C24:11D4 - GroupNames

G = C₂₄:₁₁D₄ order 192 = 2⁶·3

11^st semidirect product of C₂₄ and D₄ acting via D₄/C₂=C₂²

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C₂₄:₁₁D₄, C₃:C₈:₉D₄, (C₂xD₈):₈S₃, (C₆xD₈):₉C₂, C₈:₃(C₃:D₄), C₃:₄(C₈:₃D₄), C₂₄:C₄:₉C₂, C₄.₂₁(S₃xD₄), (C₂xC₈).₈₅D₆, C₁₂:₃D₄:₅C₂, (C₂xD₄).₆₂D₆, C₁₂.₁₆₄(C₂xD₄), C₂³.₁₂D₆:₄C₂, C₆.₂₇(C₄:₁D₄), C₂.₂₈(D₈:S₃), C₆.₄₉(C₈:C₂²), (C₂xDic₃).₆₄D₄, (C₆xD₄).₈₁C₂², C₂².₂₅₄(S₃xD₄), C₂.₁₈(C₁₂:₃D₄), (C₂xC₁₂).₄₃₁C₂³, (C₂xC₂₄).₁₄₇C₂², (C₂xD₁₂).₁₁₅C₂², (C₄xDic₃).₄₅C₂², (C₂xDic₆).₁₂₀C₂², C₄.₅(C₂xC₃:D₄), (C₂xD₄:S₃):₁₈C₂, (C₂xC₂₄:C₂):₂₃C₂, (C₂xD₄.S₃):₁₇C₂, (C₂xC₆).₃₄₄(C₂xD₄), (C₂xC₃:C₈).₁₄₈C₂², (C₂xC₄).₅₂₁(C₂²xS₃), SmallGroup(192,713)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂xC₁₂ — C₂₄:₁₁D₄

Chief series

C₁ — C₃ — C₆ — C₁₂ — C₂xC₁₂ — C₄xDic₃ — C₁₂:₃D₄ — C₂₄:₁₁D₄

Lower central

C₃ — C₆ — C₂xC₁₂ — C₂₄:₁₁D₄

Upper central

C₁ — C₂² — C₂xC₄ — C₂xD₈

Generators and relations for C₂₄:₁₁D₄
G = < a,b,c | a²⁴=b⁴=c²=1, bab^-1=a⁵, cac=a¹¹, cbc=b^-1 >

Subgroups: 504 in 144 conjugacy classes, 43 normal (31 characteristic)
C₁, C₂, C₂, C₂, C₃, C₄, C₄, C₂², C₂², S₃, C₆, C₆, C₆, C₈, C₈, C₂xC₄, C₂xC₄, D₄, Q₈, C₂³, Dic₃, C₁₂, D₆, C₂xC₆, C₂xC₆, C₄², C₂²:C₄, C₂xC₈, C₂xC₈, D₈, SD₁₆, C₂xD₄, C₂xD₄, C₂xQ₈, C₃:C₈, C₂₄, Dic₆, D₁₂, C₂xDic₃, C₂xDic₃, C₃:D₄, C₂xC₁₂, C₃xD₄, C₂²xS₃, C₂²xC₆, C₈:C₄, C₄.₄D₄, C₄:₁D₄, C₂xD₈, C₂xD₈, C₂xSD₁₆, C₂₄:C₂, C₂xC₃:C₈, C₄xDic₃, D₄:S₃, D₄.S₃, C₆.D₄, C₂xC₂₄, C₃xD₈, C₂xDic₆, C₂xD₁₂, C₂xC₃:D₄, C₆xD₄, C₈:₃D₄, C₂₄:C₄, C₂xC₂₄:C₂, C₂xD₄:S₃, C₂xD₄.S₃, C₂³.₁₂D₆, C₁₂:₃D₄, C₆xD₈, C₂₄:₁₁D₄
Quotients: C₁, C₂, C₂², S₃, D₄, C₂³, D₆, C₂xD₄, C₃:D₄, C₂²xS₃, C₄:₁D₄, C₈:C₂², S₃xD₄, C₂xC₃:D₄, C₈:₃D₄, D₈:S₃, C₁₂:₃D₄, C₂₄:₁₁D₄

