C2.D24 - GroupNames

class	1	2A	2B	2C	2D	2E	3	4A	4B	4C	4D	6A	6B	6C	8A	8B	8C	8D	12A	12B	12C	12D	24A	24B	24C	24D	24E	24F	24G	24H
size	1	1	1	1	12	12	2	2	2	12	12	2	2	2	2	2	2	2	2	2	2	2	2	2	2	2	2	2	2	2
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	i	-i	-1	1	-1	-i	-i	i	i	1	-1	-1	1	-i	i	i	-i	i	i	-i	-i	linear of order 4
ρ₆	1	-1	-1	1	1	-1	1	1	-1	i	-i	-1	1	-1	i	i	-i	-i	1	-1	-1	1	i	-i	-i	i	-i	-i	i	i	linear of order 4
ρ₇	1	-1	-1	1	-1	1	1	1	-1	-i	i	-1	1	-1	i	i	-i	-i	1	-1	-1	1	i	-i	-i	i	-i	-i	i	i	linear of order 4
ρ₈	1	-1	-1	1	1	-1	1	1	-1	-i	i	-1	1	-1	-i	-i	i	i	1	-1	-1	1	-i	i	i	-i	i	i	-i	-i	linear of order 4
ρ₉	2	-2	-2	2	0	0	2	-2	2	0	0	-2	2	-2	0	0	0	0	-2	2	2	-2	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₀	2	2	2	2	0	0	-1	2	2	0	0	-1	-1	-1	2	2	2	2	-1	-1	-1	-1	-1	-1	-1	-1	-1	-1	-1	-1	orthogonal lifted from S₃
ρ₁₁	2	2	2	2	0	0	-1	2	2	0	0	-1	-1	-1	-2	-2	-2	-2	-1	-1	-1	-1	1	1	1	1	1	1	1	1	orthogonal lifted from D₆
ρ₁₂	2	2	2	2	0	0	2	-2	-2	0	0	2	2	2	0	0	0	0	-2	-2	-2	-2	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₃	2	2	2	2	0	0	-1	-2	-2	0	0	-1	-1	-1	0	0	0	0	1	1	1	1	-√3	-√3	-√3	√3	√3	√3	-√3	√3	orthogonal lifted from D₁₂
ρ₁₄	2	2	-2	-2	0	0	2	0	0	0	0	-2	-2	2	-√2	√2	-√2	√2	0	0	0	0	√2	√2	-√2	√2	√2	-√2	-√2	-√2	orthogonal lifted from D₈
ρ₁₅	2	2	-2	-2	0	0	-1	0	0	0	0	1	1	-1	√2	-√2	√2	-√2	√3	√3	-√3	-√3	ζ₈³ζ₃+ζ₈ζ₃+ζ₈	ζ₈³ζ₃+ζ₈ζ₃+ζ₈	ζ₈³ζ₃²+ζ₈³+ζ₈ζ₃²	ζ₈⁷ζ₃+ζ₈⁷+ζ₈⁵ζ₃	ζ₈⁷ζ₃+ζ₈⁷+ζ₈⁵ζ₃	ζ₈⁷ζ₃²+ζ₈⁵ζ₃²+ζ₈⁵	ζ₈³ζ₃²+ζ₈³+ζ₈ζ₃²	ζ₈⁷ζ₃²+ζ₈⁵ζ₃²+ζ₈⁵	orthogonal lifted from D₂₄
ρ₁₆	2	2	-2	-2	0	0	2	0	0	0	0	-2	-2	2	√2	-√2	√2	-√2	0	0	0	0	-√2	-√2	√2	-√2	-√2	√2	√2	√2	orthogonal lifted from D₈
ρ₁₇	2	2	-2	-2	0	0	-1	0	0	0	0	1	1	-1	√2	-√2	√2	-√2	-√3	-√3	√3	√3	ζ₈⁷ζ₃+ζ₈⁷+ζ₈⁵ζ₃	ζ₈⁷ζ₃+ζ₈⁷+ζ₈⁵ζ₃	ζ₈⁷ζ₃²+ζ₈⁵ζ₃²+ζ₈⁵	ζ₈³ζ₃+ζ₈ζ₃+ζ₈	ζ₈³ζ₃+ζ₈ζ₃+ζ₈	ζ₈³ζ₃²+ζ₈³+ζ₈ζ₃²	ζ₈⁷ζ₃²+ζ₈⁵ζ₃²+ζ₈⁵	ζ₈³ζ₃²+ζ₈³+ζ₈ζ₃²	orthogonal lifted from D₂₄
ρ₁₈	2	2	2	2	0	0	-1	-2	-2	0	0	-1	-1	-1	0	0	0	0	1	1	1	1	√3	√3	√3	-√3	-√3	-√3	√3	-√3	orthogonal lifted from D₁₂
