C3^2xC8:C2^2 - GroupNames

G = C₃²×C₈⋊C₂² order 288 = 2⁵·3²

Direct product of C₃² and C₈⋊C₂²

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

Aliases: C₃²×C₈⋊C₂², C₈⋊C₆², D₄⋊₂C₆², Q₈⋊₃C₆², C₆².₉₉D₄, C₂₄⋊₆(C₂×C₆), (C₃×D₈)⋊₆C₆, D₈⋊₂(C₃×C₆), (C₆×D₄)⋊₁₄C₆, C₆.₉₅(C₆×D₄), (C₃×SD₁₆)⋊₅C₆, SD₁₆⋊₁(C₃×C₆), C₁₂.₈₀(C₃×D₄), C₄.₅(C₂×C₆²), (C₂×C₄).₆C₆², (C₃×C₂₄)⋊₁₈C₂², (C₃×C₁₂).₁₈₁D₄, (C₃²×D₈)⋊₁₀C₂, (C₃×M₄(2))⋊₃C₆, M₄(2)⋊₁(C₃×C₆), C₄.₁₄(D₄×C₃²), C₁₂.₅₉(C₂²×C₆), (C₃²×SD₁₆)⋊₉C₂, C₂².₅(D₄×C₃²), (C₃×C₁₂).₁₈₉C₂³, (C₆×C₁₂).₂₇₅C₂², (D₄×C₃²)⋊₂₉C₂², (C₃²×M₄(2))⋊₅C₂, (Q₈×C₃²)⋊₂₆C₂², (D₄×C₃×C₆)⋊₂₃C₂, C₂.₁₅(D₄×C₃×C₆), C₄○D₄⋊₄(C₃×C₆), (C₂×D₄)⋊₅(C₃×C₆), (C₃×C₄○D₄)⋊₁₁C₆, (C₃×D₄)⋊₁₁(C₂×C₆), (C₃×Q₈)⋊₁₂(C₂×C₆), (C₂×C₆).₃₄(C₃×D₄), (C₂×C₁₂).₇₆(C₂×C₆), (C₃×C₆).₃₁₂(C₂×D₄), (C₃²×C₄○D₄)⋊₁₂C₂, SmallGroup(288,833)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₄ — C₃²×C₈⋊C₂²

Chief series

C₁ — C₂ — C₄ — C₁₂ — C₃×C₁₂ — D₄×C₃² — C₃²×D₈ — C₃²×C₈⋊C₂²

Lower central

C₁ — C₂ — C₄ — C₃²×C₈⋊C₂²

Upper central

C₁ — C₃×C₆ — C₆×C₁₂ — C₃²×C₈⋊C₂²

Generators and relations for C₃²×C₈⋊C₂²
G = < a,b,c,d,e | a³=b³=c⁸=d²=e²=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd=c³, ece=c⁵, de=ed >

Subgroups: 348 in 204 conjugacy classes, 120 normal (24 characteristic)
C₁, C₂, C₂, C₃, C₄, C₄, C₂², C₂², C₆, C₆, C₈, C₂×C₄, C₂×C₄, D₄, D₄, D₄, Q₈, C₂³, C₃², C₁₂, C₁₂, C₂×C₆, C₂×C₆, M₄(2), D₈, SD₁₆, C₂×D₄, C₄○D₄, C₃×C₆, C₃×C₆, C₂₄, C₂×C₁₂, C₂×C₁₂, C₃×D₄, C₃×D₄, C₃×Q₈, C₂²×C₆, C₈⋊C₂², C₃×C₁₂, C₃×C₁₂, C₆², C₆², C₃×M₄(2), C₃×D₈, C₃×SD₁₆, C₆×D₄, C₃×C₄○D₄, C₃×C₂₄, C₆×C₁₂, C₆×C₁₂, D₄×C₃², D₄×C₃², D₄×C₃², Q₈×C₃², C₂×C₆², C₃×C₈⋊C₂², C₃²×M₄(2), C₃²×D₈, C₃²×SD₁₆, D₄×C₃×C₆, C₃²×C₄○D₄, C₃²×C₈⋊C₂²
Quotients: C₁, C₂, C₃, C₂², C₆, D₄, C₂³, C₃², C₂×C₆, C₂×D₄, C₃×C₆, C₃×D₄, C₂²×C₆, C₈⋊C₂², C₆², C₆×D₄, D₄×C₃², C₂×C₆², C₃×C₈⋊C₂², D₄×C₃×C₆, C₃²×C₈⋊C₂²