Character table of C₂₄:₁₁D₄

class	1	2A	2B	2C	2D	2E	2F	3	4A	4B	4C	4D	4E	6A	6B	6C	6D	6E	6F	6G	8A	8B	8C	8D	12A	12B	24A	24B	24C	24D
size	1	1	1	1	8	8	24	2	2	2	12	12	24	2	2	2	8	8	8	8	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	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	-2	0	0	0	2	2	-2	0	0	0	-2	2	-2	0	0	0	0	0	0	-2	2	2	-2	0	0	0	0	orthogonal lifted from D₄
ρ₁₀	2	2	2	2	0	0	0	2	-2	-2	2	-2	0	2	2	2	0	0	0	0	0	0	0	0	-2	-2	0	0	0	0	orthogonal lifted from D₄
ρ₁₁	2	2	2	2	-2	-2	0	-1	2	2	0	0	0	-1	-1	-1	1	1	1	1	2	2	0	0	-1	-1	-1	-1	-1	-1	orthogonal lifted from D₆
ρ₁₂	2	2	-2	-2	0	0	0	2	-2	2	0	0	0	-2	2	-2	0	0	0	0	-2	2	0	0	-2	2	-2	2	-2	2	orthogonal lifted from D₄
ρ₁₃	2	2	2	2	-2	2	0	-1	2	2	0	0	0	-1	-1	-1	1	1	-1	-1	-2	-2	0	0	-1	-1	1	1	1	1	orthogonal lifted from D₆
ρ₁₄	2	2	2	2	0	0	0	2	-2	-2	-2	2	0	2	2	2	0	0	0	0	0	0	0	0	-2	-2	0	0	0	0	orthogonal lifted from D₄
ρ₁₅	2	2	-2	-2	0	0	0	2	2	-2	0	0	0	-2	2	-2	0	0	0	0	0	0	2	-2	2	-2	0	0	0	0	orthogonal lifted from D₄
ρ₁₆	2	2	2	2	2	-2	0	-1	2	2	0	0	0	-1	-1	-1	-1	-1	1	1	-2	-2	0	0	-1	-1	1	1	1	1	orthogonal lifted from D₆
ρ₁₇	2	2	2	2	2	2	0	-1	2	2	0	0	0	-1	-1	-1	-1	-1	-1	-1	2	2	0	0	-1	-1	-1	-1	-1	-1	orthogonal lifted from S₃
ρ₁₈	2	2	-2	-2	0	0	0	2	-2	2	0	0	0	-2	2	-2	0	0	0	0	2	-2	0	0	-2	2	2	-2	2	-2	orthogonal lifted from D₄
ρ₁₉	2	2	-2	-2	0	0	0	-1	-2	2	0	0	0	1	-1	1	√-3	-√-3	√-3	-√-3	2	-2	0	0	1	-1	-1	1	-1	1	complex lifted from C₃:D₄
ρ₂₀	2	2	-2	-2	0	0	0	-1	-2	2	0	0	0	1	-1	1	√-3	-√-3	-√-3	√-3	-2	2	0	0	1	-1	1	-1	1	-1	complex lifted from C₃:D₄
ρ₂₁	2	2	-2	-2	0	0	0	-1	-2	2	0	0	0	1	-1	1	-√-3	√-3	√-3	-√-3	-2	2	0	0	1	-1	1	-1	1	-1	complex lifted from C₃:D₄
ρ₂₂	2	2	-2	-2	0	0	0	-1	-2	2	0	0	0	1	-1	1	-√-3	√-3	-√-3	√-3	2	-2	0	0	1	-1	-1	1	-1	1	complex lifted from C₃:D₄
ρ₂₃	4	-4	-4	4	0	0	0	4	0	0	0	0	0	-4	-4	4	0	0	0	0	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from C₈:C₂²
ρ₂₄	4	4	4	4	0	0	0	-2	-4	-4	0	0	0	-2	-2	-2	0	0	0	0	0	0	0	0	2	2	0	0	0	0	orthogonal lifted from S₃xD₄
ρ₂₅	4	-4	4	-4	0	0	0	4	0	0	0	0	0	4	-4	-4	0	0	0	0	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from C₈:C₂²
ρ₂₆	4	4	-4	-4	0	0	0	-2	4	-4	0	0	0	2	-2	2	0	0	0	0	0	0	0	0	-2	2	0	0	0	0	orthogonal lifted from S₃xD₄
ρ₂₇	4	-4	-4	4	0	0	0	-2	0	0	0	0	0	2	2	-2	0	0	0	0	0	0	0	0	0	0	√-6	-√-6	-√-6	√-6	complex lifted from D₈:S₃
ρ₂₈	4	-4	4	-4	0	0	0	-2	0	0	0	0	0	-2	2	2	0	0	0	0	0	0	0	0	0	0	-√-6	-√-6	√-6	√-6	complex lifted from D₈:S₃
ρ₂₉	4	-4	-4	4	0	0	0	-2	0	0	0	0	0	2	2	-2	0	0	0	0	0	0	0	0	0	0	-√-6	√-6	√-6	-√-6	complex lifted from D₈:S₃
ρ₃₀	4	-4	4	-4	0	0	0	-2	0	0	0	0	0	-2	2	2	0	0	0	0	0	0	0	0	0	0	√-6	√-6	-√-6	-√-6	complex lifted from D₈:S₃

