C12:5SD16 - GroupNames

G = C₁₂⋊₅SD₁₆ order 192 = 2⁶·3

5^th semidirect product of C₁₂ and SD₁₆ acting via SD₁₆/C₄=C₂²

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C₁₂⋊₅SD₁₆, D₁₂.₂₄D₄, C₄².₇₈D₆, C₄⋊Q₈⋊₁S₃, C₄.₅₆(S₃×D₄), C₁₂⋊C₈⋊₃₃C₂, C₁₂.₃₆(C₂×D₄), (C₂×Q₈).₆₈D₆, C₄⋊₂(Q₈⋊₂S₃), (C₄×D₁₂).₁₇C₂, (C₂×C₁₂).₁₅₄D₄, C₃⋊₄(D₄.D₄), C₆.₇₅(C₂×SD₁₆), C₁₂.₈₀(C₄○D₄), Q₈⋊₂Dic₃⋊₂₃C₂, C₄.₅(D₄⋊₂S₃), (C₆×Q₈).₆₂C₂², C₂.₁₄(D₆⋊₃D₄), C₆.₁₀₅(C₄⋊D₄), (C₂×C₁₂).₄₀₁C₂³, (C₄×C₁₂).₁₃₀C₂², C₆.₉₅(C₈.C₂²), (C₂×D₁₂).₂₄₇C₂², C₄⋊Dic₃.₃₄₆C₂², C₂.₁₆(Q₈.₁₁D₆), (C₃×C₄⋊Q₈)⋊₁C₂, (C₂×C₆).₅₃₂(C₂×D₄), (C₂×C₃⋊C₈).₁₃₅C₂², (C₂×Q₈⋊₂S₃).₆C₂, C₂.₁₃(C₂×Q₈⋊₂S₃), (C₂×C₄).₁₈₇(C₃⋊D₄), (C₂×C₄).₄₉₈(C₂²×S₃), C₂².₂₀₄(C₂×C₃⋊D₄), SmallGroup(192,642)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂×C₁₂ — C₁₂⋊₅SD₁₆

Chief series

C₁ — C₃ — C₆ — C₁₂ — C₂×C₁₂ — C₂×D₁₂ — C₄×D₁₂ — C₁₂⋊₅SD₁₆

Lower central

C₃ — C₆ — C₂×C₁₂ — C₁₂⋊₅SD₁₆

Upper central

C₁ — C₂² — C₄² — C₄⋊Q₈

Generators and relations for C₁₂⋊₅SD₁₆
G = < a,b,c | a¹²=b⁸=c²=1, bab^-1=a^-1, cac=a⁵, cbc=b³ >

Subgroups: 352 in 120 conjugacy classes, 45 normal (29 characteristic)
C₁, C₂, C₂, C₃, C₄, C₄, C₄, C₂², C₂², S₃, C₆, C₈, C₂×C₄, C₂×C₄, D₄, Q₈, C₂³, Dic₃, C₁₂, C₁₂, C₁₂, D₆, C₂×C₆, C₄², C₂²⋊C₄, C₄⋊C₄, C₂×C₈, SD₁₆, C₂²×C₄, C₂×D₄, C₂×Q₈, C₃⋊C₈, C₄×S₃, D₁₂, D₁₂, C₂×Dic₃, C₂×C₁₂, C₂×C₁₂, C₃×Q₈, C₂²×S₃, Q₈⋊C₄, C₄⋊C₈, C₄×D₄, C₄⋊Q₈, C₂×SD₁₆, C₂×C₃⋊C₈, C₄⋊Dic₃, D₆⋊C₄, Q₈⋊₂S₃, C₄×C₁₂, C₃×C₄⋊C₄, S₃×C₂×C₄, C₂×D₁₂, C₆×Q₈, D₄.D₄, C₁₂⋊C₈, Q₈⋊₂Dic₃, C₄×D₁₂, C₂×Q₈⋊₂S₃, C₃×C₄⋊Q₈, C₁₂⋊₅SD₁₆
Quotients: C₁, C₂, C₂², S₃, D₄, C₂³, D₆, SD₁₆, C₂×D₄, C₄○D₄, C₃⋊D₄, C₂²×S₃, C₄⋊D₄, C₂×SD₁₆, C₈.C₂², Q₈⋊₂S₃, S₃×D₄, D₄⋊₂S₃, C₂×C₃⋊D₄, D₄.D₄, D₆⋊₃D₄, C₂×Q₈⋊₂S₃, Q₈.₁₁D₆, C₁₂⋊₅SD₁₆

