C24.19D4 - GroupNames

G = C₂₄.₁₉D₄ order 192 = 2⁶·3

19^th non-split extension by C₂₄ of D₄ acting via D₄/C₂=C₂²

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C₂₄.₁₉D₄, C₈.₂₁D₁₂, D₁₂.₂₁D₄, Dic₆.₂₁D₄, M₄(2).₁₀D₆, C₈○D₁₂⋊₆C₂, (C₂×C₈).₇₁D₆, (C₂×D₂₄)⋊₂₁C₂, C₈⋊D₆⋊₁₀C₂, C₈.C₄⋊₆S₃, C₄.₁₃₆(S₃×D₄), C₄.₅₇(C₂×D₁₂), C₁₂.₁₃₇(C₂×D₄), C₃⋊₂(D₄.₄D₄), C₁₂.₄₆D₄⋊₃C₂, C₆.₅₀(C₄⋊D₄), C₂.₂₃(C₁₂⋊D₄), (C₂×C₂₄).₁₀₃C₂², (C₂×C₁₂).₃₁₃C₂³, C₄○D₁₂.₄₀C₂², (C₂×D₁₂).₈₈C₂², C₂².₇(Q₈⋊₃S₃), (C₃×M₄(2)).₇C₂², C₄.Dic₃.₃₈C₂², (C₃×C₈.C₄)⋊₇C₂, (C₂×C₆).₄(C₄○D₄), (C₂×C₄).₁₁₄(C₂²×S₃), SmallGroup(192,456)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂×C₁₂ — C₂₄.₁₉D₄

Chief series

C₁ — C₃ — C₆ — C₁₂ — C₂×C₁₂ — C₄○D₁₂ — C₈○D₁₂ — C₂₄.₁₉D₄

Lower central

C₃ — C₆ — C₂×C₁₂ — C₂₄.₁₉D₄

Upper central

C₁ — C₂ — C₂×C₄ — C₈.C₄

Generators and relations for C₂₄.₁₉D₄
G = < a,b,c | a²⁴=c²=1, b⁴=a¹², bab^-1=a⁷, cac=a^-1, cbc=a¹²b³ >

Subgroups: 416 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×C₈, C₂×C₈, M₄(2), M₄(2), D₈, SD₁₆, C₂×D₄, C₄○D₄, C₃⋊C₈, C₂₄, C₂₄, Dic₆, C₄×S₃, D₁₂, D₁₂, C₃⋊D₄, C₂×C₁₂, C₂²×S₃, C₄.D₄, C₈.C₄, C₈○D₄, C₂×D₈, C₈⋊C₂², S₃×C₈, C₈⋊S₃, C₂₄⋊C₂, D₂₄, C₄.Dic₃, C₂×C₂₄, C₃×M₄(2), C₂×D₁₂, C₄○D₁₂, D₄.₄D₄, C₁₂.₄₆D₄, C₃×C₈.C₄, C₈○D₁₂, C₂×D₂₄, C₈⋊D₆, C₂₄.₁₉D₄
Quotients: C₁, C₂, C₂², S₃, D₄, C₂³, D₆, C₂×D₄, C₄○D₄, D₁₂, C₂²×S₃, C₄⋊D₄, C₂×D₁₂, S₃×D₄, Q₈⋊₃S₃, D₄.₄D₄, C₁₂⋊D₄, C₂₄.₁₉D₄