ρ₁₉	2	2	-2	-2	0	0	-1	0	0	0	0	1	1	-1	-√2	√2	-√2	√2	√3	√3	-√3	-√3	ζ₈³ζ₃²+ζ₈³+ζ₈ζ₃²	ζ₈³ζ₃²+ζ₈³+ζ₈ζ₃²	ζ₈³ζ₃+ζ₈ζ₃+ζ₈	ζ₈⁷ζ₃²+ζ₈⁵ζ₃²+ζ₈⁵	ζ₈⁷ζ₃²+ζ₈⁵ζ₃²+ζ₈⁵	ζ₈⁷ζ₃+ζ₈⁷+ζ₈⁵ζ₃	ζ₈³ζ₃+ζ₈ζ₃+ζ₈	ζ₈⁷ζ₃+ζ₈⁷+ζ₈⁵ζ₃	orthogonal lifted from D₂₄
ρ₂₀	2	2	-2	-2	0	0	-1	0	0	0	0	1	1	-1	-√2	√2	-√2	√2	-√3	-√3	√3	√3	ζ₈⁷ζ₃²+ζ₈⁵ζ₃²+ζ₈⁵	ζ₈⁷ζ₃²+ζ₈⁵ζ₃²+ζ₈⁵	ζ₈⁷ζ₃+ζ₈⁷+ζ₈⁵ζ₃	ζ₈³ζ₃²+ζ₈³+ζ₈ζ₃²	ζ₈³ζ₃²+ζ₈³+ζ₈ζ₃²	ζ₈³ζ₃+ζ₈ζ₃+ζ₈	ζ₈⁷ζ₃+ζ₈⁷+ζ₈⁵ζ₃	ζ₈³ζ₃+ζ₈ζ₃+ζ₈	orthogonal lifted from D₂₄
ρ₂₁	2	-2	-2	2	0	0	-1	2	-2	0	0	1	-1	1	2i	2i	-2i	-2i	-1	1	1	-1	-i	i	i	-i	i	i	-i	-i	complex lifted from C₄×S₃
ρ₂₂	2	-2	-2	2	0	0	-1	2	-2	0	0	1	-1	1	-2i	-2i	2i	2i	-1	1	1	-1	i	-i	-i	i	-i	-i	i	i	complex lifted from C₄×S₃
ρ₂₃	2	-2	2	-2	0	0	2	0	0	0	0	2	-2	-2	√-2	-√-2	-√-2	√-2	0	0	0	0	-√-2	√-2	-√-2	-√-2	√-2	-√-2	√-2	√-2	complex lifted from SD₁₆
ρ₂₄	2	-2	2	-2	0	0	-1	0	0	0	0	-1	1	1	√-2	-√-2	-√-2	√-2	√3	-√3	√3	-√3	ζ₈³ζ₃+ζ₈³-ζ₈ζ₃	ζ₈⁷ζ₃+ζ₈⁷-ζ₈⁵ζ₃	ζ₈³ζ₃+ζ₈³-ζ₈ζ₃	ζ₈³ζ₃²+ζ₈³-ζ₈ζ₃²	ζ₈⁷ζ₃²+ζ₈⁷-ζ₈⁵ζ₃²	ζ₈³ζ₃²+ζ₈³-ζ₈ζ₃²	ζ₈⁷ζ₃+ζ₈⁷-ζ₈⁵ζ₃	ζ₈⁷ζ₃²+ζ₈⁷-ζ₈⁵ζ₃²	complex lifted from C₂₄⋊C₂
ρ₂₅	2	-2	-2	2	0	0	-1	-2	2	0	0	1	-1	1	0	0	0	0	1	-1	-1	1	-√-3	√-3	√-3	√-3	-√-3	-√-3	-√-3	√-3	complex lifted from C₃⋊D₄
ρ₂₆	2	-2	-2	2	0	0	-1	-2	2	0	0	1	-1	1	0	0	0	0	1	-1	-1	1	√-3	-√-3	-√-3	-√-3	√-3	√-3	√-3	-√-3	complex lifted from C₃⋊D₄
ρ₂₇	2	-2	2	-2	0	0	-1	0	0	0	0	-1	1	1	-√-2	√-2	√-2	-√-2	-√3	√3	-√3	√3	ζ₈⁷ζ₃²+ζ₈⁷-ζ₈⁵ζ₃²	ζ₈³ζ₃²+ζ₈³-ζ₈ζ₃²	ζ₈⁷ζ₃²+ζ₈⁷-ζ₈⁵ζ₃²	ζ₈⁷ζ₃+ζ₈⁷-ζ₈⁵ζ₃	ζ₈³ζ₃+ζ₈³-ζ₈ζ₃	ζ₈⁷ζ₃+ζ₈⁷-ζ₈⁵ζ₃	ζ₈³ζ₃²+ζ₈³-ζ₈ζ₃²	ζ₈³ζ₃+ζ₈³-ζ₈ζ₃	complex lifted from C₂₄⋊C₂
ρ₂₈	2	-2	2	-2	0	0	-1	0	0	0	0	-1	1	1	-√-2	√-2	√-2	-√-2	√3	-√3	√3	-√3	ζ₈⁷ζ₃+ζ₈⁷-ζ₈⁵ζ₃	ζ₈³ζ₃+ζ₈³-ζ₈ζ₃	ζ₈⁷ζ₃+ζ₈⁷-ζ₈⁵ζ₃	ζ₈⁷ζ₃²+ζ₈⁷-ζ₈⁵ζ₃²	ζ₈³ζ₃²+ζ₈³-ζ₈ζ₃²	ζ₈⁷ζ₃²+ζ₈⁷-ζ₈⁵ζ₃²	ζ₈³ζ₃+ζ₈³-ζ₈ζ₃	ζ₈³ζ₃²+ζ₈³-ζ₈ζ₃²	complex lifted from C₂₄⋊C₂
ρ₂₉	2	-2	2	-2	0	0	2	0	0	0	0	2	-2	-2	-√-2	√-2	√-2	-√-2	0	0	0	0	√-2	-√-2	√-2	√-2	-√-2	√-2	-√-2	-√-2	complex lifted from SD₁₆
ρ₃₀	2	-2	2	-2	0	0	-1	0	0	0	0	-1	1	1	√-2	-√-2	-√-2	√-2	-√3	√3	-√3	√3	ζ₈³ζ₃²+ζ₈³-ζ₈ζ₃²	ζ₈⁷ζ₃²+ζ₈⁷-ζ₈⁵ζ₃²	ζ₈³ζ₃²+ζ₈³-ζ₈ζ₃²	ζ₈³ζ₃+ζ₈³-ζ₈ζ₃	ζ₈⁷ζ₃+ζ₈⁷-ζ₈⁵ζ₃	ζ₈³ζ₃+ζ₈³-ζ₈ζ₃	ζ₈⁷ζ₃²+ζ₈⁷-ζ₈⁵ζ₃²	ζ₈⁷ζ₃+ζ₈⁷-ζ₈⁵ζ₃	complex lifted from C₂₄⋊C₂

