D24.C4 - GroupNames

G = D₂₄.C₄ order 192 = 2⁶·3

4^th non-split extension by D₂₄ of C₄ acting via C₄/C₂=C₂

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C₂₄.₇D₄, D₂₄.₄C₄, C₈.₁₇D₁₂, C₁₂.₄₈D₈, C₈.₂(C₄×S₃), (C₂×C₈).₄₃D₆, C₈.C₄⋊₁S₃, C₂₄.₂₂(C₂×C₄), C₄.₄(D₆⋊C₄), (C₂×C₁₂).₉₅D₄, C₁₂.C₈⋊₅C₂, (C₂×C₆).₄SD₁₆, C₄.₂₁(D₄⋊S₃), (C₂×D₂₄).₁₂C₂, C₃⋊₁(M₅(2)⋊C₂), C₆.₇(D₄⋊C₄), C₁₂.₄(C₂²⋊C₄), C₂.₉(C₆.D₈), (C₂×C₂₄).₁₀₀C₂², C₂².₃(Q₈⋊₂S₃), (C₃×C₈.C₄)⋊₉C₂, (C₂×C₄).₁₈(C₃⋊D₄), SmallGroup(192,54)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂₄ — D₂₄.C₄

Chief series

C₁ — C₃ — C₆ — C₁₂ — C₂×C₁₂ — C₂×C₂₄ — C₂×D₂₄ — D₂₄.C₄

Lower central

C₃ — C₆ — C₁₂ — C₂₄ — D₂₄.C₄

Upper central

C₁ — C₂ — C₂×C₄ — C₂×C₈ — C₈.C₄

Generators and relations for D₂₄.C₄
G = < a,b,c | a²⁴=b²=1, c⁴=a¹², bab=a^-1, cac^-1=a¹⁹, cbc^-1=a¹⁵b >

Subgroups: 288 in 62 conjugacy classes, 25 normal (23 characteristic)
C₁, C₂, C₂, C₃, C₄, C₂², C₂², S₃, C₆, C₆, C₈, C₈, C₂×C₄, D₄, C₂³, C₁₂, D₆, C₂×C₆, C₁₆, C₂×C₈, M₄(2), D₈, C₂×D₄, C₂₄, C₂₄, D₁₂, C₂×C₁₂, C₂²×S₃, C₈.C₄, M₅(2), C₂×D₈, C₃⋊C₁₆, D₂₄, D₂₄, C₂×C₂₄, C₃×M₄(2), C₂×D₁₂, M₅(2)⋊C₂, C₁₂.C₈, C₃×C₈.C₄, C₂×D₂₄, D₂₄.C₄
Quotients: C₁, C₂, C₄, C₂², S₃, C₂×C₄, D₄, D₆, C₂²⋊C₄, D₈, SD₁₆, C₄×S₃, D₁₂, C₃⋊D₄, D₄⋊C₄, D₆⋊C₄, D₄⋊S₃, Q₈⋊₂S₃, M₅(2)⋊C₂, C₆.D₈, D₂₄.C₄

