D24:8C4 - GroupNames

G = D₂₄:₈C₄ order 192 = 2⁶·3

8^th semidirect product of D₂₄ and C₄ acting via C₄/C₂=C₂

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D₂₄:₈C₄, C₈.₂₂D₁₂, C₂₄.₃₈D₄, Dic₁₂:₈C₄, C₁₂.₃SD₁₆, C₈.₁(C₄xS₃), C₄.Q₈:₁S₃, (C₂xC₆).₃₀D₈, (C₂xC₈).₄₁D₆, C₃:₁(D₈:₂C₄), C₂₄.₂₁(C₂xC₄), C₄.₁(D₆:C₄), C₄oD₂₄.₆C₂, (C₂xC₁₂).₈₈D₄, C₁₂.C₈:₄C₂, C₆.₄(D₄:C₄), C₁₂.₁(C₂²:C₄), (C₂xC₂₄).₄₇C₂², C₂².₈(D₄:S₃), C₄.₇(Q₈:₂S₃), C₂.₆(C₆.D₈), (C₃xC₄.Q₈):₁C₂, (C₂xC₄).₁₆(C₃:D₄), SmallGroup(192,47)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂₄ — D₂₄:₈C₄

Chief series

C₁ — C₃ — C₆ — C₁₂ — C₂xC₁₂ — C₂xC₂₄ — C₄oD₂₄ — D₂₄:₈C₄

Lower central

C₃ — C₆ — C₁₂ — C₂₄ — D₂₄:₈C₄

Upper central

C₁ — C₂ — C₂xC₄ — C₂xC₈ — C₄.Q₈

Generators and relations for D₂₄:₈C₄
G = < a,b,c | a²⁴=b²=c⁴=1, bab=a^-1, cac^-1=a¹⁹, cbc^-1=a¹⁵b >

Subgroups: 208 in 58 conjugacy classes, 25 normal (all characteristic)
C₁, C₂, C₂, C₃, C₄, C₄, C₂², C₂², S₃, C₆, C₆, C₈, C₂xC₄, C₂xC₄, D₄, Q₈, Dic₃, C₁₂, C₁₂, D₆, C₂xC₆, C₁₆, C₄:C₄, C₂xC₈, D₈, SD₁₆, Q₁₆, C₄oD₄, C₂₄, Dic₆, C₄xS₃, D₁₂, C₃:D₄, C₂xC₁₂, C₂xC₁₂, C₄.Q₈, M₅(2), C₄oD₈, C₃:C₁₆, C₂₄:C₂, D₂₄, Dic₁₂, C₃xC₄:C₄, C₂xC₂₄, C₄oD₁₂, D₈:₂C₄, C₁₂.C₈, C₃xC₄.Q₈, C₄oD₂₄, D₂₄:₈C₄
Quotients: C₁, C₂, C₄, C₂², S₃, C₂xC₄, D₄, D₆, C₂²:C₄, D₈, SD₁₆, C₄xS₃, D₁₂, C₃:D₄, D₄:C₄, D₆:C₄, D₄:S₃, Q₈:₂S₃, D₈:₂C₄, C₆.D₈, D₂₄:₈C₄

