C12.84S3^2 - GroupNames

G = C₁₂.₈₄S₃² order 432 = 2⁴·3³

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

Aliases: C₁₂.₈₄S₃², C₁₂⋊S₃⋊₄S₃, He₃⋊₂(C₄○D₄), He₃⋊₃D₄⋊₃C₂, He₃⋊₄D₄⋊₅C₂, He₃⋊₃Q₈⋊₅C₂, (C₃×C₁₂).₂₂D₆, C₃⋊Dic₃.₁D₆, C₃²⋊₄Q₈⋊₄S₃, C₄.₁₁(C₃²⋊D₆), (C₂×He₃).₃C₂³, C₃.₃(D₁₂⋊S₃), C₃²⋊₂(D₄⋊₂S₃), C₃²⋊₁(Q₈⋊₃S₃), C₃²⋊C₁₂.₁C₂², (C₄×He₃).₁₈C₂², He₃⋊₃C₄.₁₄C₂², C₆.₇₇(C₂×S₃²), (C₂×C₃⋊S₃).₂D₆, C₆.S₃²⋊₂C₂, C₂.₆(C₂×C₃²⋊D₆), (C₄×He₃⋊C₂)⋊₂C₂, (C₃×C₆).₃(C₂²×S₃), (C₂×C₃²⋊C₆).₁C₂², (C₂×He₃⋊C₂).₁₁C₂², SmallGroup(432,296)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃ — C₂×He₃ — C₁₂.₈₄S₃²

Chief series

C₁ — C₃ — C₃² — He₃ — C₂×He₃ — C₂×C₃²⋊C₆ — C₆.S₃² — C₁₂.₈₄S₃²

Lower central

He₃ — C₂×He₃ — C₁₂.₈₄S₃²

Upper central

C₁ — C₂ — C₄

Generators and relations for C₁₂.₈₄S₃²
G = < a,b,c,d,e | a¹²=b³=d³=e²=1, c²=a⁶, ab=ba, cac^-1=eae=a^-1, ad=da, cbc^-1=b^-1, dbd^-1=a⁸b, be=eb, cd=dc, ce=ec, ede=d^-1 >

Subgroups: 927 in 156 conjugacy classes, 35 normal (23 characteristic)
C₁, C₂, C₂, C₃, C₃, C₄, C₄, C₂², S₃, C₆, C₆, C₂×C₄, D₄, Q₈, C₃², C₃², Dic₃, C₁₂, C₁₂, D₆, C₂×C₆, C₄○D₄, C₃×S₃, C₃⋊S₃, C₃×C₆, C₃×C₆, Dic₆, C₄×S₃, D₁₂, C₂×Dic₃, C₃⋊D₄, C₂×C₁₂, C₃×D₄, C₃×Q₈, He₃, C₃×Dic₃, C₃⋊Dic₃, C₃×C₁₂, C₃×C₁₂, S₃×C₆, C₂×C₃⋊S₃, C₄○D₁₂, D₄⋊₂S₃, Q₈⋊₃S₃, C₃²⋊C₆, He₃⋊C₂, C₂×He₃, S₃×Dic₃, C₆.D₆, D₆⋊S₃, C₃⋊D₁₂, C₃×Dic₆, S₃×C₁₂, C₃×D₁₂, C₃²⋊₄Q₈, C₁₂⋊S₃, C₃²⋊C₁₂, He₃⋊₃C₄, C₄×He₃, C₂×C₃²⋊C₆, C₂×He₃⋊C₂, D₁₂⋊₅S₃, D₆.₆D₆, C₆.S₃², He₃⋊₃D₄, He₃⋊₃Q₈, He₃⋊₄D₄, C₄×He₃⋊C₂, C₁₂.₈₄S₃²
Quotients: C₁, C₂, C₂², S₃, C₂³, D₆, C₄○D₄, C₂²×S₃, S₃², D₄⋊₂S₃, Q₈⋊₃S₃, C₂×S₃², C₃²⋊D₆, D₁₂⋊S₃, C₂×C₃²⋊D₆, C₁₂.₈₄S₃²

Smallest permutation representation of C₁₂.₈₄S₃²
►On 72 points

Generators in S₇₂

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

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

(13 17 21)(14 18 22)(15 19 23)(16 20 24)(25 33 29)(26 34 30)(27 35 31)(28 36 32)(37 45 41)(38 46 42)(39 47 43)(40 48 44)(61 65 69)(62 66 70)(63 67 71)(64 68 72)

