D24:2C4 - GroupNames

G = D₂₄:₂C₄ order 192 = 2⁶·3

2^nd semidirect product of D₂₄ and C₄ acting via C₄/C₂=C₂

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D₂₄:₂C₄, C₂₄.₈₆D₄, Dic₁₂:₂C₄, M₅(2):₆S₃, C₁₂.₇SD₁₆, C₂².₃D₂₄, C₈.₆(C₄xS₃), (C₂xC₆).₂D₈, C₂₄.₃(C₂xC₄), C₈:Dic₃:₁C₂, (C₂xC₈).₄₈D₆, C₃:₂(D₈:₂C₄), C₄oD₂₄.₇C₂, (C₂xC₄).₁₁D₁₂, C₄.₂₀(D₆:C₄), (C₂xC₁₂).₁₀₁D₄, C₈.₄₃(C₃:D₄), C₄.₁₂(C₂₄:C₂), (C₃xM₅(2)):₁₀C₂, (C₂xC₂₄).₅₂C₂², C₆.₁₉(D₄:C₄), C₁₂.₄₄(C₂²:C₄), C₂.₁₁(C₂.D₂₄), SmallGroup(192,77)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂₄ — D₂₄:₂C₄

Chief series

C₁ — C₃ — C₆ — C₁₂ — C₂₄ — C₂xC₂₄ — C₄oD₂₄ — D₂₄:₂C₄

Lower central

C₃ — C₆ — C₁₂ — C₂₄ — D₂₄:₂C₄

Upper central

C₁ — C₂ — C₂xC₄ — C₂xC₈ — M₅(2)

Generators and relations for D₂₄:₂C₄
G = < a,b,c | a²⁴=b²=c⁴=1, bab=a^-1, cac^-1=a¹¹, cbc^-1=a⁷b >

Subgroups: 232 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, D₆, C₂xC₆, C₁₆, C₄:C₄, C₂xC₈, D₈, SD₁₆, Q₁₆, C₄oD₄, C₂₄, Dic₆, C₄xS₃, D₁₂, C₂xDic₃, C₃:D₄, C₂xC₁₂, C₄.Q₈, M₅(2), C₄oD₈, C₄₈, C₂₄:C₂, D₂₄, Dic₁₂, C₄:Dic₃, C₂xC₂₄, C₄oD₁₂, D₈:₂C₄, C₈:Dic₃, C₃xM₅(2), C₄oD₂₄, D₂₄:₂C₄
Quotients: C₁, C₂, C₄, C₂², S₃, C₂xC₄, D₄, D₆, C₂²:C₄, D₈, SD₁₆, C₄xS₃, D₁₂, C₃:D₄, D₄:C₄, C₂₄:C₂, D₂₄, D₆:C₄, D₈:₂C₄, C₂.D₂₄, D₂₄:₂C₄

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

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

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

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

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

36 conjugacy classes

class	1	2A	2B	2C	3	4A	4B	4C	4D	4E	6A	6B	8A	8B	8C	12A	12B	12C	16A	16B	16C	16D	24A	24B	24C	24D	24E	24F	48A	···	48H
order	1	2	2	2	3	4	4	4	4	4	6	6	8	8	8	12	12	12	16	16	16	16	24	24	24	24	24	24	48	···	48
size	1	1	2	24	2	2	2	24	24	24	2	4	2	2	4	2	2	4	4	4	4	4	2	2	2	2	4	4	4	···	4

36 irreducible representations

dim	1	1	1	1	1	1	2	2	2	2	2	2	2	2	2	2	2	4	4
type	+	+	+	+			+	+	+	+		+			+		+
image	C₁	C₂	C₂	C₂	C₄	C₄	S₃	D₄	D₄	D₆	SD₁₆	D₈	C₄xS₃	C₃:D₄	D₁₂	C₂₄:C₂	D₂₄	D₈:₂C₄	D₂₄:₂C₄
kernel	D₂₄:₂C₄	C₈:Dic₃	C₃xM₅(2)	C₄oD₂₄	D₂₄	Dic₁₂	M₅(2)	C₂₄	C₂xC₁₂	C₂xC₈	C₁₂	C₂xC₆	C₈	C₈	C₂xC₄	C₄	C₂²	C₃	C₁
# reps	1	1	1	1	2	2	1	1	1	1	2	2	2	2	2	4	4	2	4

Matrix representation of D₂₄:₂C₄ ►in GL₆(F₉₇)

0	1	0	0	0	0
96	96	0	0	0	0
0	0	17	74	0	0
0	0	59	0	0	0
0	0	81	85	40	40
0	0	89	85	57	40

0	96	0	0	0	0
96	0	0	0	0	0
0	0	73	0	23	23
0	0	8	0	0	17
0	0	55	57	12	12
0	0	17	40	12	12

22	0	0	0	0	0
75	75	0	0	0	0
0	0	1	0	0	0
0	0	64	96	0	0
0	0	76	0	57	57
0	0	68	0	57	40

G:=sub<GL(6,GF(97))| [0,96,0,0,0,0,1,96,0,0,0,0,0,0,17,59,81,89,0,0,74,0,85,85,0,0,0,0,40,57,0,0,0,0,40,40],[0,96,0,0,0,0,96,0,0,0,0,0,0,0,73,8,55,17,0,0,0,0,57,40,0,0,23,0,12,12,0,0,23,17,12,12],[22,75,0,0,0,0,0,75,0,0,0,0,0,0,1,64,76,68,0,0,0,96,0,0,0,0,0,0,57,57,0,0,0,0,57,40] >;

D₂₄:₂C₄ in GAP, Magma, Sage, TeX

D_{24}\rtimes_2C_4

% in TeX

G:=Group("D24:2C4");

// GroupNames label

G:=SmallGroup(192,77);

// by ID

G=gap.SmallGroup(192,77);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,85,92,422,387,268,570,136,1684,102,6278]);

// Polycyclic

G:=Group<a,b,c|a^24=b^2=c^4=1,b*a*b=a^-1,c*a*c^-1=a^11,c*b*c^-1=a^7*b>;

// generators/relations

G = D24:2C4 order 192 = 26·3

2nd semidirect product of D24 and C4 acting via C4/C2=C2

G = D₂₄:₂C₄ order 192 = 2⁶·3

2^nd semidirect product of D₂₄ and C₄ acting via C₄/C₂=C₂