C2^4:4ES-(3,1) - GroupNames

G = C₂⁴⋊₄3_-¹⁺² order 432 = 2⁴·3³

2^nd semidirect product of C₂⁴ and 3_-¹⁺² acting via 3_-¹⁺²/C₉=C₃

Aliases: C₂⁴⋊₄3_-¹⁺², (C₂×C₁₈)⋊₂A₄, C₉⋊(C₂²⋊A₄), C₂²⋊₂(C₉⋊A₄), (C₂³×C₁₈)⋊₃C₃, C₂⁴⋊C₉⋊₅C₃, (C₂³×C₆).₁₂C₃², (C₂×C₆).₁₃(C₃×A₄), C₃.₃(C₃×C₂²⋊A₄), (C₃×C₂²⋊A₄).₃C₃, SmallGroup(432,552)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂³×C₆ — C₂⁴⋊₄3_-¹⁺²

Chief series

C₁ — C₂² — C₂⁴ — C₂³×C₆ — C₃×C₂²⋊A₄ — C₂⁴⋊₄3_-¹⁺²

Lower central

C₂⁴ — C₂³×C₆ — C₂⁴⋊₄3_-¹⁺²

Upper central

C₁ — C₃ — C₉

Generators and relations for C₂⁴⋊₄3_-¹⁺²
G = < a,b,c,d,e,f | a²=b²=c²=d²=e⁹=f³=1, eae^-1=fbf^-1=ab=ba, ac=ca, ad=da, faf^-1=b, bc=cb, bd=db, ebe^-1=a, ece^-1=fdf^-1=cd=dc, fcf^-1=d, ede^-1=c, fef^-1=e⁴ >

Subgroups: 460 in 116 conjugacy classes, 25 normal (8 characteristic)
C₁, C₂, C₃, C₃, C₂², C₂², C₆, C₂³, C₉, C₉, C₃², A₄, C₂×C₆, C₂×C₆, C₂⁴, C₁₈, C₂²×C₆, 3_-¹⁺², C₃.A₄, C₂×C₁₈, C₂×C₁₈, C₃×A₄, C₂²⋊A₄, C₂³×C₆, C₂²×C₁₈, C₉⋊A₄, C₂⁴⋊C₉, C₂³×C₁₈, C₃×C₂²⋊A₄, C₂⁴⋊₄3_-¹⁺²
Quotients: C₁, C₃, C₃², A₄, 3_-¹⁺², C₃×A₄, C₂²⋊A₄, C₉⋊A₄, C₃×C₂²⋊A₄, C₂⁴⋊₄3_-¹⁺²

Smallest permutation representation of C₂⁴⋊₄3_-¹⁺²
►On 108 points

Generators in S₁₀₈

(1 40)(2 104)(3 95)(4 43)(5 107)(6 98)(7 37)(8 101)(9 92)(10 82)(11 63)(12 67)(13 85)(14 57)(15 70)(16 88)(17 60)(18 64)(19 31)(20 77)(21 51)(22 34)(23 80)(24 54)(25 28)(26 74)(27 48)(29 47)(30 75)(32 50)(33 78)(35 53)(36 81)(38 91)(39 102)(41 94)(42 105)(44 97)(45 108)(46 73)(49 76)(52 79)(55 84)(56 68)(58 87)(59 71)(61 90)(62 65)(66 83)(69 86)(72 89)(93 103)(96 106)(99 100)

(1 93)(2 41)(3 105)(4 96)(5 44)(6 108)(7 99)(8 38)(9 102)(10 65)(11 83)(12 55)(13 68)(14 86)(15 58)(16 71)(17 89)(18 61)(19 49)(20 32)(21 78)(22 52)(23 35)(24 81)(25 46)(26 29)(27 75)(28 73)(30 48)(31 76)(33 51)(34 79)(36 54)(37 100)(39 92)(40 103)(42 95)(43 106)(45 98)(47 74)(50 77)(53 80)(56 85)(57 69)(59 88)(60 72)(62 82)(63 66)(64 90)(67 84)(70 87)(91 101)(94 104)(97 107)

(1 40)(3 42)(4 43)(6 45)(7 37)(9 39)(10 82)(12 84)(13 85)(15 87)(16 88)(18 90)(20 50)(21 51)(23 53)(24 54)(26 47)(27 48)(29 74)(30 75)(32 77)(33 78)(35 80)(36 81)(55 67)(56 68)(58 70)(59 71)(61 64)(62 65)(92 102)(93 103)(95 105)(96 106)(98 108)(99 100)

(1 40)(2 41)(4 43)(5 44)(7 37)(8 38)(10 82)(11 83)(13 85)(14 86)(16 88)(17 89)(19 49)(21 51)(22 52)(24 54)(25 46)(27 48)(28 73)(30 75)(31 76)(33 78)(34 79)(36 81)(56 68)(57 69)(59 71)(60 72)(62 65)(63 66)(91 101)(93 103)(94 104)(96 106)(97 107)(99 100)