Character table of C₂₄.₁₉D₄

class	1	2A	2B	2C	2D	2E	3	4A	4B	4C	6A	6B	8A	8B	8C	8D	8E	8F	8G	12A	12B	12C	24A	24B	24C	24D	24E	24F	24G	24H
size	1	1	2	12	24	24	2	2	2	12	2	4	2	2	4	8	8	12	12	2	2	4	4	4	4	4	8	8	8	8
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	0	0	0	-1	2	2	0	-1	-1	-2	-2	-2	-2	2	0	0	-1	-1	-1	1	1	1	1	1	1	-1	-1	orthogonal lifted from D₆
ρ₁₀	2	2	2	0	0	0	-1	2	2	0	-1	-1	2	2	2	2	2	0	0	-1	-1	-1	-1	-1	-1	-1	-1	-1	-1	-1	orthogonal lifted from S₃
ρ₁₁	2	2	-2	-2	0	0	2	-2	2	2	2	-2	0	0	0	0	0	0	0	2	2	-2	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₂	2	2	-2	0	0	0	2	2	-2	0	2	-2	-2	-2	2	0	0	0	0	-2	-2	2	-2	2	-2	2	0	0	0	0	orthogonal lifted from D₄
ρ₁₃	2	2	-2	2	0	0	2	-2	2	-2	2	-2	0	0	0	0	0	0	0	2	2	-2	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₄	2	2	2	0	0	0	-1	2	2	0	-1	-1	2	2	2	-2	-2	0	0	-1	-1	-1	-1	-1	-1	-1	1	1	1	1	orthogonal lifted from D₆
ρ₁₅	2	2	2	0	0	0	-1	2	2	0	-1	-1	-2	-2	-2	2	-2	0	0	-1	-1	-1	1	1	1	1	-1	-1	1	1	orthogonal lifted from D₆
ρ₁₆	2	2	-2	0	0	0	2	2	-2	0	2	-2	2	2	-2	0	0	0	0	-2	-2	2	2	-2	2	-2	0	0	0	0	orthogonal lifted from D₄
ρ₁₇	2	2	-2	0	0	0	-1	2	-2	0	-1	1	2	2	-2	0	0	0	0	1	1	-1	-1	1	-1	1	-√3	√3	-√3	√3	orthogonal lifted from D₁₂
ρ₁₈	2	2	-2	0	0	0	-1	2	-2	0	-1	1	2	2	-2	0	0	0	0	1	1	-1	-1	1	-1	1	√3	-√3	√3	-√3	orthogonal lifted from D₁₂
ρ₁₉	2	2	-2	0	0	0	-1	2	-2	0	-1	1	-2	-2	2	0	0	0	0	1	1	-1	1	-1	1	-1	-√3	√3	√3	-√3	orthogonal lifted from D₁₂
ρ₂₀	2	2	-2	0	0	0	-1	2	-2	0	-1	1	-2	-2	2	0	0	0	0	1	1	-1	1	-1	1	-1	√3	-√3	-√3	√3	orthogonal lifted from D₁₂
ρ₂₁	2	2	2	0	0	0	2	-2	-2	0	2	2	0	0	0	0	0	-2i	2i	-2	-2	-2	0	0	0	0	0	0	0	0	complex lifted from C₄○D₄
ρ₂₂	2	2	2	0	0	0	2	-2	-2	0	2	2	0	0	0	0	0	2i	-2i	-2	-2	-2	0	0	0	0	0	0	0	0	complex lifted from C₄○D₄
ρ₂₃	4	4	-4	0	0	0	-2	-4	4	0	-2	2	0	0	0	0	0	0	0	-2	-2	2	0	0	0	0	0	0	0	0	orthogonal lifted from S₃×D₄
ρ₂₄	4	4	4	0	0	0	-2	-4	-4	0	-2	-2	0	0	0	0	0	0	0	2	2	2	0	0	0	0	0	0	0	0	orthogonal lifted from Q₈⋊₃S₃, Schur index 2
ρ₂₅	4	-4	0	0	0	0	4	0	0	0	-4	0	-2√2	2√2	0	0	0	0	0	0	0	0	2√2	0	-2√2	0	0	0	0	0	orthogonal lifted from D₄.₄D₄
ρ₂₆	4	-4	0	0	0	0	4	0	0	0	-4	0	2√2	-2√2	0	0	0	0	0	0	0	0	-2√2	0	2√2	0	0	0	0	0	orthogonal lifted from D₄.₄D₄
ρ₂₇	4	-4	0	0	0	0	-2	0	0	0	2	0	2√2	-2√2	0	0	0	0	0	2√3	-2√3	0	√2	√6	-√2	-√6	0	0	0	0	orthogonal faithful
ρ₂₈	4	-4	0	0	0	0	-2	0	0	0	2	0	-2√2	2√2	0	0	0	0	0	2√3	-2√3	0	-√2	-√6	√2	√6	0	0	0	0	orthogonal faithful
ρ₂₉	4	-4	0	0	0	0	-2	0	0	0	2	0	2√2	-2√2	0	0	0	0	0	-2√3	2√3	0	√2	-√6	-√2	√6	0	0	0	0	orthogonal faithful
ρ₃₀	4	-4	0	0	0	0	-2	0	0	0	2	0	-2√2	2√2	0	0	0	0	0	-2√3	2√3	0	-√2	√6	√2	-√6	0	0	0	0	orthogonal faithful

Smallest permutation representation of C₂₄.₁₉D₄
►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 44 19 26 13 32 7 38)(2 27 20 33 14 39 8 45)(3 34 21 40 15 46 9 28)(4 41 22 47 16 29 10 35)(5 48 23 30 17 36 11 42)(6 31 24 37 18 43 12 25)

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

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

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

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

Matrix representation of C₂₄.₁₉D₄ ►in GL₆(𝔽₇₃)

9	45	0	0	0	0
0	65	0	0	0	0
0	0	13	25	0	0
0	0	70	28	0	0
0	0	0	18	28	50
0	0	4	53	35	13

27	27	0	0	0	0
0	46	0	0	0	0
0	0	0	0	1	0
0	0	0	46	0	71
0	0	21	3	0	0
0	0	25	10	0	27

9	45	0	0	0	0
55	64	0	0	0	0
0	0	13	25	0	0
0	0	40	60	0	0
0	0	0	18	28	50
0	0	44	53	15	45

G:=sub<GL(6,GF(73))| [9,0,0,0,0,0,45,65,0,0,0,0,0,0,13,70,0,4,0,0,25,28,18,53,0,0,0,0,28,35,0,0,0,0,50,13],[27,0,0,0,0,0,27,46,0,0,0,0,0,0,0,0,21,25,0,0,0,46,3,10,0,0,1,0,0,0,0,0,0,71,0,27],[9,55,0,0,0,0,45,64,0,0,0,0,0,0,13,40,0,44,0,0,25,60,18,53,0,0,0,0,28,15,0,0,0,0,50,45] >;

C₂₄.₁₉D₄ in GAP, Magma, Sage, TeX

C_{24}._{19}D_4

% in TeX

G:=Group("C24.19D4");

// GroupNames label

G:=SmallGroup(192,456);

// by ID

G=gap.SmallGroup(192,456);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,120,254,219,226,1123,136,438,102,6278]);

// Polycyclic

G:=Group<a,b,c|a^24=c^2=1,b^4=a^12,b*a*b^-1=a^7,c*a*c=a^-1,c*b*c=a^12*b^3>;

// generators/relations

Export

Character table of C₂₄.₁₉D₄ in TeX

G = C24.19D4 order 192 = 26·3

19th non-split extension by C24 of D4 acting via D4/C2=C22

G = C₂₄.₁₉D₄ order 192 = 2⁶·3

19^th non-split extension by C₂₄ of D₄ acting via D₄/C₂=C₂²