C2^2xC3^2:2C8 - GroupNames

G = C₂²×C₃²⋊₂C₈ order 288 = 2⁵·3²

Direct product of C₂² and C₃²⋊₂C₈

direct product, metabelian, soluble, monomial, A-group

Aliases: C₂²×C₃²⋊₂C₈, C₆²⋊₄C₈, (C₂×C₆²).₆C₄, C₃²⋊₆(C₂²×C₈), C₆².₁₉(C₂×C₄), C₂³.₄(C₃²⋊C₄), C₃⋊Dic₃.₃₄C₂³, (C₃×C₆)⋊₅(C₂×C₈), C₂.₃(C₂²×C₃²⋊C₄), (C₂×C₃⋊Dic₃).₂₃C₄, C₃⋊Dic₃.₅₄(C₂×C₄), (C₃×C₆).₃₄(C₂²×C₄), C₂².₁₉(C₂×C₃²⋊C₄), (C₂²×C₃⋊Dic₃).₁₃C₂, (C₂×C₃⋊Dic₃).₁₇₅C₂², SmallGroup(288,939)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃² — C₂²×C₃²⋊₂C₈

Chief series

C₁ — C₃² — C₃×C₆ — C₃⋊Dic₃ — C₃²⋊₂C₈ — C₂×C₃²⋊₂C₈ — C₂²×C₃²⋊₂C₈

Lower central

C₃² — C₂²×C₃²⋊₂C₈

Upper central

C₁ — C₂³

Generators and relations for C₂²×C₃²⋊₂C₈
G = < a,b,c,d,e | a²=b²=c³=d³=e⁸=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, ede^-1=cd=dc, ece^-1=c^-1d >

Subgroups: 448 in 130 conjugacy classes, 54 normal (9 characteristic)
C₁, C₂, C₂, C₃, C₄, C₂², C₆, C₈, C₂×C₄, C₂³, C₃², Dic₃, C₂×C₆, C₂×C₈, C₂²×C₄, C₃×C₆, C₃×C₆, C₂×Dic₃, C₂²×C₆, C₂²×C₈, C₃⋊Dic₃, C₃⋊Dic₃, C₆², C₂²×Dic₃, C₃²⋊₂C₈, C₂×C₃⋊Dic₃, C₂×C₆², C₂×C₃²⋊₂C₈, C₂²×C₃⋊Dic₃, C₂²×C₃²⋊₂C₈
Quotients: C₁, C₂, C₄, C₂², C₈, C₂×C₄, C₂³, C₂×C₈, C₂²×C₄, C₂²×C₈, C₃²⋊C₄, C₃²⋊₂C₈, C₂×C₃²⋊C₄, C₂×C₃²⋊₂C₈, C₂²×C₃²⋊C₄, C₂²×C₃²⋊₂C₈

Smallest permutation representation of C₂²×C₃²⋊₂C₈
►On 96 points

Generators in S₉₆

(1 43)(2 44)(3 45)(4 46)(5 47)(6 48)(7 41)(8 42)(9 19)(10 20)(11 21)(12 22)(13 23)(14 24)(15 17)(16 18)(25 38)(26 39)(27 40)(28 33)(29 34)(30 35)(31 36)(32 37)(49 61)(50 62)(51 63)(52 64)(53 57)(54 58)(55 59)(56 60)(65 76)(66 77)(67 78)(68 79)(69 80)(70 73)(71 74)(72 75)(81 91)(82 92)(83 93)(84 94)(85 95)(86 96)(87 89)(88 90)

(1 15)(2 16)(3 9)(4 10)(5 11)(6 12)(7 13)(8 14)(17 43)(18 44)(19 45)(20 46)(21 47)(22 48)(23 41)(24 42)(25 88)(26 81)(27 82)(28 83)(29 84)(30 85)(31 86)(32 87)(33 93)(34 94)(35 95)(36 96)(37 89)(38 90)(39 91)(40 92)(49 76)(50 77)(51 78)(52 79)(53 80)(54 73)(55 74)(56 75)(57 69)(58 70)(59 71)(60 72)(61 65)(62 66)(63 67)(64 68)

(2 75 27)(4 29 77)(6 79 31)(8 25 73)(10 84 50)(12 52 86)(14 88 54)(16 56 82)(18 60 92)(20 94 62)(22 64 96)(24 90 58)(34 66 46)(36 48 68)(38 70 42)(40 44 72)

(1 74 26)(2 75 27)(3 28 76)(4 29 77)(5 78 30)(6 79 31)(7 32 80)(8 25 73)(9 83 49)(10 84 50)(11 51 85)(12 52 86)(13 87 53)(14 88 54)(15 55 81)(16 56 82)(17 59 91)(18 60 92)(19 93 61)(20 94 62)(21 63 95)(22 64 96)(23 89 57)(24 90 58)(33 65 45)(34 66 46)(35 47 67)(36 48 68)(37 69 41)(38 70 42)(39 43 71)(40 44 72)

