C2xC3^2:Q16 - GroupNames

G = C₂×C₃²⋊Q₁₆ order 288 = 2⁵·3²

Direct product of C₂ and C₃²⋊Q₁₆

direct product, non-abelian, soluble, monomial

Aliases: C₂×C₃²⋊Q₁₆, C₆².₁₆D₄, (C₃×C₆)⋊Q₁₆, C₃²⋊₂(C₂×Q₁₆), C₂².₁₆S₃≀C₂, C₃⋊Dic₃.₃₄D₄, C₃⋊Dic₃.₁₂C₂³, C₃²⋊₂C₈.₈C₂², C₃²⋊₂Q₈.₈C₂², C₂.₂₁(C₂×S₃≀C₂), (C₃×C₆).₂₁(C₂×D₄), (C₂×C₃²⋊₂C₈).₆C₂, (C₂×C₃²⋊₂Q₈).₅C₂, (C₂×C₃⋊Dic₃).₉₉C₂², SmallGroup(288,888)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃² — C₃⋊Dic₃ — C₂×C₃²⋊Q₁₆

Chief series

C₁ — C₃² — C₃×C₆ — C₃⋊Dic₃ — C₃²⋊₂Q₈ — C₃²⋊Q₁₆ — C₂×C₃²⋊Q₁₆

Lower central

C₃² — C₃×C₆ — C₃⋊Dic₃ — C₂×C₃²⋊Q₁₆

Upper central

C₁ — C₂²

Generators and relations for C₂×C₃²⋊Q₁₆
G = < a,b,c,d,e | a²=b³=c³=d⁸=1, e²=d⁴, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd^-1=ebe^-1=c, dcd^-1=b^-1, ece^-1=b, ede^-1=d^-1 >

Subgroups: 432 in 98 conjugacy classes, 27 normal (11 characteristic)
C₁, C₂, C₂, C₃, C₄, C₂², C₆, C₈, C₂×C₄, Q₈, C₃², Dic₃, C₁₂, C₂×C₆, C₂×C₈, Q₁₆, C₂×Q₈, C₃×C₆, C₃×C₆, Dic₆, C₂×Dic₃, C₂×C₁₂, C₂×Q₁₆, C₃×Dic₃, C₃⋊Dic₃, C₆², C₂×Dic₆, C₃²⋊₂C₈, C₃²⋊₂Q₈, C₃²⋊₂Q₈, C₆×Dic₃, C₂×C₃⋊Dic₃, C₃²⋊Q₁₆, C₂×C₃²⋊₂C₈, C₂×C₃²⋊₂Q₈, C₂×C₃²⋊Q₁₆
Quotients: C₁, C₂, C₂², D₄, C₂³, Q₁₆, C₂×D₄, C₂×Q₁₆, S₃≀C₂, C₃²⋊Q₁₆, C₂×S₃≀C₂, C₂×C₃²⋊Q₁₆