Smallest permutation representation of C₂₄:₁₁D₄
►On 96 points

Generators in S₉₆

(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)(49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72)(73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96)

(1 70 88 37)(2 51 89 42)(3 56 90 47)(4 61 91 28)(5 66 92 33)(6 71 93 38)(7 52 94 43)(8 57 95 48)(9 62 96 29)(10 67 73 34)(11 72 74 39)(12 53 75 44)(13 58 76 25)(14 63 77 30)(15 68 78 35)(16 49 79 40)(17 54 80 45)(18 59 81 26)(19 64 82 31)(20 69 83 36)(21 50 84 41)(22 55 85 46)(23 60 86 27)(24 65 87 32)

(2 12)(3 23)(4 10)(5 21)(6 8)(7 19)(9 17)(11 15)(14 24)(16 22)(18 20)(25 58)(26 69)(27 56)(28 67)(29 54)(30 65)(31 52)(32 63)(33 50)(34 61)(35 72)(36 59)(37 70)(38 57)(39 68)(40 55)(41 66)(42 53)(43 64)(44 51)(45 62)(46 49)(47 60)(48 71)(73 91)(74 78)(75 89)(77 87)(79 85)(80 96)(81 83)(82 94)(84 92)(86 90)(93 95)

G:=sub<Sym(96)| (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)(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96), (1,70,88,37)(2,51,89,42)(3,56,90,47)(4,61,91,28)(5,66,92,33)(6,71,93,38)(7,52,94,43)(8,57,95,48)(9,62,96,29)(10,67,73,34)(11,72,74,39)(12,53,75,44)(13,58,76,25)(14,63,77,30)(15,68,78,35)(16,49,79,40)(17,54,80,45)(18,59,81,26)(19,64,82,31)(20,69,83,36)(21,50,84,41)(22,55,85,46)(23,60,86,27)(24,65,87,32), (2,12)(3,23)(4,10)(5,21)(6,8)(7,19)(9,17)(11,15)(14,24)(16,22)(18,20)(25,58)(26,69)(27,56)(28,67)(29,54)(30,65)(31,52)(32,63)(33,50)(34,61)(35,72)(36,59)(37,70)(38,57)(39,68)(40,55)(41,66)(42,53)(43,64)(44,51)(45,62)(46,49)(47,60)(48,71)(73,91)(74,78)(75,89)(77,87)(79,85)(80,96)(81,83)(82,94)(84,92)(86,90)(93,95)>;