Smallest permutation representation of C₁₂⋊₅SD₁₆
►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 40 89 28 20 71 50 81)(2 39 90 27 21 70 51 80)(3 38 91 26 22 69 52 79)(4 37 92 25 23 68 53 78)(5 48 93 36 24 67 54 77)(6 47 94 35 13 66 55 76)(7 46 95 34 14 65 56 75)(8 45 96 33 15 64 57 74)(9 44 85 32 16 63 58 73)(10 43 86 31 17 62 59 84)(11 42 87 30 18 61 60 83)(12 41 88 29 19 72 49 82)

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

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

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

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

33 conjugacy classes

class	1	2A	2B	2C	2D	2E	3	4A	4B	4C	4D	4E	4F	4G	4H	4I	6A	6B	6C	8A	8B	8C	8D	12A	···	12F	12G	12H	12I	12J
order	1	2	2	2	2	2	3	4	4	4	4	4	4	4	4	4	6	6	6	8	8	8	8	12	···	12	12	12	12	12
size	1	1	1	1	12	12	2	2	2	2	2	4	8	8	12	12	2	2	2	12	12	12	12	4	···	4	8	8	8	8

33 irreducible representations

dim	1	1	1	1	1	1	2	2	2	2	2	2	2	2	4	4	4	4	4
type	+	+	+	+	+	+	+	+	+	+	+				-	+	+	-
image	C₁	C₂	C₂	C₂	C₂	C₂	S₃	D₄	D₄	D₆	D₆	SD₁₆	C₄○D₄	C₃⋊D₄	C₈.C₂²	Q₈⋊₂S₃	S₃×D₄	D₄⋊₂S₃	Q₈.₁₁D₆
kernel	C₁₂⋊₅SD₁₆	C₁₂⋊C₈	Q₈⋊₂Dic₃	C₄×D₁₂	C₂×Q₈⋊₂S₃	C₃×C₄⋊Q₈	C₄⋊Q₈	D₁₂	C₂×C₁₂	C₄²	C₂×Q₈	C₁₂	C₁₂	C₂×C₄	C₆	C₄	C₄	C₄	C₂
# reps	1	1	2	1	2	1	1	2	2	1	2	4	2	4	1	2	1	1	2

Matrix representation of C₁₂⋊₅SD₁₆ ►in GL₆(𝔽₇₃)

1	72	0	0	0	0
1	0	0	0	0	0
0	0	20	6	0	0
0	0	67	53	0	0
0	0	0	0	1	0
0	0	0	0	0	1

0	72	0	0	0	0
72	0	0	0	0	0
0	0	0	1	0	0
0	0	1	0	0	0
0	0	0	0	61	12
0	0	0	0	67	0

0	1	0	0	0	0
1	0	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	1	72

G:=sub<GL(6,GF(73))| [1,1,0,0,0,0,72,0,0,0,0,0,0,0,20,67,0,0,0,0,6,53,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,72,0,0,0,0,72,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,61,67,0,0,0,0,12,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,0,72] >;

C₁₂⋊₅SD₁₆ in GAP, Magma, Sage, TeX

C_{12}\rtimes_5{\rm SD}_{16}

% in TeX

G:=Group("C12:5SD16");

// GroupNames label

G:=SmallGroup(192,642);

// by ID

G=gap.SmallGroup(192,642);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,120,254,219,184,1123,297,136,6278]);

// Polycyclic

G:=Group<a,b,c|a^12=b^8=c^2=1,b*a*b^-1=a^-1,c*a*c=a^5,c*b*c=b^3>;

// generators/relations

G = C12⋊5SD16 order 192 = 26·3

5th semidirect product of C12 and SD16 acting via SD16/C4=C22

G = C₁₂⋊₅SD₁₆ order 192 = 2⁶·3

5^th semidirect product of C₁₂ and SD₁₆ acting via SD₁₆/C₄=C₂²