Character table of D₂₄:₈C₄

class	1	2A	2B	2C	3	4A	4B	4C	4D	4E	6A	6B	6C	8A	8B	8C	12A	12B	12C	12D	12E	12F	16A	16B	16C	16D	24A	24B	24C	24D
size	1	1	2	24	2	2	2	8	8	24	2	2	2	2	2	4	4	4	8	8	8	8	12	12	12	12	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	i	-i	-1	-1	1	-1	-1	-1	1	1	-1	i	i	-i	-i	i	-i	i	-i	1	-1	1	-1	linear of order 4
ρ₆	1	1	-1	-1	1	1	-1	-i	i	1	-1	1	-1	-1	-1	1	1	-1	-i	-i	i	i	i	-i	i	-i	1	-1	1	-1	linear of order 4
ρ₇	1	1	-1	1	1	1	-1	-i	i	-1	-1	1	-1	-1	-1	1	1	-1	-i	-i	i	i	-i	i	-i	i	1	-1	1	-1	linear of order 4
ρ₈	1	1	-1	-1	1	1	-1	i	-i	1	-1	1	-1	-1	-1	1	1	-1	i	i	-i	-i	-i	i	-i	i	1	-1	1	-1	linear of order 4
ρ₉	2	2	2	0	-1	2	2	-2	-2	0	-1	-1	-1	2	2	2	-1	-1	1	1	1	1	0	0	0	0	-1	-1	-1	-1	orthogonal lifted from D₆
ρ₁₀	2	2	2	0	-1	2	2	2	2	0	-1	-1	-1	2	2	2	-1	-1	-1	-1	-1	-1	0	0	0	0	-1	-1	-1	-1	orthogonal lifted from S₃
ρ₁₁	2	2	2	0	2	2	2	0	0	0	2	2	2	-2	-2	-2	2	2	0	0	0	0	0	0	0	0	-2	-2	-2	-2	orthogonal lifted from D₄
ρ₁₂	2	2	-2	0	2	2	-2	0	0	0	-2	2	-2	2	2	-2	2	-2	0	0	0	0	0	0	0	0	-2	2	-2	2	orthogonal lifted from D₄
ρ₁₃	2	2	2	0	2	-2	-2	0	0	0	2	2	2	0	0	0	-2	-2	0	0	0	0	-√2	-√2	√2	√2	0	0	0	0	orthogonal lifted from D₈
ρ₁₄	2	2	2	0	2	-2	-2	0	0	0	2	2	2	0	0	0	-2	-2	0	0	0	0	√2	√2	-√2	-√2	0	0	0	0	orthogonal lifted from D₈
ρ₁₅	2	2	-2	0	-1	2	-2	0	0	0	1	-1	1	2	2	-2	-1	1	√3	-√3	√3	-√3	0	0	0	0	1	-1	1	-1	orthogonal lifted from D₁₂
ρ₁₆	2	2	-2	0	-1	2	-2	0	0	0	1	-1	1	2	2	-2	-1	1	-√3	√3	-√3	√3	0	0	0	0	1	-1	1	-1	orthogonal lifted from D₁₂
ρ₁₇	2	2	-2	0	-1	2	-2	2i	-2i	0	1	-1	1	-2	-2	2	-1	1	-i	-i	i	i	0	0	0	0	-1	1	-1	1	complex lifted from C₄xS₃
ρ₁₈	2	2	-2	0	-1	2	-2	-2i	2i	0	1	-1	1	-2	-2	2	-1	1	i	i	-i	-i	0	0	0	0	-1	1	-1	1	complex lifted from C₄xS₃
ρ₁₉	2	2	-2	0	2	-2	2	0	0	0	-2	2	-2	0	0	0	-2	2	0	0	0	0	-√-2	√-2	√-2	-√-2	0	0	0	0	complex lifted from SD₁₆
ρ₂₀	2	2	-2	0	2	-2	2	0	0	0	-2	2	-2	0	0	0	-2	2	0	0	0	0	√-2	-√-2	-√-2	√-2	0	0	0	0	complex lifted from SD₁₆
ρ₂₁	2	2	2	0	-1	2	2	0	0	0	-1	-1	-1	-2	-2	-2	-1	-1	√-3	-√-3	-√-3	√-3	0	0	0	0	1	1	1	1	complex lifted from C₃:D₄
ρ₂₂	2	2	2	0	-1	2	2	0	0	0	-1	-1	-1	-2	-2	-2	-1	-1	-√-3	√-3	√-3	-√-3	0	0	0	0	1	1	1	1	complex lifted from C₃:D₄
ρ₂₃	4	4	4	0	-2	-4	-4	0	0	0	-2	-2	-2	0	0	0	2	2	0	0	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from D₄:S₃, Schur index 2
ρ₂₄	4	4	-4	0	-2	-4	4	0	0	0	2	-2	2	0	0	0	2	-2	0	0	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from Q₈:₂S₃
ρ₂₅	4	-4	0	0	4	0	0	0	0	0	0	-4	0	-2√-2	2√-2	0	0	0	0	0	0	0	0	0	0	0	0	-2√-2	0	2√-2	complex lifted from D₈:₂C₄
ρ₂₆	4	-4	0	0	4	0	0	0	0	0	0	-4	0	2√-2	-2√-2	0	0	0	0	0	0	0	0	0	0	0	0	2√-2	0	-2√-2	complex lifted from D₈:₂C₄
ρ₂₇	4	-4	0	0	-2	0	0	0	0	0	-2√-3	2	2√-3	-2√-2	2√-2	0	0	0	0	0	0	0	0	0	0	0	-√6	√-2	√6	-√-2	complex faithful
ρ₂₈	4	-4	0	0	-2	0	0	0	0	0	2√-3	2	-2√-3	-2√-2	2√-2	0	0	0	0	0	0	0	0	0	0	0	√6	√-2	-√6	-√-2	complex faithful
ρ₂₉	4	-4	0	0	-2	0	0	0	0	0	2√-3	2	-2√-3	2√-2	-2√-2	0	0	0	0	0	0	0	0	0	0	0	-√6	-√-2	√6	√-2	complex faithful
ρ₃₀	4	-4	0	0	-2	0	0	0	0	0	-2√-3	2	2√-3	2√-2	-2√-2	0	0	0	0	0	0	0	0	0	0	0	√6	-√-2	-√6	√-2	complex faithful