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)(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96), (1,70,88,37)(2,51,89,42)(3,56,90,47)(4,61,91,28)(5,66,92,33)(6,71,93,38)(7,52,94,43)(8,57,95,48)(9,62,96,29)(10,67,73,34)(11,72,74,39)(12,53,75,44)(13,58,76,25)(14,63,77,30)(15,68,78,35)(16,49,79,40)(17,54,80,45)(18,59,81,26)(19,64,82,31)(20,69,83,36)(21,50,84,41)(22,55,85,46)(23,60,86,27)(24,65,87,32), (2,12)(3,23)(4,10)(5,21)(6,8)(7,19)(9,17)(11,15)(14,24)(16,22)(18,20)(25,58)(26,69)(27,56)(28,67)(29,54)(30,65)(31,52)(32,63)(33,50)(34,61)(35,72)(36,59)(37,70)(38,57)(39,68)(40,55)(41,66)(42,53)(43,64)(44,51)(45,62)(46,49)(47,60)(48,71)(73,91)(74,78)(75,89)(77,87)(79,85)(80,96)(81,83)(82,94)(84,92)(86,90)(93,95) );

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),(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72),(73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96)], [(1,70,88,37),(2,51,89,42),(3,56,90,47),(4,61,91,28),(5,66,92,33),(6,71,93,38),(7,52,94,43),(8,57,95,48),(9,62,96,29),(10,67,73,34),(11,72,74,39),(12,53,75,44),(13,58,76,25),(14,63,77,30),(15,68,78,35),(16,49,79,40),(17,54,80,45),(18,59,81,26),(19,64,82,31),(20,69,83,36),(21,50,84,41),(22,55,85,46),(23,60,86,27),(24,65,87,32)], [(2,12),(3,23),(4,10),(5,21),(6,8),(7,19),(9,17),(11,15),(14,24),(16,22),(18,20),(25,58),(26,69),(27,56),(28,67),(29,54),(30,65),(31,52),(32,63),(33,50),(34,61),(35,72),(36,59),(37,70),(38,57),(39,68),(40,55),(41,66),(42,53),(43,64),(44,51),(45,62),(46,49),(47,60),(48,71),(73,91),(74,78),(75,89),(77,87),(79,85),(80,96),(81,83),(82,94),(84,92),(86,90),(93,95)]])

Matrix representation of C₂₄:₁₁D₄ ►in GL₆(F₇₃)

72	0	0	0	0	0
0	72	0	0	0	0
0	0	42	31	31	42
0	0	42	11	31	62
0	0	42	31	42	31
0	0	42	11	42	11

0	72	0	0	0	0
1	0	0	0	0	0
0	0	17	34	24	0
0	0	17	56	49	49
0	0	49	0	17	34
0	0	24	24	17	56

72	0	0	0	0	0
0	1	0	0	0	0
0	0	1	0	0	0
0	0	72	72	0	0
0	0	0	0	72	0
0	0	0	0	1	1

G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,42,42,42,42,0,0,31,11,31,11,0,0,31,31,42,42,0,0,42,62,31,11],[0,1,0,0,0,0,72,0,0,0,0,0,0,0,17,17,49,24,0,0,34,56,0,24,0,0,24,49,17,17,0,0,0,49,34,56],[72,0,0,0,0,0,0,1,0,0,0,0,0,0,1,72,0,0,0,0,0,72,0,0,0,0,0,0,72,1,0,0,0,0,0,1] >;

C₂₄:₁₁D₄ in GAP, Magma, Sage, TeX

C_{24}\rtimes_{11}D_4

% in TeX

G:=Group("C24:11D4");

// GroupNames label

G:=SmallGroup(192,713);

// by ID

G=gap.SmallGroup(192,713);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,253,1094,135,570,297,136,6278]);

// Polycyclic

G:=Group<a,b,c|a^24=b^4=c^2=1,b*a*b^-1=a^5,c*a*c=a^11,c*b*c=b^-1>;

// generators/relations

Export

Character table of C₂₄:₁₁D₄ in TeX

G = C24:11D4 order 192 = 26·3

11st semidirect product of C24 and D4 acting via D4/C2=C22

G = C₂₄:₁₁D₄ order 192 = 2⁶·3

11^st semidirect product of C₂₄ and D₄ acting via D₄/C₂=C₂²