(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 97 98 99)(100 101 102 103 104 105 106 107 108)

(1 23 63)(2 21 58)(3 19 62)(4 26 57)(5 24 61)(6 22 56)(7 20 60)(8 27 55)(9 25 59)(10 105 31)(11 103 35)(12 101 30)(13 108 34)(14 106 29)(15 104 33)(16 102 28)(17 100 32)(18 107 36)(37 50 72)(38 48 67)(39 46 71)(40 53 66)(41 51 70)(42 49 65)(43 47 69)(44 54 64)(45 52 68)(73 88 92)(74 86 96)(75 84 91)(76 82 95)(77 89 99)(78 87 94)(79 85 98)(80 83 93)(81 90 97)

G:=sub<Sym(108)| (1,40)(2,104)(3,95)(4,43)(5,107)(6,98)(7,37)(8,101)(9,92)(10,82)(11,63)(12,67)(13,85)(14,57)(15,70)(16,88)(17,60)(18,64)(19,31)(20,77)(21,51)(22,34)(23,80)(24,54)(25,28)(26,74)(27,48)(29,47)(30,75)(32,50)(33,78)(35,53)(36,81)(38,91)(39,102)(41,94)(42,105)(44,97)(45,108)(46,73)(49,76)(52,79)(55,84)(56,68)(58,87)(59,71)(61,90)(62,65)(66,83)(69,86)(72,89)(93,103)(96,106)(99,100), (1,93)(2,41)(3,105)(4,96)(5,44)(6,108)(7,99)(8,38)(9,102)(10,65)(11,83)(12,55)(13,68)(14,86)(15,58)(16,71)(17,89)(18,61)(19,49)(20,32)(21,78)(22,52)(23,35)(24,81)(25,46)(26,29)(27,75)(28,73)(30,48)(31,76)(33,51)(34,79)(36,54)(37,100)(39,92)(40,103)(42,95)(43,106)(45,98)(47,74)(50,77)(53,80)(56,85)(57,69)(59,88)(60,72)(62,82)(63,66)(64,90)(67,84)(70,87)(91,101)(94,104)(97,107), (1,40)(3,42)(4,43)(6,45)(7,37)(9,39)(10,82)(12,84)(13,85)(15,87)(16,88)(18,90)(20,50)(21,51)(23,53)(24,54)(26,47)(27,48)(29,74)(30,75)(32,77)(33,78)(35,80)(36,81)(55,67)(56,68)(58,70)(59,71)(61,64)(62,65)(92,102)(93,103)(95,105)(96,106)(98,108)(99,100), (1,40)(2,41)(4,43)(5,44)(7,37)(8,38)(10,82)(11,83)(13,85)(14,86)(16,88)(17,89)(19,49)(21,51)(22,52)(24,54)(25,46)(27,48)(28,73)(30,75)(31,76)(33,78)(34,79)(36,81)(56,68)(57,69)(59,71)(60,72)(62,65)(63,66)(91,101)(93,103)(94,104)(96,106)(97,107)(99,100), (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)(91,92,93,94,95,96,97,98,99)(100,101,102,103,104,105,106,107,108), (1,23,63)(2,21,58)(3,19,62)(4,26,57)(5,24,61)(6,22,56)(7,20,60)(8,27,55)(9,25,59)(10,105,31)(11,103,35)(12,101,30)(13,108,34)(14,106,29)(15,104,33)(16,102,28)(17,100,32)(18,107,36)(37,50,72)(38,48,67)(39,46,71)(40,53,66)(41,51,70)(42,49,65)(43,47,69)(44,54,64)(45,52,68)(73,88,92)(74,86,96)(75,84,91)(76,82,95)(77,89,99)(78,87,94)(79,85,98)(80,83,93)(81,90,97)>;

