂, C₆².₈₉(C₂×C₄), C₂².₃(C₆×C₁₂), (C₆×C₁₂).₂₅₈C₂², (C₃²×M₄(2))⋊₁₅C₂, (D₄×C₃×C₆).₁₄C₂, (C₂×D₄).₂(C₃×C₆), (C₂×C₁₂).₆₇(C₂×C₆), (C₂×C₆).₃₀(C₂×C₁₂), C₆.₃₁(C₃×C₂²⋊C₄), C₂.₄(C₃²×C₂²⋊C₄), (C₃×C₆).₈₀(C₂²⋊C₄), SmallGroup(288,318)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂² — C₃²×C₄.D₄

Chief series

C₁ — C₂ — C₄ — C₂×C₄ — C₂×C₁₂ — C₆×C₁₂ — C₃²×M₄(2) — C₃²×C₄.D₄

Lower central

C₁ — C₂ — C₂² — C₃²×C₄.D₄

Upper central

C₁ — C₃×C₆ — C₆×C₁₂ — C₃²×C₄.D₄

Generators and relations for C₃²×C₄.D₄
G = < a,b,c,d,e | a³=b³=c⁴=1, d⁴=c², e²=c, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd^-1=c^-1, ce=ec, ede^-1=c^-1d³ >

Subgroups: 252 in 138 conjugacy classes, 72 normal (12 characteristic)
C₁, C₂, C₂, C₃, C₄, C₂², C₂², C₆, C₆, C₈, C₂×C₄, D₄, C₂³, C₃², C₁₂, C₂×C₆, C₂×C₆, M₄(2), C₂×D₄, C₃×C₆, C₃×C₆, C₂₄, C₂×C₁₂, C₃×D₄, C₂²×C₆, C₄.D₄, C₃×C₁₂, C₆², C₆², C₃×M₄(2), C₆×D₄, C₃×C₂₄, C₆×C₁₂, D₄×C₃², C₂×C₆², C₃×C₄.D₄, C₃²×M₄(2), D₄×C₃×C₆, C₃²×C₄.D₄
Quotients: C₁, C₂, C₃, C₄, C₂², C₆, C₂×C₄, D₄, C₃², C₁₂, C₂×C₆, C₂²⋊C₄, C₃×C₆, C₂×C₁₂, C₃×D₄, C₄.D₄, C₃×C₁₂, C₆², C₃×C₂²⋊C₄, C₆×C₁₂, D₄×C₃², C₃×C₄.D₄, C₃²×C₂²⋊C₄, C₃²×C₄.D₄

Smallest permutation representation of C₃²×C₄.D₄
►On 72 points

Generators in S₇₂

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

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

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

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

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

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

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

99 conjugacy classes

class	1	2A	2B	2C	2D	3A	···	3H	4A	4B	6A	···	6H	6I	···	6P	6Q	···	6AF	8A	8B	8C	8D	12A	···	12P	24A	···	24AF
order	1	2	2	2	2	3	···	3	4	4	6	···	6	6	···	6	6	···	6	8	8	8	8	12	···	12	24	···	24
size	1	1	2	4	4	1	···	1	2	2	1	···	1	2	···	2	4	···	4	4	4	4	4	2	···	2	4	···	4

99 irreducible representations

dim	1	1	1	1	1	1	1	1	2	2	4	4
type	+	+	+						+		+
image	C₁	C₂	C₂	C₃	C₄	C₆	C₆	C₁₂	D₄	C₃×D₄	C₄.D₄	C₃×C₄.D₄
kernel	C₃²×C₄.D₄	C₃²×M₄(2)	D₄×C₃×C₆	C₃×C₄.D₄	C₂×C₆²	C₃×M₄(2)	C₆×D₄	C₂²×C₆	C₃×C₁₂	C₁₂	C₃²	C₃
# reps	1	2	1	8	4	16	8	32	2	16	1	8

Matrix representation of C₃²×C₄.D₄ ►in GL₆(𝔽₇₃)

64	0	0	0	0	0
0	64	0	0	0	0
0	0	8	0	0	0
0	0	0	8	0	0
0	0	0	0	8	0
0	0	0	0	0	8

8	0	0	0	0	0
0	8	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	1

72	0	0	0	0	0
0	72	0	0	0	0
0	0	0	1	0	0
0	0	72	0	0	0
0	0	21	52	1	71
0	0	21	0	1	72

72	0	0	0	0	0
2	1	0	0	0	0
0	0	21	52	1	71
0	0	0	0	1	0
0	0	72	0	0	0
0	0	1	71	0	52

72	72	0	0	0	0
2	1	0	0	0	0
0	0	0	0	1	0
0	0	21	52	1	71
0	0	0	1	0	0
0	0	72	2	0	21

G:=sub<GL(6,GF(73))| [64,0,0,0,0,0,0,64,0,0,0,0,0,0,8,0,0,0,0,0,0,8,0,0,0,0,0,0,8,0,0,0,0,0,0,8],[8,0,0,0,0,0,0,8,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,72,21,21,0,0,1,0,52,0,0,0,0,0,1,1,0,0,0,0,71,72],[72,2,0,0,0,0,0,1,0,0,0,0,0,0,21,0,72,1,0,0,52,0,0,71,0,0,1,1,0,0,0,0,71,0,0,52],[72,2,0,0,0,0,72,1,0,0,0,0,0,0,0,21,0,72,0,0,0,52,1,2,0,0,1,1,0,0,0,0,0,71,0,21] >;

C₃²×C₄.D₄ in GAP, Magma, Sage, TeX

C_3^2\times C_4.D_4

% in TeX

G:=Group("C3^2xC4.D4");

// GroupNames label

G:=SmallGroup(288,318);

// by ID

G=gap.SmallGroup(288,318);

# by ID

G:=PCGroup([7,-2,-2,-3,-3,-2,-2,-2,504,533,6304,4548,124]);

// Polycyclic

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

// generators/relations

G = C32×C4.D4 order 288 = 25·32

Direct product of C32 and C4.D4

G = C₃²×C₄.D₄ order 288 = 2⁵·3²

Direct product of C₃² and C₄.D₄