Smallest permutation representation of D₂₄:₈C₄
►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 42)(2 41)(3 40)(4 39)(5 38)(6 37)(7 36)(8 35)(9 34)(10 33)(11 32)(12 31)(13 30)(14 29)(15 28)(16 27)(17 26)(18 25)(19 48)(20 47)(21 46)(22 45)(23 44)(24 43)

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

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

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

Matrix representation of D₂₄:₈C₄ ►in GL₆(F₉₇)

0	1	0	0	0	0
96	96	0	0	0	0
0	0	40	40	0	0
0	0	57	40	0	0
0	0	0	0	57	40
0	0	0	0	57	57

27	42	0	0	0	0
15	70	0	0	0	0
0	0	0	0	57	40
0	0	0	0	57	57
0	0	40	40	0	0
0	0	57	40	0	0

75	0	0	0	0	0
0	75	0	0	0	0
0	0	1	0	0	0
0	0	0	96	0	0
0	0	0	0	40	57
0	0	0	0	57	57

G:=sub<GL(6,GF(97))| [0,96,0,0,0,0,1,96,0,0,0,0,0,0,40,57,0,0,0,0,40,40,0,0,0,0,0,0,57,57,0,0,0,0,40,57],[27,15,0,0,0,0,42,70,0,0,0,0,0,0,0,0,40,57,0,0,0,0,40,40,0,0,57,57,0,0,0,0,40,57,0,0],[75,0,0,0,0,0,0,75,0,0,0,0,0,0,1,0,0,0,0,0,0,96,0,0,0,0,0,0,40,57,0,0,0,0,57,57] >;

D₂₄:₈C₄ in GAP, Magma, Sage, TeX

D_{24}\rtimes_8C_4

% in TeX

G:=Group("D24:8C4");

// GroupNames label

G:=SmallGroup(192,47);

// by ID

G=gap.SmallGroup(192,47);

# by ID

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

// Polycyclic

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

// generators/relations

Export

Character table of D₂₄:₈C₄ in TeX

G = D24:8C4 order 192 = 26·3

8th semidirect product of D24 and C4 acting via C4/C2=C2

G = D₂₄:₈C₄ order 192 = 2⁶·3

8^th semidirect product of D₂₄ and C₄ acting via C₄/C₂=C₂