C12.69S3^2 - GroupNames

G = C₁₂.₆₉S₃² order 432 = 2⁴·3³

26^th non-split extension by C₁₂ of S₃² acting via S₃²/C₃×S₃=C₂

metabelian, supersoluble, monomial, A-group

Aliases: C₁₂.₆₉S₃², C₃²⋊₈(S₃×C₈), C₃³⋊₁₂(C₂×C₈), C₃³⋊C₂⋊₃C₈, C₃²⋊₄C₈⋊₁₇S₃, (C₃×C₁₂).₁₆₄D₆, C₃³⋊₅C₄.₃C₄, C₆.₇(C₆.D₆), C₃⋊₁(C₁₂.₂₉D₆), (C₃²×C₁₂).₆₆C₂², C₃⋊₁(C₈×C₃⋊S₃), (C₃×C₃⋊C₈)⋊₆S₃, C₃⋊C₈⋊₆(C₃⋊S₃), C₆.₁(C₄×C₃⋊S₃), C₄.₂₄(S₃×C₃⋊S₃), C₁₂.₃₉(C₂×C₃⋊S₃), (C₃²×C₃⋊C₈)⋊₁₁C₂, (C₃×C₆).₄₅(C₄×S₃), C₂.₁(C₃³⋊₈(C₂×C₄)), (C₃×C₃²⋊₄C₈)⋊₁₀C₂, (C₃²×C₆).₃₃(C₂×C₄), (C₂×C₃³⋊C₂).₃C₄, (C₄×C₃³⋊C₂).₃C₂, SmallGroup(432,432)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃³ — C₁₂.₆₉S₃²

Chief series

C₁ — C₃ — C₃² — C₃³ — C₃²×C₆ — C₃²×C₁₂ — C₃²×C₃⋊C₈ — C₁₂.₆₉S₃²

Lower central

C₃³ — C₁₂.₆₉S₃²

Upper central

C₁ — C₄

Generators and relations for C₁₂.₆₉S₃²
G = < a,b,c,d,e | a³=b³=c⁸=d³=e²=1, ab=ba, ac=ca, ad=da, eae=a^-1, cbc^-1=ebe=b^-1, bd=db, cd=dc, ce=ec, ede=d^-1 >

Subgroups: 1160 in 196 conjugacy classes, 50 normal (20 characteristic)
C₁, C₂, C₂, C₃, C₃, C₃, C₄, C₄, C₂², S₃, C₆, C₆, C₆, C₈, C₂×C₄, C₃², C₃², C₃², Dic₃, C₁₂, C₁₂, C₁₂, D₆, C₂×C₈, C₃⋊S₃, C₃×C₆, C₃×C₆, C₃×C₆, C₃⋊C₈, C₃⋊C₈, C₂₄, C₄×S₃, C₃³, C₃⋊Dic₃, C₃×C₁₂, C₃×C₁₂, C₃×C₁₂, C₂×C₃⋊S₃, S₃×C₈, C₃³⋊C₂, C₃²×C₆, C₃×C₃⋊C₈, C₃×C₃⋊C₈, C₃²⋊₄C₈, C₃×C₂₄, C₄×C₃⋊S₃, C₃³⋊₅C₄, C₃²×C₁₂, C₂×C₃³⋊C₂, C₁₂.₂₉D₆, C₈×C₃⋊S₃, C₃²×C₃⋊C₈, C₃×C₃²⋊₄C₈, C₄×C₃³⋊C₂, C₁₂.₆₉S₃²
Quotients: C₁, C₂, C₄, C₂², S₃, C₈, C₂×C₄, D₆, C₂×C₈, C₃⋊S₃, C₄×S₃, S₃², C₂×C₃⋊S₃, S₃×C₈, C₆.D₆, C₄×C₃⋊S₃, S₃×C₃⋊S₃, C₁₂.₂₉D₆, C₈×C₃⋊S₃, C₃³⋊₈(C₂×C₄), C₁₂.₆₉S₃²

Smallest permutation representation of C₁₂.₆₉S₃²
►On 72 points

Generators in S₇₂

(1 24 48)(2 17 41)(3 18 42)(4 19 43)(5 20 44)(6 21 45)(7 22 46)(8 23 47)(9 25 64)(10 26 57)(11 27 58)(12 28 59)(13 29 60)(14 30 61)(15 31 62)(16 32 63)(33 71 51)(34 72 52)(35 65 53)(36 66 54)(37 67 55)(38 68 56)(39 69 49)(40 70 50)

