C24.23D4 - GroupNames

G = C₂₄.₂₃D₄ order 192 = 2⁶·3

23^rd non-split extension by C₂₄ of D₄ acting via D₄/C₂=C₂²

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C₂₄.₂₃D₄, D₁₂.₂₅D₄, Dic₆.₂₅D₄, (C₆×D₈)⋊₂C₂, C₈○D₁₂⋊₁C₂, (C₂×D₈)⋊₁₀S₃, (C₂×C₈).₈₇D₆, C₄.₆₁(S₃×D₄), (C₂×D₄).₆₇D₆, C₂₄.C₄⋊₃C₂, D₁₂⋊₆C₂²⋊₄C₂, C₁₂.₁₆₉(C₂×D₄), C₁₂.D₄⋊₇C₂, C₃⋊₄(D₄.₄D₄), C₈.₂₇(C₃⋊D₄), (C₂×C₂₄).₃₂C₂², (C₆×D₄).₈₆C₂², C₂.₁₈(D₆⋊₃D₄), C₆.₁₁₁(C₄⋊D₄), (C₂×C₁₂).₄₃₇C₂³, C₄○D₁₂.₄₆C₂², C₄.Dic₃.₁₉C₂², C₂².₂₀(D₄⋊₂S₃), C₄.₈₁(C₂×C₃⋊D₄), (C₂×C₆).₁₅₈(C₄○D₄), (C₂×C₄).₁₂₆(C₂²×S₃), SmallGroup(192,719)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂×C₁₂ — C₂₄.₂₃D₄

Chief series

C₁ — C₃ — C₆ — C₁₂ — C₂×C₁₂ — C₄○D₁₂ — C₈○D₁₂ — C₂₄.₂₃D₄

Lower central

C₃ — C₆ — C₂×C₁₂ — C₂₄.₂₃D₄

Upper central

C₁ — C₂ — C₂×C₄ — C₂×D₈

Generators and relations for C₂₄.₂₃D₄
G = < a,b,c | a²⁴=c²=1, b⁴=a¹², bab^-1=a^-1, cac=a¹⁷, cbc=a¹²b³ >

Subgroups: 312 in 108 conjugacy classes, 37 normal (27 characteristic)
C₁, C₂, C₂, C₃, C₄, C₄, C₂², C₂², S₃, C₆, C₆, C₈, C₈, C₂×C₄, C₂×C₄, D₄, Q₈, C₂³, Dic₃, C₁₂, D₆, C₂×C₆, C₂×C₆, C₂×C₈, C₂×C₈, M₄(2), D₈, SD₁₆, C₂×D₄, C₄○D₄, C₃⋊C₈, C₂₄, Dic₆, C₄×S₃, D₁₂, C₃⋊D₄, C₂×C₁₂, C₃×D₄, C₂²×C₆, C₄.D₄, C₈.C₄, C₈○D₄, C₂×D₈, C₈⋊C₂², S₃×C₈, C₈⋊S₃, C₄.Dic₃, C₄.Dic₃, D₄⋊S₃, D₄.S₃, C₂×C₂₄, C₃×D₈, C₄○D₁₂, C₆×D₄, D₄.₄D₄, C₂₄.C₄, C₁₂.D₄, C₈○D₁₂, D₁₂⋊₆C₂², C₆×D₈, C₂₄.₂₃D₄
Quotients: C₁, C₂, C₂², S₃, D₄, C₂³, D₆, C₂×D₄, C₄○D₄, C₃⋊D₄, C₂²×S₃, C₄⋊D₄, S₃×D₄, D₄⋊₂S₃, C₂×C₃⋊D₄, D₄.₄D₄, D₆⋊₃D₄, C₂₄.₂₃D₄