Character table of C₂×C₃²⋊Q₁₆

class	1	2A	2B	2C	3A	3B	4A	4B	4C	4D	4E	4F	6A	6B	6C	6D	6E	6F	8A	8B	8C	8D	12A	12B	12C	12D	12E	12F	12G	12H
size	1	1	1	1	4	4	12	12	12	12	18	18	4	4	4	4	4	4	18	18	18	18	12	12	12	12	12	12	12	12
ρ₁	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	trivial
ρ₂	1	-1	-1	1	1	1	-1	-1	1	1	1	-1	1	-1	-1	1	-1	-1	-1	1	-1	1	1	1	-1	-1	-1	-1	1	1	linear of order 2
ρ₃	1	-1	-1	1	1	1	-1	1	1	-1	1	-1	1	-1	-1	1	-1	-1	1	-1	1	-1	1	-1	-1	-1	1	1	-1	1	linear of order 2
ρ₄	1	1	1	1	1	1	1	-1	1	-1	1	1	1	1	1	1	1	1	-1	-1	-1	-1	1	-1	1	1	-1	-1	-1	1	linear of order 2
ρ₅	1	-1	-1	1	1	1	1	1	-1	-1	1	-1	1	-1	-1	1	-1	-1	-1	1	-1	1	-1	-1	1	1	1	1	-1	-1	linear of order 2
ρ₆	1	1	1	1	1	1	-1	-1	-1	-1	1	1	1	1	1	1	1	1	1	1	1	1	-1	-1	-1	-1	-1	-1	-1	-1	linear of order 2
ρ₇	1	1	1	1	1	1	-1	1	-1	1	1	1	1	1	1	1	1	1	-1	-1	-1	-1	-1	1	-1	-1	1	1	1	-1	linear of order 2
ρ₈	1	-1	-1	1	1	1	1	-1	-1	1	1	-1	1	-1	-1	1	-1	-1	1	-1	1	-1	-1	1	1	1	-1	-1	1	-1	linear of order 2
ρ₉	2	2	2	2	2	2	0	0	0	0	-2	-2	2	2	2	2	2	2	0	0	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₀	2	-2	-2	2	2	2	0	0	0	0	-2	2	2	-2	-2	2	-2	-2	0	0	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₁	2	-2	2	-2	2	2	0	0	0	0	0	0	-2	-2	-2	-2	2	2	√2	√2	-√2	-√2	0	0	0	0	0	0	0	0	symplectic lifted from Q₁₆, Schur index 2
ρ₁₂	2	2	-2	-2	2	2	0	0	0	0	0	0	-2	2	2	-2	-2	-2	-√2	√2	√2	-√2	0	0	0	0	0	0	0	0	symplectic lifted from Q₁₆, Schur index 2
ρ₁₃	2	-2	2	-2	2	2	0	0	0	0	0	0	-2	-2	-2	-2	2	2	-√2	-√2	√2	√2	0	0	0	0	0	0	0	0	symplectic lifted from Q₁₆, Schur index 2
ρ₁₄	2	2	-2	-2	2	2	0	0	0	0	0	0	-2	2	2	-2	-2	-2	√2	-√2	-√2	√2	0	0	0	0	0	0	0	0	symplectic lifted from Q₁₆, Schur index 2
ρ₁₅	4	-4	-4	4	-2	1	-2	0	2	0	0	0	-2	-1	2	1	-1	2	0	0	0	0	-1	0	1	1	0	0	0	-1	orthogonal lifted from C₂×S₃≀C₂
ρ₁₆	4	-4	-4	4	1	-2	0	2	0	-2	0	0	1	2	-1	-2	2	-1	0	0	0	0	0	1	0	0	-1	-1	1	0	orthogonal lifted from C₂×S₃≀C₂
ρ₁₇	4	-4	-4	4	-2	1	2	0	-2	0	0	0	-2	-1	2	1	-1	2	0	0	0	0	1	0	-1	-1	0	0	0	1	orthogonal lifted from C₂×S₃≀C₂
ρ₁₈	4	4	4	4	1	-2	0	2	0	2	0	0	1	-2	1	-2	-2	1	0	0	0	0	0	-1	0	0	-1	-1	-1	0	orthogonal lifted from S₃≀C₂
ρ₁₉	4	4	4	4	1	-2	0	-2	0	-2	0	0	1	-2	1	-2	-2	1	0	0	0	0	0	1	0	0	1	1	1	0	orthogonal lifted from S₃≀C₂
ρ₂₀	4	4	4	4	-2	1	2	0	2	0	0	0	-2	1	-2	1	1	-2	0	0	0	0	-1	0	-1	-1	0	0	0	-1	orthogonal lifted from S₃≀C₂
ρ₂₁	4	-4	-4	4	1	-2	0	-2	0	2	0	0	1	2	-1	-2	2	-1	0	0	0	0	0	-1	0	0	1	1	-1	0	orthogonal lifted from C₂×S₃≀C₂
ρ₂₂	4	4	4	4	-2	1	-2	0	-2	0	0	0	-2	1	-2	1	1	-2	0	0	0	0	1	0	1	1	0	0	0	1	orthogonal lifted from S₃≀C₂
ρ₂₃	4	4	-4	-4	-2	1	0	0	0	0	0	0	2	1	-2	-1	-1	2	0	0	0	0	-√3	0	-√3	√3	0	0	0	√3	symplectic lifted from C₃²⋊Q₁₆, Schur index 2
ρ₂₄	4	-4	4	-4	1	-2	0	0	0	0	0	0	-1	2	-1	2	-2	1	0	0	0	0	0	√3	0	0	√3	-√3	-√3	0	symplectic lifted from C₃²⋊Q₁₆, Schur index 2
ρ₂₅	4	-4	4	-4	-2	1	0	0	0	0	0	0	2	-1	2	-1	1	-2	0	0	0	0	√3	0	-√3	√3	0	0	0	-√3	symplectic lifted from C₃²⋊Q₁₆, Schur index 2
ρ₂₆	4	4	-4	-4	1	-2	0	0	0	0	0	0	-1	-2	1	2	2	-1	0	0	0	0	0	√3	0	0	-√3	√3	-√3	0	symplectic lifted from C₃²⋊Q₁₆, Schur index 2
ρ₂₇	4	-4	4	-4	1	-2	0	0	0	0	0	0	-1	2	-1	2	-2	1	0	0	0	0	0	-√3	0	0	-√3	√3	√3	0	symplectic lifted from C₃²⋊Q₁₆, Schur index 2
ρ₂₈	4	4	-4	-4	1	-2	0	0	0	0	0	0	-1	-2	1	2	2	-1	0	0	0	0	0	-√3	0	0	√3	-√3	√3	0	symplectic lifted from C₃²⋊Q₁₆, Schur index 2
ρ₂₉	4	-4	4	-4	-2	1	0	0	0	0	0	0	2	-1	2	-1	1	-2	0	0	0	0	-√3	0	√3	-√3	0	0	0	√3	symplectic lifted from C₃²⋊Q₁₆, Schur index 2
ρ₃₀	4	4	-4	-4	-2	1	0	0	0	0	0	0	2	1	-2	-1	-1	2	0	0	0	0	√3	0	√3	-√3	0	0	0	-√3	symplectic lifted from C₃²⋊Q₁₆, Schur index 2

