C6^2.110D4 - GroupNames

G = C₆².₁₁₀D₄ order 288 = 2⁵·3²

15^th non-split extension by C₆² of D₄ acting via D₄/C₂²=C₂

Aliases: C₆².₁₁₀D₄, (C₂xC₆).₁₀D₁₂, C₆²:₅C₄:₂C₂, C₆.₂₁(D₆:C₄), C₃²:₇(C₂³:C₄), C₆².₃₉(C₂xC₄), (C₂²xC₆).₅₆D₆, C₃:₂(C₂³.₆D₆), (C₂xC₆²).₁₁C₂², C₂².₂(C₁₂:S₃), C₂.₄(C₆.₁₁D₁₂), C₂².₈(C₃²:₇D₄), (C₂²xC₃:S₃):₂C₄, (C₃xC₂²:C₄):₂S₃, (C₂xC₆).₁₃(C₄xS₃), (C₂xC₃:Dic₃):₂C₄, C₂³.₆(C₂xC₃:S₃), C₂².₃(C₄xC₃:S₃), C₂²:C₄:₁(C₃:S₃), (C₂xC₆).₈₆(C₃:D₄), (C₃²xC₂²:C₄):₂C₂, (C₂xC₃²:₇D₄).₂C₂, (C₃xC₆).₅₂(C₂²:C₄), SmallGroup(288,281)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₆² — C₆².₁₁₀D₄

Chief series

C₁ — C₃ — C₃² — C₃xC₆ — C₆² — C₂xC₆² — C₂xC₃²:₇D₄ — C₆².₁₁₀D₄

Lower central

C₃² — C₃xC₆ — C₆² — C₆².₁₁₀D₄

Upper central

C₁ — C₂ — C₂³ — C₂²:C₄

Generators and relations for C₆².₁₁₀D₄
G = < a,b,c,d | a⁶=b⁶=c⁴=d²=1, ab=ba, cac^-1=dad=a^-1b³, cbc^-1=dbd=b^-1, dcd=a³c^-1 >

Subgroups: 732 in 156 conjugacy classes, 47 normal (19 characteristic)
C₁, C₂, C₂, C₃, C₄, C₂², C₂², S₃, C₆, C₆, C₂xC₄, D₄, C₂³, C₂³, C₃², Dic₃, C₁₂, D₆, C₂xC₆, C₂xC₆, C₂²:C₄, C₂²:C₄, C₂xD₄, C₃:S₃, C₃xC₆, C₃xC₆, C₂xDic₃, C₃:D₄, C₂xC₁₂, C₂²xS₃, C₂²xC₆, C₂³:C₄, C₃:Dic₃, C₃xC₁₂, C₂xC₃:S₃, C₆², C₆², C₆.D₄, C₃xC₂²:C₄, C₂xC₃:D₄, C₂xC₃:Dic₃, C₂xC₃:Dic₃, C₃²:₇D₄, C₆xC₁₂, C₂²xC₃:S₃, C₂xC₆², C₂³.₆D₆, C₆²:₅C₄, C₃²xC₂²:C₄, C₂xC₃²:₇D₄, C₆².₁₁₀D₄
Quotients: C₁, C₂, C₄, C₂², S₃, C₂xC₄, D₄, D₆, C₂²:C₄, C₃:S₃, C₄xS₃, D₁₂, C₃:D₄, C₂³:C₄, C₂xC₃:S₃, D₆:C₄, C₄xC₃:S₃, C₁₂:S₃, C₃²:₇D₄, C₂³.₆D₆, C₆.₁₁D₁₂, C₆².₁₁₀D₄

Smallest permutation representation of C₆².₁₁₀D₄
►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 51 46 17 22 58)(2 52 47 18 23 59)(3 53 48 13 24 60)(4 54 43 14 19 55)(5 49 44 15 20 56)(6 50 45 16 21 57)(7 31 68 61 26 42)(8 32 69 62 27 37)(9 33 70 63 28 38)(10 34 71 64 29 39)(11 35 72 65 30 40)(12 36 67 66 25 41)

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

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

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

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

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

51 conjugacy classes

class	1	2A	2B	2C	2D	2E	3A	3B	3C	3D	4A	4B	4C	4D	4E	6A	···	6L	6M	···	6T	12A	···	12P
order	1	2	2	2	2	2	3	3	3	3	4	4	4	4	4	6	···	6	6	···	6	12	···	12
size	1	1	2	2	2	36	2	2	2	2	4	4	36	36	36	2	···	2	4	···	4	4	···	4

51 irreducible representations

dim	1	1	1	1	1	1	2	2	2	2	2	2	4	4
type	+	+	+	+			+	+	+		+		+
image	C₁	C₂	C₂	C₂	C₄	C₄	S₃	D₄	D₆	C₄xS₃	D₁₂	C₃:D₄	C₂³:C₄	C₂³.₆D₆
kernel	C₆².₁₁₀D₄	C₆²:₅C₄	C₃²xC₂²:C₄	C₂xC₃²:₇D₄	C₂xC₃:Dic₃	C₂²xC₃:S₃	C₃xC₂²:C₄	C₆²	C₂²xC₆	C₂xC₆	C₂xC₆	C₂xC₆	C₃²	C₃
# reps	1	1	1	1	2	2	4	2	4	8	8	8	1	8

Matrix representation of C₆².₁₁₀D₄ ►in GL₆(F₁₃)

0	1	0	0	0	0
12	1	0	0	0	0
0	0	2	4	2	7
0	0	9	11	7	4
0	0	0	0	11	4
0	0	0	0	9	2

0	12	0	0	0	0
1	12	0	0	0	0
0	0	1	1	7	5
0	0	12	0	5	1
0	0	0	0	0	1
0	0	0	0	12	1

8	5	0	0	0	0
0	5	0	0	0	0
0	0	1	1	6	9
0	0	0	12	9	10
0	0	0	0	9	2
0	0	0	0	11	4

12	1	0	0	0	0
0	1	0	0	0	0
0	0	5	5	12	4
0	0	2	4	8	3
0	0	12	1	7	10
0	0	1	2	6	10

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

C₆².₁₁₀D₄ in GAP, Magma, Sage, TeX

C_6^2._{110}D_4

% in TeX

G:=Group("C6^2.110D4");

// GroupNames label

G:=SmallGroup(288,281);

// by ID

G=gap.SmallGroup(288,281);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,141,36,422,346,2693,9414]);

// Polycyclic

G:=Group<a,b,c,d|a^6=b^6=c^4=d^2=1,a*b=b*a,c*a*c^-1=d*a*d=a^-1*b^3,c*b*c^-1=d*b*d=b^-1,d*c*d=a^3*c^-1>;

// generators/relations

G = C62.110D4 order 288 = 25·32

15th non-split extension by C62 of D4 acting via D4/C22=C2

G = C₆².₁₁₀D₄ order 288 = 2⁵·3²

15^th non-split extension by C₆² of D₄ acting via D₄/C₂²=C₂