C3^2:3Q32 - GroupNames

G = C₃²⋊₃Q₃₂ order 288 = 2⁵·3²

2^nd semidirect product of C₃² and Q₃₂ acting via Q₃₂/C₈=C₂²

Aliases: C₃²⋊₃Q₃₂, C₃⋊₂Dic₂₄, C₆.₁₅D₂₄, C₂₄.₅₀D₆, C₁₂.₁₂D₁₂, Dic₁₂.₁S₃, C₈.₇S₃², C₃⋊C₁₆.S₃, (C₃×C₆).₁₂D₈, C₆.₄(D₄⋊S₃), (C₃×C₁₂).₂₇D₄, C₃⋊₁(C₃⋊Q₃₂), C₄.₄(C₃⋊D₁₂), C₂.₇(C₃⋊D₂₄), C₃²⋊₅Q₁₆.₂C₂, C₁₂.₆₉(C₃⋊D₄), (C₃×C₂₄).₁₂C₂², (C₃×Dic₁₂).₃C₂, (C₃×C₃⋊C₁₆).₁C₂, SmallGroup(288,199)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃×C₂₄ — C₃²⋊₃Q₃₂

Chief series

C₁ — C₃ — C₃² — C₃×C₆ — C₃×C₁₂ — C₃×C₂₄ — C₃×Dic₁₂ — C₃²⋊₃Q₃₂

Lower central

C₃² — C₃×C₆ — C₃×C₁₂ — C₃×C₂₄ — C₃²⋊₃Q₃₂

Upper central

C₁ — C₂ — C₄ — C₈

Generators and relations for C₃²⋊₃Q₃₂
G = < a,b,c,d | a³=b³=c¹⁶=1, d²=c⁸, ab=ba, cac^-1=a^-1, ad=da, bc=cb, dbd^-1=b^-1, dcd^-1=c^-1 >

Subgroups: 258 in 57 conjugacy classes, 22 normal (all characteristic)
C₁, C₂, C₃, C₃, C₄, C₄, C₆, C₆, C₈, Q₈, C₃², Dic₃, C₁₂, C₁₂, C₁₆, Q₁₆, C₃×C₆, C₂₄, C₂₄, Dic₆, C₃×Q₈, Q₃₂, C₃×Dic₃, C₃⋊Dic₃, C₃×C₁₂, C₃⋊C₁₆, C₄₈, Dic₁₂, Dic₁₂, C₃×Q₁₆, C₃×C₂₄, C₃×Dic₆, C₃²⋊₄Q₈, Dic₂₄, C₃⋊Q₃₂, C₃×C₃⋊C₁₆, C₃×Dic₁₂, C₃²⋊₅Q₁₆, C₃²⋊₃Q₃₂
Quotients: C₁, C₂, C₂², S₃, D₄, D₆, D₈, D₁₂, C₃⋊D₄, Q₃₂, S₃², D₂₄, D₄⋊S₃, C₃⋊D₁₂, Dic₂₄, C₃⋊Q₃₂, C₃⋊D₂₄, C₃²⋊₃Q₃₂

Smallest permutation representation of C₃²⋊₃Q₃₂
►On 96 points

Generators in S₉₆

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

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

(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)

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

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

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

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

42 conjugacy classes

class	1	2	3A	3B	3C	4A	4B	4C	6A	6B	6C	8A	8B	12A	12B	12C	12D	12E	12F	12G	16A	16B	16C	16D	24A	24B	24C	24D	24E	···	24J	48A	···	48H
order	1	2	3	3	3	4	4	4	6	6	6	8	8	12	12	12	12	12	12	12	16	16	16	16	24	24	24	24	24	···	24	48	···	48
size	1	1	2	2	4	2	24	72	2	2	4	2	2	2	2	4	4	4	24	24	6	6	6	6	2	2	2	2	4	···	4	6	···	6

42 irreducible representations

dim	1	1	1	1	2	2	2	2	2	2	2	2	2	2	4	4	4	4	4	4
type	+	+	+	+	+	+	+	+	+	+		-	+	-	+	+	+	-	+	-
image	C₁	C₂	C₂	C₂	S₃	S₃	D₄	D₆	D₈	D₁₂	C₃⋊D₄	Q₃₂	D₂₄	Dic₂₄	S₃²	D₄⋊S₃	C₃⋊D₁₂	C₃⋊Q₃₂	C₃⋊D₂₄	C₃²⋊₃Q₃₂
kernel	C₃²⋊₃Q₃₂	C₃×C₃⋊C₁₆	C₃×Dic₁₂	C₃²⋊₅Q₁₆	C₃⋊C₁₆	Dic₁₂	C₃×C₁₂	C₂₄	C₃×C₆	C₁₂	C₁₂	C₃²	C₆	C₃	C₈	C₆	C₄	C₃	C₂	C₁
# reps	1	1	1	1	1	1	1	2	2	2	2	4	4	8	1	1	1	2	2	4

Matrix representation of C₃²⋊₃Q₃₂ ►in GL₆(𝔽₉₇)

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	96
0	0	0	0	1	96

1	0	0	0	0	0
0	1	0	0	0	0
0	0	0	96	0	0
0	0	1	96	0	0
0	0	0	0	1	0
0	0	0	0	0	1

24	4	0	0	0	0
95	28	0	0	0	0
0	0	29	39	0	0
0	0	58	68	0	0
0	0	0	0	82	56
0	0	0	0	41	15

95	75	0	0	0	0
84	2	0	0	0	0
0	0	39	29	0	0
0	0	68	58	0	0
0	0	0	0	41	15
0	0	0	0	82	56

G:=sub<GL(6,GF(97))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,96,96],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,96,96,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[24,95,0,0,0,0,4,28,0,0,0,0,0,0,29,58,0,0,0,0,39,68,0,0,0,0,0,0,82,41,0,0,0,0,56,15],[95,84,0,0,0,0,75,2,0,0,0,0,0,0,39,68,0,0,0,0,29,58,0,0,0,0,0,0,41,82,0,0,0,0,15,56] >;

C₃²⋊₃Q₃₂ in GAP, Magma, Sage, TeX

C_3^2\rtimes_3Q_{32}

% in TeX

G:=Group("C3^2:3Q32");

// GroupNames label

G:=SmallGroup(288,199);

// by ID

G=gap.SmallGroup(288,199);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,112,85,92,254,142,675,80,1356,9414]);

// Polycyclic

G:=Group<a,b,c,d|a^3=b^3=c^16=1,d^2=c^8,a*b=b*a,c*a*c^-1=a^-1,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^-1>;

// generators/relations

G = C32⋊3Q32 order 288 = 25·32

2nd semidirect product of C32 and Q32 acting via Q32/C8=C22

G = C₃²⋊₃Q₃₂ order 288 = 2⁵·3²

2^nd semidirect product of C₃² and Q₃₂ acting via Q₃₂/C₈=C₂²