Character table of D₂₄.C₄

class	1	2A	2B	2C	2D	3	4A	4B	6A	6B	8A	8B	8C	8D	8E	12A	12B	12C	16A	16B	16C	16D	24A	24B	24C	24D	24E	24F	24G	24H
size	1	1	2	24	24	2	2	2	2	4	2	2	4	8	8	2	2	4	12	12	12	12	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	i	-i	-1	-1	1	-i	i	-i	i	-1	1	-1	1	-i	i	i	-i	linear of order 4
ρ₆	1	1	-1	1	-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	linear of order 4
ρ₇	1	1	-1	1	-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	linear of order 4
ρ₈	1	1	-1	-1	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	linear of order 4
ρ₉	2	2	2	0	0	2	2	2	2	2	-2	-2	-2	0	0	2	2	2	0	0	0	0	-2	-2	-2	-2	0	0	0	0	orthogonal lifted from D₄
ρ₁₀	2	2	-2	0	0	2	-2	2	2	-2	2	2	-2	0	0	-2	-2	2	0	0	0	0	2	-2	2	-2	0	0	0	0	orthogonal lifted from D₄
ρ₁₁	2	2	2	0	0	-1	2	2	-1	-1	2	2	2	2	2	-1	-1	-1	0	0	0	0	-1	-1	-1	-1	-1	-1	-1	-1	orthogonal lifted from S₃
ρ₁₂	2	2	2	0	0	-1	2	2	-1	-1	2	2	2	-2	-2	-1	-1	-1	0	0	0	0	-1	-1	-1	-1	1	1	1	1	orthogonal lifted from D₆
ρ₁₃	2	2	-2	0	0	2	2	-2	2	-2	0	0	0	0	0	2	2	-2	√2	√2	-√2	-√2	0	0	0	0	0	0	0	0	orthogonal lifted from D₈
ρ₁₄	2	2	-2	0	0	2	2	-2	2	-2	0	0	0	0	0	2	2	-2	-√2	-√2	√2	√2	0	0	0	0	0	0	0	0	orthogonal lifted from D₈
ρ₁₅	2	2	-2	0	0	-1	-2	2	-1	1	2	2	-2	0	0	1	1	-1	0	0	0	0	-1	1	-1	1	-√3	-√3	√3	√3	orthogonal lifted from D₁₂
ρ₁₆	2	2	-2	0	0	-1	-2	2	-1	1	2	2	-2	0	0	1	1	-1	0	0	0	0	-1	1	-1	1	√3	√3	-√3	-√3	orthogonal lifted from D₁₂
ρ₁₇	2	2	-2	0	0	-1	-2	2	-1	1	-2	-2	2	2i	-2i	1	1	-1	0	0	0	0	1	-1	1	-1	i	-i	-i	i	complex lifted from C₄×S₃
ρ₁₈	2	2	-2	0	0	-1	-2	2	-1	1	-2	-2	2	-2i	2i	1	1	-1	0	0	0	0	1	-1	1	-1	-i	i	i	-i	complex lifted from C₄×S₃
ρ₁₉	2	2	2	0	0	2	-2	-2	2	2	0	0	0	0	0	-2	-2	-2	-√-2	√-2	√-2	-√-2	0	0	0	0	0	0	0	0	complex lifted from SD₁₆
ρ₂₀	2	2	2	0	0	2	-2	-2	2	2	0	0	0	0	0	-2	-2	-2	√-2	-√-2	-√-2	√-2	0	0	0	0	0	0	0	0	complex lifted from SD₁₆
ρ₂₁	2	2	2	0	0	-1	2	2	-1	-1	-2	-2	-2	0	0	-1	-1	-1	0	0	0	0	1	1	1	1	-√-3	√-3	-√-3	√-3	complex lifted from C₃⋊D₄
ρ₂₂	2	2	2	0	0	-1	2	2	-1	-1	-2	-2	-2	0	0	-1	-1	-1	0	0	0	0	1	1	1	1	√-3	-√-3	√-3	-√-3	complex lifted from C₃⋊D₄
ρ₂₃	4	4	4	0	0	-2	-4	-4	-2	-2	0	0	0	0	0	2	2	2	0	0	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from Q₈⋊₂S₃
ρ₂₄	4	4	-4	0	0	-2	4	-4	-2	2	0	0	0	0	0	-2	-2	2	0	0	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from D₄⋊S₃, Schur index 2
ρ₂₅	4	-4	0	0	0	4	0	0	-4	0	2√2	-2√2	0	0	0	0	0	0	0	0	0	0	2√2	0	-2√2	0	0	0	0	0	orthogonal lifted from M₅(2)⋊C₂
ρ₂₆	4	-4	0	0	0	4	0	0	-4	0	-2√2	2√2	0	0	0	0	0	0	0	0	0	0	-2√2	0	2√2	0	0	0	0	0	orthogonal lifted from M₅(2)⋊C₂
ρ₂₇	4	-4	0	0	0	-2	0	0	2	0	2√2	-2√2	0	0	0	-2√3	2√3	0	0	0	0	0	-√2	√6	√2	-√6	0	0	0	0	orthogonal faithful
ρ₂₈	4	-4	0	0	0	-2	0	0	2	0	-2√2	2√2	0	0	0	2√3	-2√3	0	0	0	0	0	√2	√6	-√2	-√6	0	0	0	0	orthogonal faithful
ρ₂₉	4	-4	0	0	0	-2	0	0	2	0	2√2	-2√2	0	0	0	2√3	-2√3	0	0	0	0	0	-√2	-√6	√2	√6	0	0	0	0	orthogonal faithful
ρ₃₀	4	-4	0	0	0	-2	0	0	2	0	-2√2	2√2	0	0	0	-2√3	2√3	0	0	0	0	0	√2	-√6	-√2	√6	0	0	0	0	orthogonal faithful

Smallest permutation representation of D₂₄.C₄
►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 18)(2 17)(3 16)(4 15)(5 14)(6 13)(7 12)(8 11)(9 10)(19 24)(20 23)(21 22)(25 27)(28 48)(29 47)(30 46)(31 45)(32 44)(33 43)(34 42)(35 41)(36 40)(37 39)

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

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

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

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

Matrix representation of D₂₄.C₄ ►in GL₄(𝔽₉₇) generated by

18	16	0	0
81	2	0	0
12	0	16	2
12	12	95	18

79	95	0	0
16	18	0	0
60	25	68	58
25	62	29	29

56	15	33	33
82	41	31	33
53	54	56	82
96	53	15	41

G:=sub<GL(4,GF(97))| [18,81,12,12,16,2,0,12,0,0,16,95,0,0,2,18],[79,16,60,25,95,18,25,62,0,0,68,29,0,0,58,29],[56,82,53,96,15,41,54,53,33,31,56,15,33,33,82,41] >;

D₂₄.C₄ in GAP, Magma, Sage, TeX

D_{24}.C_4

% in TeX

G:=Group("D24.C4");

// GroupNames label

G:=SmallGroup(192,54);

// by ID

G=gap.SmallGroup(192,54);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,141,36,758,184,675,794,192,1684,851,102,6278]);

// Polycyclic

G:=Group<a,b,c|a^24=b^2=1,c^4=a^12,b*a*b=a^-1,c*a*c^-1=a^19,c*b*c^-1=a^15*b>;

// generators/relations

Export

Character table of D₂₄.C₄ in TeX

G = D24.C4 order 192 = 26·3

4th non-split extension by D24 of C4 acting via C4/C2=C2

G = D₂₄.C₄ order 192 = 2⁶·3

4^th non-split extension by D₂₄ of C₄ acting via C₄/C₂=C₂