Smallest permutation representation of C₂×C₃²⋊Q₁₆
►On 96 points

Generators in S₉₆

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

(2 50 82)(4 84 52)(6 54 86)(8 88 56)(10 32 77)(12 79 26)(14 28 73)(16 75 30)(17 91 36)(19 38 93)(21 95 40)(23 34 89)(42 65 60)(44 62 67)(46 69 64)(48 58 71)

(1 49 81)(3 83 51)(5 53 85)(7 87 55)(9 31 76)(11 78 25)(13 27 80)(15 74 29)(18 37 92)(20 94 39)(22 33 96)(24 90 35)(41 72 59)(43 61 66)(45 68 63)(47 57 70)

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

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

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

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

Matrix representation of C₂×C₃²⋊Q₁₆ ►in GL₆(𝔽₇₃)

1	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	0	0	72	72

1	0	0	0	0	0
0	1	0	0	0	0
0	0	0	1	0	0
0	0	72	72	0	0
0	0	0	0	1	0
0	0	0	0	0	1

55	48	0	0	0	0
25	59	0	0	0	0
0	0	0	0	14	19
0	0	0	0	5	59
0	0	66	59	0	0
0	0	14	7	0	0

57	4	0	0	0	0
27	16	0	0	0	0
0	0	0	0	1	0
0	0	0	0	0	1
0	0	72	0	0	0
0	0	0	72	0	0

G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,72,0,0,0,0,1,72],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,72,0,0,0,0,1,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[55,25,0,0,0,0,48,59,0,0,0,0,0,0,0,0,66,14,0,0,0,0,59,7,0,0,14,5,0,0,0,0,19,59,0,0],[57,27,0,0,0,0,4,16,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,1,0,0,0,0,0,0,1,0,0] >;

C₂×C₃²⋊Q₁₆ in GAP, Magma, Sage, TeX

C_2\times C_3^2\rtimes Q_{16}

% in TeX

G:=Group("C2xC3^2:Q16");

// GroupNames label

G:=SmallGroup(288,888);

// by ID

G=gap.SmallGroup(288,888);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,112,141,120,675,346,80,2693,2028,362,797,1203]);

// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^3=c^3=d^8=1,e^2=d^4,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,d*b*d^-1=e*b*e^-1=c,d*c*d^-1=b^-1,e*c*e^-1=b,e*d*e^-1=d^-1>;

// generators/relations

Export

Character table of C₂×C₃²⋊Q₁₆ in TeX

G = C2×C32⋊Q16 order 288 = 25·32

Direct product of C2 and C32⋊Q16

G = C₂×C₃²⋊Q₁₆ order 288 = 2⁵·3²

Direct product of C₂ and C₃²⋊Q₁₆