Smallest permutation representation of C₃²×C₈⋊C₂²
►On 72 points

Generators in S₇₂

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

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

(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)

(2 4)(3 7)(6 8)(10 12)(11 15)(14 16)(17 23)(19 21)(20 24)(25 31)(27 29)(28 32)(33 39)(35 37)(36 40)(41 47)(43 45)(44 48)(49 53)(50 56)(52 54)(57 61)(58 64)(60 62)(65 69)(66 72)(68 70)

(2 6)(4 8)(10 14)(12 16)(17 21)(19 23)(25 29)(27 31)(33 37)(35 39)(41 45)(43 47)(50 54)(52 56)(58 62)(60 64)(66 70)(68 72)

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

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

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

99 conjugacy classes

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

99 irreducible representations

dim	1	1	1	1	1	1	1	1	1	1	1	1	2	2	2	2	4	4
type	+	+	+	+	+	+							+	+			+
image	C₁	C₂	C₂	C₂	C₂	C₂	C₃	C₆	C₆	C₆	C₆	C₆	D₄	D₄	C₃×D₄	C₃×D₄	C₈⋊C₂²	C₃×C₈⋊C₂²
kernel	C₃²×C₈⋊C₂²	C₃²×M₄(2)	C₃²×D₈	C₃²×SD₁₆	D₄×C₃×C₆	C₃²×C₄○D₄	C₃×C₈⋊C₂²	C₃×M₄(2)	C₃×D₈	C₃×SD₁₆	C₆×D₄	C₃×C₄○D₄	C₃×C₁₂	C₆²	C₁₂	C₂×C₆	C₃²	C₃
# reps	1	1	2	2	1	1	8	8	16	16	8	8	1	1	8	8	1	8

Matrix representation of C₃²×C₈⋊C₂² ►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

64	0	0	0	0	0
0	64	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

3	71	0	0	0	0
5	70	0	0	0	0
0	0	15	0	1	39
0	0	56	0	0	19
0	0	72	1	0	57
0	0	71	0	0	58

1	0	0	0	0	0
3	72	0	0	0	0
0	0	1	0	58	15
0	0	0	72	17	0
0	0	0	0	1	0
0	0	0	0	2	72

1	0	0	0	0	0
0	1	0	0	0	0
0	0	1	0	0	15
0	0	0	1	0	56
0	0	0	0	72	0
0	0	0	0	0	72

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],[64,0,0,0,0,0,0,64,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[3,5,0,0,0,0,71,70,0,0,0,0,0,0,15,56,72,71,0,0,0,0,1,0,0,0,1,0,0,0,0,0,39,19,57,58],[1,3,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,58,17,1,2,0,0,15,0,0,72],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,15,56,0,72] >;

C₃²×C₈⋊C₂² in GAP, Magma, Sage, TeX

C_3^2\times C_8\rtimes C_2^2

% in TeX

G:=Group("C3^2xC8:C2^2");

// GroupNames label

G:=SmallGroup(288,833);

// by ID

G=gap.SmallGroup(288,833);

# by ID

G:=PCGroup([7,-2,-2,-2,-3,-3,-2,-2,1037,3110,9077,4548,124]);

// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^3=c^8=d^2=e^2=1,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=c^3,e*c*e=c^5,d*e=e*d>;

// generators/relations

G = C32×C8⋊C22 order 288 = 25·32

Direct product of C32 and C8⋊C22

G = C₃²×C₈⋊C₂² order 288 = 2⁵·3²

Direct product of C₃² and C₈⋊C₂²