Smallest permutation representation of C₂.D₂₄
►On 48 points

Generators in S₄₈

(1 43)(2 44)(3 45)(4 46)(5 47)(6 48)(7 25)(8 26)(9 27)(10 28)(11 29)(12 30)(13 31)(14 32)(15 33)(16 34)(17 35)(18 36)(19 37)(20 38)(21 39)(22 40)(23 41)(24 42)

(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)

(1 42 43 24)(2 23 44 41)(3 40 45 22)(4 21 46 39)(5 38 47 20)(6 19 48 37)(7 36 25 18)(8 17 26 35)(9 34 27 16)(10 15 28 33)(11 32 29 14)(12 13 30 31)

G:=sub<Sym(48)| (1,43)(2,44)(3,45)(4,46)(5,47)(6,48)(7,25)(8,26)(9,27)(10,28)(11,29)(12,30)(13,31)(14,32)(15,33)(16,34)(17,35)(18,36)(19,37)(20,38)(21,39)(22,40)(23,41)(24,42), (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), (1,42,43,24)(2,23,44,41)(3,40,45,22)(4,21,46,39)(5,38,47,20)(6,19,48,37)(7,36,25,18)(8,17,26,35)(9,34,27,16)(10,15,28,33)(11,32,29,14)(12,13,30,31)>;

G:=Group( (1,43)(2,44)(3,45)(4,46)(5,47)(6,48)(7,25)(8,26)(9,27)(10,28)(11,29)(12,30)(13,31)(14,32)(15,33)(16,34)(17,35)(18,36)(19,37)(20,38)(21,39)(22,40)(23,41)(24,42), (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), (1,42,43,24)(2,23,44,41)(3,40,45,22)(4,21,46,39)(5,38,47,20)(6,19,48,37)(7,36,25,18)(8,17,26,35)(9,34,27,16)(10,15,28,33)(11,32,29,14)(12,13,30,31) );

G=PermutationGroup([[(1,43),(2,44),(3,45),(4,46),(5,47),(6,48),(7,25),(8,26),(9,27),(10,28),(11,29),(12,30),(13,31),(14,32),(15,33),(16,34),(17,35),(18,36),(19,37),(20,38),(21,39),(22,40),(23,41),(24,42)], [(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)], [(1,42,43,24),(2,23,44,41),(3,40,45,22),(4,21,46,39),(5,38,47,20),(6,19,48,37),(7,36,25,18),(8,17,26,35),(9,34,27,16),(10,15,28,33),(11,32,29,14),(12,13,30,31)]])