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

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

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

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

32 conjugacy classes

class	1	2A	2B	2C	2D	3A	3B	3C	3D	4A	4B	4C	4D	4E	6A	6B	6C	6D	6E	6F	6G	6H	12A	12B	12C	12D	12E	12F	12G	12H	12I	12J
order	1	2	2	2	2	3	3	3	3	4	4	4	4	4	6	6	6	6	6	6	6	6	12	12	12	12	12	12	12	12	12	12
size	1	1	18	18	18	2	6	6	12	2	9	9	18	18	2	6	6	12	18	18	36	36	2	2	12	12	12	12	18	18	36	36

32 irreducible representations

dim	1	1	1	1	1	1	2	2	2	2	2	2	4	4	4	4	4	6	6	6
type	+	+	+	+	+	+	+	+	+	+	+		+	-	+	+		+	+
image	C₁	C₂	C₂	C₂	C₂	C₂	S₃	S₃	D₆	D₆	D₆	C₄○D₄	S₃²	D₄⋊₂S₃	Q₈⋊₃S₃	C₂×S₃²	D₁₂⋊S₃	C₃²⋊D₆	C₂×C₃²⋊D₆	C₁₂.₈₄S₃²
kernel	C₁₂.₈₄S₃²	C₆.S₃²	He₃⋊₃D₄	He₃⋊₃Q₈	He₃⋊₄D₄	C₄×He₃⋊C₂	C₃²⋊₄Q₈	C₁₂⋊S₃	C₃⋊Dic₃	C₃×C₁₂	C₂×C₃⋊S₃	He₃	C₁₂	C₃²	C₃²	C₆	C₃	C₄	C₂	C₁
# reps	1	2	2	1	1	1	1	1	2	2	2	2	1	1	1	1	2	2	2	4

Matrix representation of C₁₂.₈₄S₃² ►in GL₆(𝔽₁₃)

6	0	0	0	0	0
0	6	0	0	0	0
0	0	6	0	0	0
0	0	0	11	0	0
0	0	0	0	11	0
0	0	0	0	0	11

0	1	0	0	0	0
0	0	1	0	0	0
1	0	0	0	0	0
0	0	0	0	0	1
0	0	0	1	0	0
0	0	0	0	1	0

0	0	0	1	0	0
0	0	0	0	1	0
0	0	0	0	0	1
12	0	0	0	0	0
0	12	0	0	0	0
0	0	12	0	0	0

1	0	0	0	0	0
0	3	0	0	0	0
0	0	9	0	0	0
0	0	0	1	0	0
0	0	0	0	3	0
0	0	0	0	0	9

0	0	0	5	0	0
0	0	0	0	0	5
0	0	0	0	5	0
8	0	0	0	0	0
0	0	8	0	0	0
0	8	0	0	0	0

G:=sub<GL(6,GF(13))| [6,0,0,0,0,0,0,6,0,0,0,0,0,0,6,0,0,0,0,0,0,11,0,0,0,0,0,0,11,0,0,0,0,0,0,11],[0,0,1,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,1,0,0],[0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0],[1,0,0,0,0,0,0,3,0,0,0,0,0,0,9,0,0,0,0,0,0,1,0,0,0,0,0,0,3,0,0,0,0,0,0,9],[0,0,0,8,0,0,0,0,0,0,0,8,0,0,0,0,8,0,5,0,0,0,0,0,0,0,5,0,0,0,0,5,0,0,0,0] >;

C₁₂.₈₄S₃² in GAP, Magma, Sage, TeX

C_{12}._{84}S_3^2

% in TeX

G:=Group("C12.84S3^2");

// GroupNames label

G:=SmallGroup(432,296);

// by ID

G=gap.SmallGroup(432,296);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,64,254,135,58,571,4037,537,14118,7069]);

// Polycyclic

G:=Group<a,b,c,d,e|a^12=b^3=d^3=e^2=1,c^2=a^6,a*b=b*a,c*a*c^-1=e*a*e=a^-1,a*d=d*a,c*b*c^-1=b^-1,d*b*d^-1=a^8*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;

// generators/relations

G = C12.84S32 order 432 = 24·33

4th non-split extension by C12 of S32 acting via S32/C3⋊S3=C2

G = C₁₂.₈₄S₃² order 432 = 2⁴·3³

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