(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)(73 74 75 76 77 78 79 80)(81 82 83 84 85 86 87 88)(89 90 91 92 93 94 95 96)

G:=sub<Sym(96)| (1,43)(2,44)(3,45)(4,46)(5,47)(6,48)(7,41)(8,42)(9,19)(10,20)(11,21)(12,22)(13,23)(14,24)(15,17)(16,18)(25,38)(26,39)(27,40)(28,33)(29,34)(30,35)(31,36)(32,37)(49,61)(50,62)(51,63)(52,64)(53,57)(54,58)(55,59)(56,60)(65,76)(66,77)(67,78)(68,79)(69,80)(70,73)(71,74)(72,75)(81,91)(82,92)(83,93)(84,94)(85,95)(86,96)(87,89)(88,90), (1,15)(2,16)(3,9)(4,10)(5,11)(6,12)(7,13)(8,14)(17,43)(18,44)(19,45)(20,46)(21,47)(22,48)(23,41)(24,42)(25,88)(26,81)(27,82)(28,83)(29,84)(30,85)(31,86)(32,87)(33,93)(34,94)(35,95)(36,96)(37,89)(38,90)(39,91)(40,92)(49,76)(50,77)(51,78)(52,79)(53,80)(54,73)(55,74)(56,75)(57,69)(58,70)(59,71)(60,72)(61,65)(62,66)(63,67)(64,68), (2,75,27)(4,29,77)(6,79,31)(8,25,73)(10,84,50)(12,52,86)(14,88,54)(16,56,82)(18,60,92)(20,94,62)(22,64,96)(24,90,58)(34,66,46)(36,48,68)(38,70,42)(40,44,72), (1,74,26)(2,75,27)(3,28,76)(4,29,77)(5,78,30)(6,79,31)(7,32,80)(8,25,73)(9,83,49)(10,84,50)(11,51,85)(12,52,86)(13,87,53)(14,88,54)(15,55,81)(16,56,82)(17,59,91)(18,60,92)(19,93,61)(20,94,62)(21,63,95)(22,64,96)(23,89,57)(24,90,58)(33,65,45)(34,66,46)(35,47,67)(36,48,68)(37,69,41)(38,70,42)(39,43,71)(40,44,72), (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)(73,74,75,76,77,78,79,80)(81,82,83,84,85,86,87,88)(89,90,91,92,93,94,95,96)>;

G:=Group( (1,43)(2,44)(3,45)(4,46)(5,47)(6,48)(7,41)(8,42)(9,19)(10,20)(11,21)(12,22)(13,23)(14,24)(15,17)(16,18)(25,38)(26,39)(27,40)(28,33)(29,34)(30,35)(31,36)(32,37)(49,61)(50,62)(51,63)(52,64)(53,57)(54,58)(55,59)(56,60)(65,76)(66,77)(67,78)(68,79)(69,80)(70,73)(71,74)(72,75)(81,91)(82,92)(83,93)(84,94)(85,95)(86,96)(87,89)(88,90), (1,15)(2,16)(3,9)(4,10)(5,11)(6,12)(7,13)(8,14)(17,43)(18,44)(19,45)(20,46)(21,47)(22,48)(23,41)(24,42)(25,88)(26,81)(27,82)(28,83)(29,84)(30,85)(31,86)(32,87)(33,93)(34,94)(35,95)(36,96)(37,89)(38,90)(39,91)(40,92)(49,76)(50,77)(51,78)(52,79)(53,80)(54,73)(55,74)(56,75)(57,69)(58,70)(59,71)(60,72)(61,65)(62,66)(63,67)(64,68), (2,75,27)(4,29,77)(6,79,31)(8,25,73)(10,84,50)(12,52,86)(14,88,54)(16,56,82)(18,60,92)(20,94,62)(22,64,96)(24,90,58)(34,66,46)(36,48,68)(38,70,42)(40,44,72), (1,74,26)(2,75,27)(3,28,76)(4,29,77)(5,78,30)(6,79,31)(7,32,80)(8,25,73)(9,83,49)(10,84,50)(11,51,85)(12,52,86)(13,87,53)(14,88,54)(15,55,81)(16,56,82)(17,59,91)(18,60,92)(19,93,61)(20,94,62)(21,63,95)(22,64,96)(23,89,57)(24,90,58)(33,65,45)(34,66,46)(35,47,67)(36,48,68)(37,69,41)(38,70,42)(39,43,71)(40,44,72), (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)(73,74,75,76,77,78,79,80)(81,82,83,84,85,86,87,88)(89,90,91,92,93,94,95,96) );

