C6^2.37D4 - GroupNames

G = C₆².₃₇D₄ order 288 = 2⁵·3²

21^st non-split extension by C₆² of D₄ acting via D₄/C₂=C₂²

Aliases: C₆².₃₇D₄, C₃²:₁₁C₄wrC₂, C₁₂:S₃:₉C₄, C₁₂.₂₉(C₄xS₃), (C₂xC₁₂).₉₂D₆, (C₂xC₆).₁₁D₁₂, C₃²:₄Q₈:₉C₄, C₆.₂₉(D₆:C₄), (C₃xC₁₂).₁₇₁D₄, C₃:₂(D₁₂:C₄), M₄(2):₄(C₃:S₃), (C₃xM₄(2)):₁₀S₃, (C₆xC₁₂).₅₉C₂², C₁₂.₁₃₂(C₃:D₄), C₁₂.₅₉D₆.₃C₂, C₄.₂₉(C₃²:₇D₄), C₂².₃(C₁₂:S₃), (C₃²xM₄(2)):₁₄C₂, C₂.₁₁(C₆.₁₁D₁₂), C₄.₃(C₄xC₃:S₃), (C₄xC₃:Dic₃):₂C₂, (C₃xC₁₂).₅₁(C₂xC₄), (C₃xC₆).₆₀(C₂²:C₄), (C₂xC₄).₃₉(C₂xC₃:S₃), SmallGroup(288,300)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃xC₁₂ — C₆².₃₇D₄

Chief series

C₁ — C₃ — C₃² — C₃xC₆ — C₃xC₁₂ — C₆xC₁₂ — C₁₂.₅₉D₆ — C₆².₃₇D₄

Lower central

C₃² — C₃xC₆ — C₃xC₁₂ — C₆².₃₇D₄

Upper central

C₁ — C₄ — C₂xC₄ — M₄(2)

Generators and relations for C₆².₃₇D₄
G = < a,b,c,d | a⁶=b⁶=1, c⁴=b³, d²=a³, ab=ba, cac^-1=ab³, dad^-1=a^-1, bc=cb, dbd^-1=b^-1, dcd^-1=a³b³c³ >

Subgroups: 572 in 132 conjugacy classes, 47 normal (19 characteristic)
C₁, C₂, C₂, C₃, C₄, C₄, C₂², C₂², S₃, C₆, C₆, C₈, C₂xC₄, C₂xC₄, D₄, Q₈, C₃², Dic₃, C₁₂, D₆, C₂xC₆, C₄², M₄(2), C₄oD₄, C₃:S₃, C₃xC₆, C₃xC₆, C₂₄, Dic₆, C₄xS₃, D₁₂, C₂xDic₃, C₃:D₄, C₂xC₁₂, C₄wrC₂, C₃:Dic₃, C₃xC₁₂, C₂xC₃:S₃, C₆², C₄xDic₃, C₃xM₄(2), C₄oD₁₂, C₃xC₂₄, C₃²:₄Q₈, C₄xC₃:S₃, C₁₂:S₃, C₂xC₃:Dic₃, C₃²:₇D₄, C₆xC₁₂, D₁₂:C₄, C₄xC₃:Dic₃, C₃²xM₄(2), C₁₂.₅₉D₆, C₆².₃₇D₄
Quotients: C₁, C₂, C₄, C₂², S₃, C₂xC₄, D₄, D₆, C₂²:C₄, C₃:S₃, C₄xS₃, D₁₂, C₃:D₄, C₄wrC₂, C₂xC₃:S₃, D₆:C₄, C₄xC₃:S₃, C₁₂:S₃, C₃²:₇D₄, D₁₂:C₄, C₆.₁₁D₁₂, C₆².₃₇D₄

Smallest permutation representation of C₆².₃₇D₄
►On 72 points

Generators in S₇₂

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

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

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

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

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

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

54 conjugacy classes

class	1	2A	2B	2C	3A	3B	3C	3D	4A	4B	4C	4D	4E	4F	4G	4H	6A	6B	6C	6D	6E	6F	6G	6H	8A	8B	12A	···	12H	12I	12J	12K	12L	24A	···	24P
order	1	2	2	2	3	3	3	3	4	4	4	4	4	4	4	4	6	6	6	6	6	6	6	6	8	8	12	···	12	12	12	12	12	24	···	24
size	1	1	2	36	2	2	2	2	1	1	2	18	18	18	18	36	2	2	2	2	4	4	4	4	4	4	2	···	2	4	4	4	4	4	···	4

54 irreducible representations

dim	1	1	1	1	1	1	2	2	2	2	2	2	2	2	4
type	+	+	+	+			+	+	+	+			+
image	C₁	C₂	C₂	C₂	C₄	C₄	S₃	D₄	D₄	D₆	C₄xS₃	C₃:D₄	D₁₂	C₄wrC₂	D₁₂:C₄
kernel	C₆².₃₇D₄	C₄xC₃:Dic₃	C₃²xM₄(2)	C₁₂.₅₉D₆	C₃²:₄Q₈	C₁₂:S₃	C₃xM₄(2)	C₃xC₁₂	C₆²	C₂xC₁₂	C₁₂	C₁₂	C₂xC₆	C₃²	C₃
# reps	1	1	1	1	2	2	4	1	1	4	8	8	8	4	8

Matrix representation of C₆².₃₇D₄ ►in GL₆(F₇₃)

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	72	0
0	0	0	0	0	1

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

7	59	0	0	0	0
14	66	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	46	0

0	1	0	0	0	0
1	0	0	0	0	0
0	0	1	0	0	0
0	0	72	72	0	0
0	0	0	0	27	0
0	0	0	0	0	72

G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,72,0,0,0,0,1,72,0,0,0,0,0,0,72,0,0,0,0,0,0,1],[0,1,0,0,0,0,72,72,0,0,0,0,0,0,72,1,0,0,0,0,72,0,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[7,14,0,0,0,0,59,66,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,46,0,0,0,0,1,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,72,0,0,0,0,0,72,0,0,0,0,0,0,27,0,0,0,0,0,0,72] >;

C₆².₃₇D₄ in GAP, Magma, Sage, TeX

C_6^2._{37}D_4

% in TeX

G:=Group("C6^2.37D4");

// GroupNames label

G:=SmallGroup(288,300);

// by ID

G=gap.SmallGroup(288,300);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,141,36,100,675,346,80,2693,9414]);

// Polycyclic

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

// generators/relations

G = C62.37D4 order 288 = 25·32

21st non-split extension by C62 of D4 acting via D4/C2=C22

G = C₆².₃₇D₄ order 288 = 2⁵·3²

21^st non-split extension by C₆² of D₄ acting via D₄/C₂=C₂²