D48:C2 - GroupNames

G = D₄₈:C₂ order 192 = 2⁶·3

6^th semidirect product of D₄₈ and C₂ acting faithfully

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C₁₆:₃D₆, D₄₈:₆C₂, Q₁₆:₂D₆, D₈.₃D₆, D₆.₇D₈, C₄₈:₃C₂², SD₃₂:₁S₃, D₂₄:₆C₂², Dic₃.₉D₈, C₂₄.₁₇C₂³, C₃:C₈.₃D₄, (S₃xD₈):₅C₂, C₄.₅(S₃xD₄), C₃:D₁₆:₃C₂, C₃:C₁₆:₂C₂², D₆.C₈:₁C₂, (C₄xS₃).₈D₄, C₂.₂₀(S₃xD₈), C₆.₃₆(C₂xD₈), C₃:₃(C₁₆:C₂²), (C₃xSD₃₂):₁C₂, C₁₂.₁₁(C₂xD₄), C₈.₆D₆:₂C₂, D₂₄:C₂:₃C₂, (S₃xC₈).₄C₂², C₈.₂₃(C₂²xS₃), (C₃xQ₁₆):₅C₂², (C₃xD₈).₃C₂², SmallGroup(192,473)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂₄ — D₄₈:C₂

Chief series

C₁ — C₃ — C₆ — C₁₂ — C₂₄ — S₃xC₈ — S₃xD₈ — D₄₈:C₂

Lower central

C₃ — C₆ — C₁₂ — C₂₄ — D₄₈:C₂

Upper central

C₁ — C₂ — C₄ — C₈ — SD₃₂

Generators and relations for D₄₈:C₂
G = < a,b,c | a⁴⁸=b²=c²=1, bab=a^-1, cac=a⁷, bc=cb >

Subgroups: 396 in 90 conjugacy classes, 31 normal (all characteristic)
C₁, C₂, C₂, C₃, C₄, C₄, C₂², S₃, C₆, C₆, C₈, C₈, C₂xC₄, D₄, Q₈, C₂³, Dic₃, C₁₂, C₁₂, D₆, D₆, C₂xC₆, C₁₆, C₁₆, C₂xC₈, D₈, D₈, SD₁₆, Q₁₆, C₂xD₄, C₄oD₄, C₃:C₈, C₂₄, C₄xS₃, C₄xS₃, D₁₂, C₃:D₄, C₃xD₄, C₃xQ₈, C₂²xS₃, M₅(2), D₁₆, SD₃₂, SD₃₂, C₂xD₈, C₄oD₈, C₃:C₁₆, C₄₈, S₃xC₈, D₂₄, D₄:S₃, Q₈:₂S₃, C₃xD₈, C₃xQ₁₆, S₃xD₄, Q₈:₃S₃, C₁₆:C₂², D₆.C₈, D₄₈, C₃:D₁₆, C₈.₆D₆, C₃xSD₃₂, S₃xD₈, D₂₄:C₂, D₄₈:C₂
Quotients: C₁, C₂, C₂², S₃, D₄, C₂³, D₆, D₈, C₂xD₄, C₂²xS₃, C₂xD₈, S₃xD₄, C₁₆:C₂², S₃xD₈, D₄₈:C₂