G=PermutationGroup([[(1,43),(2,44),(3,45),(4,46),(5,47),(6,48),(7,41),(8,42),(9,19),(10,20),(11,21),(12,22),(13,23),(14,24),(15,17),(16,18),(25,38),(26,39),(27,40),(28,33),(29,34),(30,35),(31,36),(32,37),(49,61),(50,62),(51,63),(52,64),(53,57),(54,58),(55,59),(56,60),(65,76),(66,77),(67,78),(68,79),(69,80),(70,73),(71,74),(72,75),(81,91),(82,92),(83,93),(84,94),(85,95),(86,96),(87,89),(88,90)], [(1,15),(2,16),(3,9),(4,10),(5,11),(6,12),(7,13),(8,14),(17,43),(18,44),(19,45),(20,46),(21,47),(22,48),(23,41),(24,42),(25,88),(26,81),(27,82),(28,83),(29,84),(30,85),(31,86),(32,87),(33,93),(34,94),(35,95),(36,96),(37,89),(38,90),(39,91),(40,92),(49,76),(50,77),(51,78),(52,79),(53,80),(54,73),(55,74),(56,75),(57,69),(58,70),(59,71),(60,72),(61,65),(62,66),(63,67),(64,68)], [(2,75,27),(4,29,77),(6,79,31),(8,25,73),(10,84,50),(12,52,86),(14,88,54),(16,56,82),(18,60,92),(20,94,62),(22,64,96),(24,90,58),(34,66,46),(36,48,68),(38,70,42),(40,44,72)], [(1,74,26),(2,75,27),(3,28,76),(4,29,77),(5,78,30),(6,79,31),(7,32,80),(8,25,73),(9,83,49),(10,84,50),(11,51,85),(12,52,86),(13,87,53),(14,88,54),(15,55,81),(16,56,82),(17,59,91),(18,60,92),(19,93,61),(20,94,62),(21,63,95),(22,64,96),(23,89,57),(24,90,58),(33,65,45),(34,66,46),(35,47,67),(36,48,68),(37,69,41),(38,70,42),(39,43,71),(40,44,72)], [(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),(73,74,75,76,77,78,79,80),(81,82,83,84,85,86,87,88),(89,90,91,92,93,94,95,96)]])

48 conjugacy classes

class	1	2A	···	2G	3A	3B	4A	···	4H	6A	···	6N	8A	···	8P
order	1	2	···	2	3	3	4	···	4	6	···	6	8	···	8
size	1	1	···	1	4	4	9	···	9	4	···	4	9	···	9

48 irreducible representations

dim	1	1	1	1	1	1	4	4	4
type	+	+	+				+	-	+
image	C₁	C₂	C₂	C₄	C₄	C₈	C₃²⋊C₄	C₃²⋊₂C₈	C₂×C₃²⋊C₄
kernel	C₂²×C₃²⋊₂C₈	C₂×C₃²⋊₂C₈	C₂²×C₃⋊Dic₃	C₂×C₃⋊Dic₃	C₂×C₆²	C₆²	C₂³	C₂²	C₂²
# reps	1	6	1	6	2	16	2	8	6

Matrix representation of C₂²×C₃²⋊₂C₈ ►in GL₆(𝔽₇₃)

72	0	0	0	0	0
0	72	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	1	0	0	0	0
0	0	72	0	0	0
0	0	0	72	0	0
0	0	0	0	72	0
0	0	0	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	0	1
0	0	46	46	72	72

1	0	0	0	0	0
0	1	0	0	0	0
0	0	0	72	0	0
0	0	1	72	0	0
0	0	0	27	0	1
0	0	46	0	72	72

63	0	0	0	0	0
0	72	0	0	0	0
0	0	0	0	72	1
0	0	46	46	71	72
0	0	32	19	27	0
0	0	19	21	27	0

G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[72,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,46,0,0,0,1,0,46,0,0,0,0,0,72,0,0,0,0,1,72],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,46,0,0,72,72,27,0,0,0,0,0,0,72,0,0,0,0,1,72],[63,0,0,0,0,0,0,72,0,0,0,0,0,0,0,46,32,19,0,0,0,46,19,21,0,0,72,71,27,27,0,0,1,72,0,0] >;

C₂²×C₃²⋊₂C₈ in GAP, Magma, Sage, TeX

C_2^2\times C_3^2\rtimes_2C_8

% in TeX

G:=Group("C2^2xC3^2:2C8");

// GroupNames label

G:=SmallGroup(288,939);

// by ID

G=gap.SmallGroup(288,939);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,56,80,9413,362,12550,1203]);

// Polycyclic

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

// generators/relations

G = C22×C32⋊2C8 order 288 = 25·32

Direct product of C22 and C32⋊2C8

G = C₂²×C₃²⋊₂C₈ order 288 = 2⁵·3²

Direct product of C₂² and C₃²⋊₂C₈