G:=Group( (1,40)(2,104)(3,95)(4,43)(5,107)(6,98)(7,37)(8,101)(9,92)(10,82)(11,63)(12,67)(13,85)(14,57)(15,70)(16,88)(17,60)(18,64)(19,31)(20,77)(21,51)(22,34)(23,80)(24,54)(25,28)(26,74)(27,48)(29,47)(30,75)(32,50)(33,78)(35,53)(36,81)(38,91)(39,102)(41,94)(42,105)(44,97)(45,108)(46,73)(49,76)(52,79)(55,84)(56,68)(58,87)(59,71)(61,90)(62,65)(66,83)(69,86)(72,89)(93,103)(96,106)(99,100), (1,93)(2,41)(3,105)(4,96)(5,44)(6,108)(7,99)(8,38)(9,102)(10,65)(11,83)(12,55)(13,68)(14,86)(15,58)(16,71)(17,89)(18,61)(19,49)(20,32)(21,78)(22,52)(23,35)(24,81)(25,46)(26,29)(27,75)(28,73)(30,48)(31,76)(33,51)(34,79)(36,54)(37,100)(39,92)(40,103)(42,95)(43,106)(45,98)(47,74)(50,77)(53,80)(56,85)(57,69)(59,88)(60,72)(62,82)(63,66)(64,90)(67,84)(70,87)(91,101)(94,104)(97,107), (1,40)(3,42)(4,43)(6,45)(7,37)(9,39)(10,82)(12,84)(13,85)(15,87)(16,88)(18,90)(20,50)(21,51)(23,53)(24,54)(26,47)(27,48)(29,74)(30,75)(32,77)(33,78)(35,80)(36,81)(55,67)(56,68)(58,70)(59,71)(61,64)(62,65)(92,102)(93,103)(95,105)(96,106)(98,108)(99,100), (1,40)(2,41)(4,43)(5,44)(7,37)(8,38)(10,82)(11,83)(13,85)(14,86)(16,88)(17,89)(19,49)(21,51)(22,52)(24,54)(25,46)(27,48)(28,73)(30,75)(31,76)(33,78)(34,79)(36,81)(56,68)(57,69)(59,71)(60,72)(62,65)(63,66)(91,101)(93,103)(94,104)(96,106)(97,107)(99,100), (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)(91,92,93,94,95,96,97,98,99)(100,101,102,103,104,105,106,107,108), (1,23,63)(2,21,58)(3,19,62)(4,26,57)(5,24,61)(6,22,56)(7,20,60)(8,27,55)(9,25,59)(10,105,31)(11,103,35)(12,101,30)(13,108,34)(14,106,29)(15,104,33)(16,102,28)(17,100,32)(18,107,36)(37,50,72)(38,48,67)(39,46,71)(40,53,66)(41,51,70)(42,49,65)(43,47,69)(44,54,64)(45,52,68)(73,88,92)(74,86,96)(75,84,91)(76,82,95)(77,89,99)(78,87,94)(79,85,98)(80,83,93)(81,90,97) );

G=PermutationGroup([[(1,40),(2,104),(3,95),(4,43),(5,107),(6,98),(7,37),(8,101),(9,92),(10,82),(11,63),(12,67),(13,85),(14,57),(15,70),(16,88),(17,60),(18,64),(19,31),(20,77),(21,51),(22,34),(23,80),(24,54),(25,28),(26,74),(27,48),(29,47),(30,75),(32,50),(33,78),(35,53),(36,81),(38,91),(39,102),(41,94),(42,105),(44,97),(45,108),(46,73),(49,76),(52,79),(55,84),(56,68),(58,87),(59,71),(61,90),(62,65),(66,83),(69,86),(72,89),(93,103),(96,106),(99,100)], [(1,93),(2,41),(3,105),(4,96),(5,44),(6,108),(7,99),(8,38),(9,102),(10,65),(11,83),(12,55),(13,68),(14,86),(15,58),(16,71),(17,89),(18,61),(19,49),(20,32),(21,78),(22,52),(23,35),(24,81),(25,46),(26,29),(27,75),(28,73),(30,48),(31,76),(33,51),(34,79),(36,54),(37,100),(39,92),(40,103),(42,95),(43,106),(45,98),(47,74),(50,77),(53,80),(56,85),(57,69),(59,88),(60,72),(62,82),(63,66),(64,90),(67,84),(70,87),(91,101),(94,104),(97,107)], [(1,40),(3,42),(4,43),(6,45),(7,37),(9,39),(10,82),(12,84),(13,85),(15,87),(16,88),(18,90),(20,50),(21,51),(23,53),(24,54),(26,47),(27,48),(29,74),(30,75),(32,77),(33,78),(35,80),(36,81),(55,67),(56,68),(58,70),(59,71),(61,64),(62,65),(92,102),(93,103),(95,105),(96,106),(98,108),(99,100)], [(1,40),(2,41),(4,43),(5,44),(7,37),(8,38),(10,82),(11,83),(13,85),(14,86),(16,88),(17,89),(19,49),(21,51),(22,52),(24,54),(25,46),(27,48),(28,73),(30,75),(31,76),(33,78),(34,79),(36,81),(56,68),(57,69),(59,71),(60,72),(62,65),(63,66),(91,101),(93,103),(94,104),(96,106),(97,107),(99,100)], [(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),(91,92,93,94,95,96,97,98,99),(100,101,102,103,104,105,106,107,108)], [(1,23,63),(2,21,58),(3,19,62),(4,26,57),(5,24,61),(6,22,56),(7,20,60),(8,27,55),(9,25,59),(10,105,31),(11,103,35),(12,101,30),(13,108,34),(14,106,29),(15,104,33),(16,102,28),(17,100,32),(18,107,36),(37,50,72),(38,48,67),(39,46,71),(40,53,66),(41,51,70),(42,49,65),(43,47,69),(44,54,64),(45,52,68),(73,88,92),(74,86,96),(75,84,91),(76,82,95),(77,89,99),(78,87,94),(79,85,98),(80,83,93),(81,90,97)]])