Character table of D₄₈:C₂

class	1	2A	2B	2C	2D	2E	3	4A	4B	4C	6A	6B	8A	8B	8C	12A	12B	16A	16B	16C	16D	24A	24B	48A	48B	48C	48D
size	1	1	6	8	24	24	2	2	6	8	2	16	2	2	12	4	16	4	4	12	12	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	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	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	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	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	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	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	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	linear of order 2
ρ₉	2	2	0	2	0	0	-1	2	0	-2	-1	-1	2	2	0	-1	1	-2	-2	0	0	-1	-1	1	1	1	1	orthogonal lifted from D₆
ρ₁₀	2	2	0	2	0	0	-1	2	0	2	-1	-1	2	2	0	-1	-1	2	2	0	0	-1	-1	-1	-1	-1	-1	orthogonal lifted from S₃
ρ₁₁	2	2	2	0	0	0	2	2	2	0	2	0	-2	-2	-2	2	0	0	0	0	0	-2	-2	0	0	0	0	orthogonal lifted from D₄
ρ₁₂	2	2	0	-2	0	0	-1	2	0	2	-1	1	2	2	0	-1	-1	-2	-2	0	0	-1	-1	1	1	1	1	orthogonal lifted from D₆
ρ₁₃	2	2	0	-2	0	0	-1	2	0	-2	-1	1	2	2	0	-1	1	2	2	0	0	-1	-1	-1	-1	-1	-1	orthogonal lifted from D₆
ρ₁₄	2	2	-2	0	0	0	2	2	-2	0	2	0	-2	-2	2	2	0	0	0	0	0	-2	-2	0	0	0	0	orthogonal lifted from D₄
ρ₁₅	2	2	2	0	0	0	2	-2	-2	0	2	0	0	0	0	-2	0	√2	-√2	-√2	√2	0	0	√2	-√2	-√2	√2	orthogonal lifted from D₈
ρ₁₆	2	2	-2	0	0	0	2	-2	2	0	2	0	0	0	0	-2	0	√2	-√2	√2	-√2	0	0	√2	-√2	-√2	√2	orthogonal lifted from D₈
ρ₁₇	2	2	-2	0	0	0	2	-2	2	0	2	0	0	0	0	-2	0	-√2	√2	-√2	√2	0	0	-√2	√2	√2	-√2	orthogonal lifted from D₈
ρ₁₈	2	2	2	0	0	0	2	-2	-2	0	2	0	0	0	0	-2	0	-√2	√2	√2	-√2	0	0	-√2	√2	√2	-√2	orthogonal lifted from D₈
ρ₁₉	4	4	0	0	0	0	-2	4	0	0	-2	0	-4	-4	0	-2	0	0	0	0	0	2	2	0	0	0	0	orthogonal lifted from S₃xD₄
ρ₂₀	4	-4	0	0	0	0	4	0	0	0	-4	0	2√2	-2√2	0	0	0	0	0	0	0	-2√2	2√2	0	0	0	0	orthogonal lifted from C₁₆:C₂²
ρ₂₁	4	4	0	0	0	0	-2	-4	0	0	-2	0	0	0	0	2	0	2√2	-2√2	0	0	0	0	-√2	√2	√2	-√2	orthogonal lifted from S₃xD₈
ρ₂₂	4	-4	0	0	0	0	4	0	0	0	-4	0	-2√2	2√2	0	0	0	0	0	0	0	2√2	-2√2	0	0	0	0	orthogonal lifted from C₁₆:C₂²
ρ₂₃	4	4	0	0	0	0	-2	-4	0	0	-2	0	0	0	0	2	0	-2√2	2√2	0	0	0	0	√2	-√2	-√2	√2	orthogonal lifted from S₃xD₈
ρ₂₄	4	-4	0	0	0	0	-2	0	0	0	2	0	-2√2	2√2	0	0	0	0	0	0	0	-√2	√2	2ζ₁₆⁷ζ₃+ζ₁₆⁷+2ζ₁₆ζ₃+ζ₁₆	2ζ₁₆⁵ζ₃²+ζ₁₆⁵+2ζ₁₆³ζ₃²+ζ₁₆³	2ζ₁₆¹³ζ₃²+ζ₁₆¹³+2ζ₁₆¹¹ζ₃²+ζ₁₆¹¹	2ζ₁₆⁷ζ₃²+ζ₁₆⁷+2ζ₁₆ζ₃²+ζ₁₆	orthogonal faithful
ρ₂₅	4	-4	0	0	0	0	-2	0	0	0	2	0	2√2	-2√2	0	0	0	0	0	0	0	√2	-√2	2ζ₁₆⁵ζ₃²+ζ₁₆⁵+2ζ₁₆³ζ₃²+ζ₁₆³	2ζ₁₆⁷ζ₃²+ζ₁₆⁷+2ζ₁₆ζ₃²+ζ₁₆	2ζ₁₆⁷ζ₃+ζ₁₆⁷+2ζ₁₆ζ₃+ζ₁₆	2ζ₁₆¹³ζ₃²+ζ₁₆¹³+2ζ₁₆¹¹ζ₃²+ζ₁₆¹¹	orthogonal faithful
ρ₂₆	4	-4	0	0	0	0	-2	0	0	0	2	0	-2√2	2√2	0	0	0	0	0	0	0	-√2	√2	2ζ₁₆⁷ζ₃²+ζ₁₆⁷+2ζ₁₆ζ₃²+ζ₁₆	2ζ₁₆¹³ζ₃²+ζ₁₆¹³+2ζ₁₆¹¹ζ₃²+ζ₁₆¹¹	2ζ₁₆⁵ζ₃²+ζ₁₆⁵+2ζ₁₆³ζ₃²+ζ₁₆³	2ζ₁₆⁷ζ₃+ζ₁₆⁷+2ζ₁₆ζ₃+ζ₁₆	orthogonal faithful
ρ₂₇	4	-4	0	0	0	0	-2	0	0	0	2	0	2√2	-2√2	0	0	0	0	0	0	0	√2	-√2	2ζ₁₆¹³ζ₃²+ζ₁₆¹³+2ζ₁₆¹¹ζ₃²+ζ₁₆¹¹	2ζ₁₆⁷ζ₃+ζ₁₆⁷+2ζ₁₆ζ₃+ζ₁₆	2ζ₁₆⁷ζ₃²+ζ₁₆⁷+2ζ₁₆ζ₃²+ζ₁₆	2ζ₁₆⁵ζ₃²+ζ₁₆⁵+2ζ₁₆³ζ₃²+ζ₁₆³	orthogonal faithful

Smallest permutation representation of D₄₈:C₂
►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 33)(2 32)(3 31)(4 30)(5 29)(6 28)(7 27)(8 26)(9 25)(10 24)(11 23)(12 22)(13 21)(14 20)(15 19)(16 18)(34 48)(35 47)(36 46)(37 45)(38 44)(39 43)(40 42)

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

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

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

Matrix representation of D₄₈:C₂ ►in GL₄(F₇) generated by

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

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

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

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

D₄₈:C₂ in GAP, Magma, Sage, TeX

D_{48}\rtimes C_2

% in TeX

G:=Group("D48:C2");

// GroupNames label

G:=SmallGroup(192,473);

// by ID

G=gap.SmallGroup(192,473);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,758,135,184,346,185,192,851,438,102,6278]);

// Polycyclic

G:=Group<a,b,c|a^48=b^2=c^2=1,b*a*b=a^-1,c*a*c=a^7,b*c=c*b>;

// generators/relations

Export

Character table of D₄₈:C₂ in TeX

G = D48:C2 order 192 = 26·3

6th semidirect product of D48 and C2 acting faithfully

G = D₄₈:C₂ order 192 = 2⁶·3

6^th semidirect product of D₄₈ and C₂ acting faithfully