C₂.D₂₄ is a maximal subgroup of
C₄×C₂₄⋊C₂ C₄×D₂₄ C₄.₅D₂₄ C₄².₂₆₄D₆ C₄².₁₆D₆ D₂₄⋊C₄ C₄².₁₉D₆ C₄².₂₀D₆ D₁₂.₃₁D₄ D₁₂⋊₁₃D₄ D₁₂.₃₂D₄ D₁₂⋊₁₄D₄ C₂³.₄₃D₁₂ C₂².D₂₄ C₂³.₁₈D₁₂ Dic₃⋊₄D₈ Dic₃.SD₁₆ C₄⋊C₄.D₆ S₃×D₄⋊C₄ C₄⋊C₄⋊₁₉D₆ D₆⋊D₈ C₃⋊C₈⋊D₄ D₄⋊S₃⋊C₄ Dic₃⋊₇SD₁₆ (C₂×C₈).D₆ Q₈⋊C₄⋊S₃ Q₈⋊₇(C₄×S₃) C₄⋊C₄.₁₅₀D₆ D₆⋊₂SD₁₆ C₃⋊(C₈⋊D₄) Q₈⋊₃(C₄×S₃) D₁₂⋊₃Q₈ C₄⋊D₂₄ D₁₂.₁₉D₄ C₄².₃₆D₆ D₁₂⋊₄Q₈ D₁₂.₃Q₈ Dic₆⋊₈D₄ D₆.₄SD₁₆ C₄.Q₈⋊S₃ D₁₂⋊Q₈ D₁₂.Q₈ D₆.₅D₈ C₂.D₈⋊S₃ D₁₂⋊₂Q₈ D₁₂.₂Q₈ C₂³.₂₈D₁₂ C₂₄⋊₃₀D₄ C₂₄⋊₂₉D₄ C₂³.₅₃D₁₂ C₂³.₅₄D₁₂ C₂₄⋊₂D₄ C₂₄⋊₃D₄ Dic₃⋊D₈ D₁₂⋊D₄ Dic₃⋊₅SD₁₆ (C₃×D₄).D₄ D₆⋊₆SD₁₆ D₁₂⋊₇D₄ (C₂×Q₁₆)⋊S₃ D₁₂.₁₇D₄ C₂.D₇₂ C₆.₁₆D₂₄ C₆.₁₇D₂₄ C₆².₈₄D₄ C₁₀.D₂₄ D₆₀⋊₁₅C₄ D₆₀⋊₈C₄ D₁₂⋊F₅
C₂.D₂₄ is a maximal quotient of
C₄.₁₇D₂₄ C₂².₂D₂₄ C₄.D₂₄ C₁₂.₂D₈ C₂.Dic₂₄ C₂.D₄₈ D₂₄.₁C₄ M₅(2)⋊S₃ C₁₂.₄D₈ D₂₄⋊₂C₄ C₁₂.₉C₄² C₂.D₇₂ C₆.₁₆D₂₄ C₆.₁₇D₂₄ C₆².₈₄D₄ C₁₀.D₂₄ D₆₀⋊₁₅C₄ D₆₀⋊₈C₄ D₁₂⋊F₅

Matrix representation of C₂.D₂₄ ►in GL₅(𝔽₇₃)

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

46	0	0	0	0
0	0	72	0	0
0	1	1	0	0
0	0	0	0	25
0	0	0	35	32

27	0	0	0	0
0	0	72	0	0
0	72	0	0	0
0	0	0	0	25
0	0	0	38	0

G:=sub<GL(5,GF(73))| [72,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[46,0,0,0,0,0,0,1,0,0,0,72,1,0,0,0,0,0,0,35,0,0,0,25,32],[27,0,0,0,0,0,0,72,0,0,0,72,0,0,0,0,0,0,0,38,0,0,0,25,0] >;

C₂.D₂₄ in GAP, Magma, Sage, TeX

C_2.D_{24}

% in TeX

G:=Group("C2.D24");

// GroupNames label

G:=SmallGroup(96,28);

// by ID

G=gap.SmallGroup(96,28);

# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-3,73,79,362,86,2309]);

// Polycyclic

G:=Group<a,b,c|a^2=b^24=1,c^2=a,a*b=b*a,a*c=c*a,c*b*c^-1=a*b^-1>;

// generators/relations

Export

Subgroup lattice of C₂.D₂₄ in TeX
Character table of C₂.D₂₄ in TeX

G = C2.D24 order 96 = 25·3

2nd central extension by C2 of D24

G = C₂.D₂₄ order 96 = 2⁵·3

2^nd central extension by C₂ of D₂₄