56 conjugacy classes

class	1	2A	···	2E	3A	3B	3C	3D	6A	···	6J	9A	9B	9C	9D	9E	9F	18A	···	18AD
order	1	2	···	2	3	3	3	3	6	···	6	9	9	9	9	9	9	18	···	18
size	1	3	···	3	1	1	48	48	3	···	3	3	3	48	48	48	48	3	···	3

56 irreducible representations

dim	1	1	1	1	3	3	3	3
type	+				+
image	C₁	C₃	C₃	C₃	A₄	3_-¹⁺²	C₃×A₄	C₉⋊A₄
kernel	C₂⁴⋊₄3_-¹⁺²	C₂⁴⋊C₉	C₂³×C₁₈	C₃×C₂²⋊A₄	C₂×C₁₈	C₂⁴	C₂×C₆	C₂²
# reps	1	4	2	2	5	2	10	30

Matrix representation of C₂⁴⋊₄3_-¹⁺² ►in GL₆(𝔽₁₉)

18	0	0	0	0	0
14	1	0	0	0	0
0	0	18	0	0	0
0	0	0	1	0	0
0	0	0	0	18	0
0	0	0	17	0	18

1	0	0	0	0	0
5	18	0	0	0	0
1	0	18	0	0	0
0	0	0	18	0	0
0	0	0	0	18	0
0	0	0	2	3	1

18	0	0	0	0	0
0	18	0	0	0	0
18	0	1	0	0	0
0	0	0	1	0	0
0	0	0	0	1	0
0	0	0	0	0	1

18	0	0	0	0	0
14	1	0	0	0	0
0	0	18	0	0	0
0	0	0	1	0	0
0	0	0	0	1	0
0	0	0	0	0	1

6	9	0	0	0	0
16	13	17	0	0	0
14	14	0	0	0	0
0	0	0	0	5	0
0	0	0	2	3	2
0	0	0	0	0	16

1	0	17	0	0	0
2	0	14	0	0	0
17	1	18	0	0	0
0	0	0	18	8	18
0	0	0	1	0	0
0	0	0	9	10	1

G:=sub<GL(6,GF(19))| [18,14,0,0,0,0,0,1,0,0,0,0,0,0,18,0,0,0,0,0,0,1,0,17,0,0,0,0,18,0,0,0,0,0,0,18],[1,5,1,0,0,0,0,18,0,0,0,0,0,0,18,0,0,0,0,0,0,18,0,2,0,0,0,0,18,3,0,0,0,0,0,1],[18,0,18,0,0,0,0,18,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[18,14,0,0,0,0,0,1,0,0,0,0,0,0,18,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[6,16,14,0,0,0,9,13,14,0,0,0,0,17,0,0,0,0,0,0,0,0,2,0,0,0,0,5,3,0,0,0,0,0,2,16],[1,2,17,0,0,0,0,0,1,0,0,0,17,14,18,0,0,0,0,0,0,18,1,9,0,0,0,8,0,10,0,0,0,18,0,1] >;

C₂⁴⋊₄3_-¹⁺² in GAP, Magma, Sage, TeX

C_2^4\rtimes_43_-^{1+2}

% in TeX

G:=Group("C2^4:4ES-(3,1)");

// GroupNames label

G:=SmallGroup(432,552);

// by ID

G=gap.SmallGroup(432,552);

# by ID

G:=PCGroup([7,-3,-3,-3,-2,2,-2,2,169,50,1515,2839,9077,15882]);

// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=e^9=f^3=1,e*a*e^-1=f*b*f^-1=a*b=b*a,a*c=c*a,a*d=d*a,f*a*f^-1=b,b*c=c*b,b*d=d*b,e*b*e^-1=a,e*c*e^-1=f*d*f^-1=c*d=d*c,f*c*f^-1=d,e*d*e^-1=c,f*e*f^-1=e^4>;

// generators/relations

G = C24⋊43- 1+2 order 432 = 24·33

2nd semidirect product of C24 and 3- 1+2 acting via 3- 1+2/C9=C3

G = C₂⁴⋊₄3_-¹⁺² order 432 = 2⁴·3³

2^nd semidirect product of C₂⁴ and 3_-¹⁺² acting via 3_-¹⁺²/C₉=C₃