(1 27 40)(2 33 28)(3 29 34)(4 35 30)(5 31 36)(6 37 32)(7 25 38)(8 39 26)(9 56 46)(10 47 49)(11 50 48)(12 41 51)(13 52 42)(14 43 53)(15 54 44)(16 45 55)(17 71 59)(18 60 72)(19 65 61)(20 62 66)(21 67 63)(22 64 68)(23 69 57)(24 58 70)

(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)

(1 70 11)(2 71 12)(3 72 13)(4 65 14)(5 66 15)(6 67 16)(7 68 9)(8 69 10)(17 51 28)(18 52 29)(19 53 30)(20 54 31)(21 55 32)(22 56 25)(23 49 26)(24 50 27)(33 59 41)(34 60 42)(35 61 43)(36 62 44)(37 63 45)(38 64 46)(39 57 47)(40 58 48)

(9 68)(10 69)(11 70)(12 71)(13 72)(14 65)(15 66)(16 67)(17 41)(18 42)(19 43)(20 44)(21 45)(22 46)(23 47)(24 48)(25 38)(26 39)(27 40)(28 33)(29 34)(30 35)(31 36)(32 37)(49 57)(50 58)(51 59)(52 60)(53 61)(54 62)(55 63)(56 64)

G:=sub<Sym(72)| (1,24,48)(2,17,41)(3,18,42)(4,19,43)(5,20,44)(6,21,45)(7,22,46)(8,23,47)(9,25,64)(10,26,57)(11,27,58)(12,28,59)(13,29,60)(14,30,61)(15,31,62)(16,32,63)(33,71,51)(34,72,52)(35,65,53)(36,66,54)(37,67,55)(38,68,56)(39,69,49)(40,70,50), (1,27,40)(2,33,28)(3,29,34)(4,35,30)(5,31,36)(6,37,32)(7,25,38)(8,39,26)(9,56,46)(10,47,49)(11,50,48)(12,41,51)(13,52,42)(14,43,53)(15,54,44)(16,45,55)(17,71,59)(18,60,72)(19,65,61)(20,62,66)(21,67,63)(22,64,68)(23,69,57)(24,58,70), (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), (1,70,11)(2,71,12)(3,72,13)(4,65,14)(5,66,15)(6,67,16)(7,68,9)(8,69,10)(17,51,28)(18,52,29)(19,53,30)(20,54,31)(21,55,32)(22,56,25)(23,49,26)(24,50,27)(33,59,41)(34,60,42)(35,61,43)(36,62,44)(37,63,45)(38,64,46)(39,57,47)(40,58,48), (9,68)(10,69)(11,70)(12,71)(13,72)(14,65)(15,66)(16,67)(17,41)(18,42)(19,43)(20,44)(21,45)(22,46)(23,47)(24,48)(25,38)(26,39)(27,40)(28,33)(29,34)(30,35)(31,36)(32,37)(49,57)(50,58)(51,59)(52,60)(53,61)(54,62)(55,63)(56,64)>;

G:=Group( (1,24,48)(2,17,41)(3,18,42)(4,19,43)(5,20,44)(6,21,45)(7,22,46)(8,23,47)(9,25,64)(10,26,57)(11,27,58)(12,28,59)(13,29,60)(14,30,61)(15,31,62)(16,32,63)(33,71,51)(34,72,52)(35,65,53)(36,66,54)(37,67,55)(38,68,56)(39,69,49)(40,70,50), (1,27,40)(2,33,28)(3,29,34)(4,35,30)(5,31,36)(6,37,32)(7,25,38)(8,39,26)(9,56,46)(10,47,49)(11,50,48)(12,41,51)(13,52,42)(14,43,53)(15,54,44)(16,45,55)(17,71,59)(18,60,72)(19,65,61)(20,62,66)(21,67,63)(22,64,68)(23,69,57)(24,58,70), (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), (1,70,11)(2,71,12)(3,72,13)(4,65,14)(5,66,15)(6,67,16)(7,68,9)(8,69,10)(17,51,28)(18,52,29)(19,53,30)(20,54,31)(21,55,32)(22,56,25)(23,49,26)(24,50,27)(33,59,41)(34,60,42)(35,61,43)(36,62,44)(37,63,45)(38,64,46)(39,57,47)(40,58,48), (9,68)(10,69)(11,70)(12,71)(13,72)(14,65)(15,66)(16,67)(17,41)(18,42)(19,43)(20,44)(21,45)(22,46)(23,47)(24,48)(25,38)(26,39)(27,40)(28,33)(29,34)(30,35)(31,36)(32,37)(49,57)(50,58)(51,59)(52,60)(53,61)(54,62)(55,63)(56,64) );