Character table of C₂₄.₂₃D₄

class	1	2A	2B	2C	2D	2E	3	4A	4B	4C	6A	6B	6C	6D	6E	6F	6G	8A	8B	8C	8D	8E	8F	8G	12A	12B	24A	24B	24C	24D
size	1	1	2	8	8	12	2	2	2	12	2	2	2	8	8	8	8	2	2	4	12	12	24	24	4	4	4	4	4	4
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	0	-1	2	2	0	-1	-1	-1	1	-1	1	-1	-2	-2	-2	0	0	0	0	-1	-1	1	1	1	1	orthogonal lifted from D₆
ρ₁₀	2	2	2	2	2	0	-1	2	2	0	-1	-1	-1	-1	-1	-1	-1	2	2	2	0	0	0	0	-1	-1	-1	-1	-1	-1	orthogonal lifted from S₃
ρ₁₁	2	2	-2	0	0	0	2	2	-2	0	-2	-2	2	0	0	0	0	-2	-2	2	0	0	0	0	2	-2	-2	-2	2	2	orthogonal lifted from D₄
ρ₁₂	2	2	-2	0	0	2	2	-2	2	-2	-2	-2	2	0	0	0	0	0	0	0	0	0	0	0	-2	2	0	0	0	0	orthogonal lifted from D₄
ρ₁₃	2	2	-2	0	0	0	2	2	-2	0	-2	-2	2	0	0	0	0	2	2	-2	0	0	0	0	2	-2	2	2	-2	-2	orthogonal lifted from D₄
ρ₁₄	2	2	2	-2	-2	0	-1	2	2	0	-1	-1	-1	1	1	1	1	2	2	2	0	0	0	0	-1	-1	-1	-1	-1	-1	orthogonal lifted from D₆
ρ₁₅	2	2	-2	0	0	-2	2	-2	2	2	-2	-2	2	0	0	0	0	0	0	0	0	0	0	0	-2	2	0	0	0	0	orthogonal lifted from D₄
ρ₁₆	2	2	2	2	-2	0	-1	2	2	0	-1	-1	-1	-1	1	-1	1	-2	-2	-2	0	0	0	0	-1	-1	1	1	1	1	orthogonal lifted from D₆
ρ₁₇	2	2	-2	0	0	0	-1	2	-2	0	1	1	-1	√-3	-√-3	-√-3	√-3	-2	-2	2	0	0	0	0	-1	1	1	1	-1	-1	complex lifted from C₃⋊D₄
ρ₁₈	2	2	-2	0	0	0	-1	2	-2	0	1	1	-1	√-3	√-3	-√-3	-√-3	2	2	-2	0	0	0	0	-1	1	-1	-1	1	1	complex lifted from C₃⋊D₄
ρ₁₉	2	2	-2	0	0	0	-1	2	-2	0	1	1	-1	-√-3	-√-3	√-3	√-3	2	2	-2	0	0	0	0	-1	1	-1	-1	1	1	complex lifted from C₃⋊D₄
ρ₂₀	2	2	-2	0	0	0	-1	2	-2	0	1	1	-1	-√-3	√-3	√-3	-√-3	-2	-2	2	0	0	0	0	-1	1	1	1	-1	-1	complex lifted from C₃⋊D₄
ρ₂₁	2	2	2	0	0	0	2	-2	-2	0	2	2	2	0	0	0	0	0	0	0	-2i	2i	0	0	-2	-2	0	0	0	0	complex lifted from C₄○D₄
ρ₂₂	2	2	2	0	0	0	2	-2	-2	0	2	2	2	0	0	0	0	0	0	0	2i	-2i	0	0	-2	-2	0	0	0	0	complex lifted from C₄○D₄
ρ₂₃	4	4	-4	0	0	0	-2	-4	4	0	2	2	-2	0	0	0	0	0	0	0	0	0	0	0	2	-2	0	0	0	0	orthogonal lifted from S₃×D₄
ρ₂₄	4	-4	0	0	0	0	4	0	0	0	0	0	-4	0	0	0	0	-2√2	2√2	0	0	0	0	0	0	0	2√2	-2√2	0	0	orthogonal lifted from D₄.₄D₄
ρ₂₅	4	-4	0	0	0	0	4	0	0	0	0	0	-4	0	0	0	0	2√2	-2√2	0	0	0	0	0	0	0	-2√2	2√2	0	0	orthogonal lifted from D₄.₄D₄
ρ₂₆	4	4	4	0	0	0	-2	-4	-4	0	-2	-2	-2	0	0	0	0	0	0	0	0	0	0	0	2	2	0	0	0	0	symplectic lifted from D₄⋊₂S₃, Schur index 2
ρ₂₇	4	-4	0	0	0	0	-2	0	0	0	2√-3	-2√-3	2	0	0	0	0	2√2	-2√2	0	0	0	0	0	0	0	√2	-√2	√-6	-√-6	complex faithful
ρ₂₈	4	-4	0	0	0	0	-2	0	0	0	-2√-3	2√-3	2	0	0	0	0	2√2	-2√2	0	0	0	0	0	0	0	√2	-√2	-√-6	√-6	complex faithful
ρ₂₉	4	-4	0	0	0	0	-2	0	0	0	2√-3	-2√-3	2	0	0	0	0	-2√2	2√2	0	0	0	0	0	0	0	-√2	√2	-√-6	√-6	complex faithful
ρ₃₀	4	-4	0	0	0	0	-2	0	0	0	-2√-3	2√-3	2	0	0	0	0	-2√2	2√2	0	0	0	0	0	0	0	-√2	√2	√-6	-√-6	complex faithful

Smallest permutation representation of C₂₄.₂₃D₄
►On 48 points

Generators in S₄₈

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

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

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

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

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

Matrix representation of C₂₄.₂₃D₄ ►in GL₄(𝔽₇) generated by

2	1	2	1
6	1	3	1
6	4	3	3
2	2	0	4

1	3	6	1
3	0	1	5
1	3	5	0
3	5	2	1

5	1	0	0
4	2	0	0
6	3	5	1
5	1	4	2

G:=sub<GL(4,GF(7))| [2,6,6,2,1,1,4,2,2,3,3,0,1,1,3,4],[1,3,1,3,3,0,3,5,6,1,5,2,1,5,0,1],[5,4,6,5,1,2,3,1,0,0,5,4,0,0,1,2] >;

C₂₄.₂₃D₄ in GAP, Magma, Sage, TeX

C_{24}._{23}D_4

% in TeX

G:=Group("C24.23D4");

// GroupNames label

G:=SmallGroup(192,719);

// by ID

G=gap.SmallGroup(192,719);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,120,254,219,1123,297,136,438,102,6278]);

// Polycyclic

G:=Group<a,b,c|a^24=c^2=1,b^4=a^12,b*a*b^-1=a^-1,c*a*c=a^17,c*b*c=a^12*b^3>;

// generators/relations

Export

Character table of C₂₄.₂₃D₄ in TeX

G = C24.23D4 order 192 = 26·3

23rd non-split extension by C24 of D4 acting via D4/C2=C22

G = C₂₄.₂₃D₄ order 192 = 2⁶·3

23^rd non-split extension by C₂₄ of D₄ acting via D₄/C₂=C₂²