C6^2.40C2^3 - GroupNames

G = C₆².₄₀C₂³ order 288 = 2⁵·3²

35^th non-split extension by C₆² of C₂³ acting via C₂³/C₂=C₂²

Aliases: C₆².₄₀C₂³, C₆.₆(S₃×Q₈), (C₂×C₁₂).₁₉₃D₆, Dic₃⋊C₄.₅S₃, C₃⋊Dic₃.₁₀Q₈, C₆.₂₈(C₄○D₁₂), C₃⋊₄(Dic₃.Q₈), (C₂×Dic₃).₁₈D₆, C₆.₆₁(D₄⋊₂S₃), C₃²⋊₇(C₄².C₂), (C₆×C₁₂).₂₂₁C₂², C₂.₉(D₆.₄D₆), C₆².C₂².₅C₂, Dic₃⋊Dic₃.₁₄C₂, C₂.₈(Dic₃.D₆), C₂.₁₅(D₆.D₆), (C₆×Dic₃).₇₈C₂², (C₂×C₄).₉₂S₃², C₂².₉₇(C₂×S₃²), (C₃×C₆).₂₂(C₂×Q₈), (C₃×C₆).₂₄(C₄○D₄), (C₃×Dic₃⋊C₄).₉C₂, (C₄×C₃⋊Dic₃).₁₉C₂, (C₂×C₆).₅₉(C₂²×S₃), (C₂×C₃⋊Dic₃).₁₂₁C₂², SmallGroup(288,518)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₆² — C₆².₄₀C₂³

Chief series

C₁ — C₃ — C₃² — C₃×C₆ — C₆² — C₆×Dic₃ — C₆².C₂² — C₆².₄₀C₂³

Lower central

C₃² — C₆² — C₆².₄₀C₂³

Upper central

C₁ — C₂² — C₂×C₄

Generators and relations for C₆².₄₀C₂³
G = < a,b,c,d,e | a⁶=b⁶=1, c²=d²=b³, e²=a³b³, ab=ba, ac=ca, dad^-1=a^-1, ae=ea, cbc^-1=b^-1, bd=db, be=eb, dcd^-1=a³c, ece^-1=b³c, ede^-1=b³d >

Subgroups: 394 in 127 conjugacy classes, 46 normal (18 characteristic)
C₁, C₂, C₃, C₃, C₄, C₂², C₆, C₆, C₂×C₄, C₂×C₄, C₃², Dic₃, C₁₂, C₂×C₆, C₂×C₆, C₄², C₄⋊C₄, C₃×C₆, C₂×Dic₃, C₂×Dic₃, C₂×C₁₂, C₂×C₁₂, C₄².C₂, C₃×Dic₃, C₃⋊Dic₃, C₃⋊Dic₃, C₃×C₁₂, C₆², C₄×Dic₃, Dic₃⋊C₄, Dic₃⋊C₄, C₄⋊Dic₃, C₃×C₄⋊C₄, C₆×Dic₃, C₂×C₃⋊Dic₃, C₆×C₁₂, Dic₃.Q₈, Dic₃⋊Dic₃, C₆².C₂², C₃×Dic₃⋊C₄, C₄×C₃⋊Dic₃, C₆².₄₀C₂³
Quotients: C₁, C₂, C₂², S₃, Q₈, C₂³, D₆, C₂×Q₈, C₄○D₄, C₂²×S₃, C₄².C₂, S₃², C₄○D₁₂, D₄⋊₂S₃, S₃×Q₈, C₂×S₃², Dic₃.Q₈, Dic₃.D₆, D₆.D₆, D₆.₄D₆, C₆².₄₀C₂³

Smallest permutation representation of C₆².₄₀C₂³
►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 18 5 16 3 14)(2 13 6 17 4 15)(7 93 11 91 9 95)(8 94 12 92 10 96)(19 28 23 26 21 30)(20 29 24 27 22 25)(31 42 33 38 35 40)(32 37 34 39 36 41)(43 49 45 51 47 53)(44 50 46 52 48 54)(55 61 57 63 59 65)(56 62 58 64 60 66)(67 78 69 74 71 76)(68 73 70 75 72 77)(79 88 83 86 81 90)(80 89 84 87 82 85)

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

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

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

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

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

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

42 conjugacy classes

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

42 irreducible representations

dim	1	1	1	1	1	2	2	2	2	2	2	4	4	4	4	4	4	4
type	+	+	+	+	+	+	-	+	+			+	-	-	+			-
image	C₁	C₂	C₂	C₂	C₂	S₃	Q₈	D₆	D₆	C₄○D₄	C₄○D₁₂	S₃²	D₄⋊₂S₃	S₃×Q₈	C₂×S₃²	Dic₃.D₆	D₆.D₆	D₆.₄D₆
kernel	C₆².₄₀C₂³	Dic₃⋊Dic₃	C₆².C₂²	C₃×Dic₃⋊C₄	C₄×C₃⋊Dic₃	Dic₃⋊C₄	C₃⋊Dic₃	C₂×Dic₃	C₂×C₁₂	C₃×C₆	C₆	C₂×C₄	C₆	C₆	C₂²	C₂	C₂	C₂
# reps	1	2	2	2	1	2	2	4	2	4	8	1	2	2	1	2	2	2

Matrix representation of C₆².₄₀C₂³ ►in GL₆(𝔽₁₃)

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

12	0	0	0	0	0
0	12	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	0	0	0	9	0
0	0	0	0	3	3

10	3	0	0	0	0
1	3	0	0	0	0
0	0	12	0	0	0
0	0	0	12	0	0
0	0	0	0	4	5
0	0	0	0	10	9

2	1	0	0	0	0
8	11	0	0	0	0
0	0	12	0	0	0
0	0	1	1	0	0
0	0	0	0	12	0
0	0	0	0	12	1

12	7	0	0	0	0
0	1	0	0	0	0
0	0	12	0	0	0
0	0	0	12	0	0
0	0	0	0	8	0
0	0	0	0	0	8

G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,1,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,9,3,0,0,0,0,0,3],[10,1,0,0,0,0,3,3,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,4,10,0,0,0,0,5,9],[2,8,0,0,0,0,1,11,0,0,0,0,0,0,12,1,0,0,0,0,0,1,0,0,0,0,0,0,12,12,0,0,0,0,0,1],[12,0,0,0,0,0,7,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,8,0,0,0,0,0,0,8] >;

C₆².₄₀C₂³ in GAP, Magma, Sage, TeX

C_6^2._{40}C_2^3

% in TeX

G:=Group("C6^2.40C2^3");

// GroupNames label

G:=SmallGroup(288,518);

// by ID

G=gap.SmallGroup(288,518);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,112,141,120,422,219,142,1356,9414]);

// Polycyclic

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

// generators/relations

G = C62.40C23 order 288 = 25·32

35th non-split extension by C62 of C23 acting via C23/C2=C22

G = C₆².₄₀C₂³ order 288 = 2⁵·3²

35^th non-split extension by C₆² of C₂³ acting via C₂³/C₂=C₂²