G=PermutationGroup([[(1,24,48),(2,17,41),(3,18,42),(4,19,43),(5,20,44),(6,21,45),(7,22,46),(8,23,47),(9,25,64),(10,26,57),(11,27,58),(12,28,59),(13,29,60),(14,30,61),(15,31,62),(16,32,63),(33,71,51),(34,72,52),(35,65,53),(36,66,54),(37,67,55),(38,68,56),(39,69,49),(40,70,50)], [(1,27,40),(2,33,28),(3,29,34),(4,35,30),(5,31,36),(6,37,32),(7,25,38),(8,39,26),(9,56,46),(10,47,49),(11,50,48),(12,41,51),(13,52,42),(14,43,53),(15,54,44),(16,45,55),(17,71,59),(18,60,72),(19,65,61),(20,62,66),(21,67,63),(22,64,68),(23,69,57),(24,58,70)], [(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)], [(1,70,11),(2,71,12),(3,72,13),(4,65,14),(5,66,15),(6,67,16),(7,68,9),(8,69,10),(17,51,28),(18,52,29),(19,53,30),(20,54,31),(21,55,32),(22,56,25),(23,49,26),(24,50,27),(33,59,41),(34,60,42),(35,61,43),(36,62,44),(37,63,45),(38,64,46),(39,57,47),(40,58,48)], [(9,68),(10,69),(11,70),(12,71),(13,72),(14,65),(15,66),(16,67),(17,41),(18,42),(19,43),(20,44),(21,45),(22,46),(23,47),(24,48),(25,38),(26,39),(27,40),(28,33),(29,34),(30,35),(31,36),(32,37),(49,57),(50,58),(51,59),(52,60),(53,61),(54,62),(55,63),(56,64)]])

72 conjugacy classes

class	1	2A	2B	2C	3A	···	3E	3F	3G	3H	3I	4A	4B	4C	4D	6A	···	6E	6F	6G	6H	6I	8A	8B	8C	8D	8E	8F	8G	8H	12A	···	12J	12K	···	12R	24A	···	24P	24Q	24R	24S	24T
order	1	2	2	2	3	···	3	3	3	3	3	4	4	4	4	6	···	6	6	6	6	6	8	8	8	8	8	8	8	8	12	···	12	12	···	12	24	···	24	24	24	24	24
size	1	1	27	27	2	···	2	4	4	4	4	1	1	27	27	2	···	2	4	4	4	4	3	3	3	3	9	9	9	9	2	···	2	4	···	4	6	···	6	18	18	18	18

72 irreducible representations

dim	1	1	1	1	1	1	1	2	2	2	2	2	4	4	4
type	+	+	+	+				+	+	+			+	+
image	C₁	C₂	C₂	C₂	C₄	C₄	C₈	S₃	S₃	D₆	C₄×S₃	S₃×C₈	S₃²	C₆.D₆	C₁₂.₂₉D₆
kernel	C₁₂.₆₉S₃²	C₃²×C₃⋊C₈	C₃×C₃²⋊₄C₈	C₄×C₃³⋊C₂	C₃³⋊₅C₄	C₂×C₃³⋊C₂	C₃³⋊C₂	C₃×C₃⋊C₈	C₃²⋊₄C₈	C₃×C₁₂	C₃×C₆	C₃²	C₁₂	C₆	C₃
# reps	1	1	1	1	2	2	8	4	1	5	10	20	4	4	8

Matrix representation of C₁₂.₆₉S₃² ►in GL₆(𝔽₇₃)

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

1	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	0	72
0	0	0	0	1	72

22	0	0	0	0	0
0	22	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	0	0	0	0	1
0	0	0	0	1	0

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

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

G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,72,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,72,72],[22,0,0,0,0,0,0,22,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[72,1,0,0,0,0,72,0,0,0,0,0,0,0,0,1,0,0,0,0,72,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,72,0,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;

C₁₂.₆₉S₃² in GAP, Magma, Sage, TeX

C_{12}._{69}S_3^2

% in TeX

G:=Group("C12.69S3^2");

// GroupNames label

G:=SmallGroup(432,432);

// by ID

G=gap.SmallGroup(432,432);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,28,36,58,571,2028,14118]);

// Polycyclic

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

// generators/relations

G = C12.69S32 order 432 = 24·33

26th non-split extension by C12 of S32 acting via S32/C3×S3=C2

G = C₁₂.₆₉S₃² order 432 = 2⁴·3³

26^th non-split extension by C₁₂ of S₃² acting via S₃²/C₃×S₃=C₂