C12.78D12 - GroupNames

G = C₁₂.₇₈D₁₂ order 288 = 2⁵·3²

9^th non-split extension by C₁₂ of D₁₂ acting via D₁₂/D₆=C₂

Aliases: C₁₂.₇₈D₁₂, C₆.₄(S₃xC₈), C₃:₁(D₆:C₈), C₆.₁(D₆:C₄), C₆.₂(C₈:S₃), (C₃xC₁₂).₁₀₈D₄, (C₂xC₁₂).₂₉₆D₆, C₃²:₅(C₂²:C₈), C₆².₂₆(C₂xC₄), (C₃xC₆).₆M₄(2), C₁₂.₇₇(C₃:D₄), C₄.₂₇(C₃:D₁₂), (C₆xC₁₂).₂₀₁C₂², C₂.₄(C₁₂.₂₉D₆), C₂.₂(C₁₂.₃₁D₆), C₂.₁(C₆.D₁₂), C₂².₉(C₆.D₆), (C₆xC₃:C₈):₂C₂, (C₂xC₃:S₃):₃C₈, (C₂xC₃:C₈):₁₀S₃, (C₂xC₄).₁₂₉S₃², (C₂xC₆).₂₇(C₄xS₃), (C₃xC₆).₁₉(C₂xC₈), (C₂²xC₃:S₃).₆C₄, (C₂xC₃:Dic₃).₈C₄, (C₃xC₆).₂₄(C₂²:C₄), (C₂xC₄xC₃:S₃).₁₁C₂, SmallGroup(288,205)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃xC₆ — C₁₂.₇₈D₁₂

Chief series

C₁ — C₃ — C₃² — C₃xC₆ — C₃xC₁₂ — C₆xC₁₂ — C₆xC₃:C₈ — C₁₂.₇₈D₁₂

Lower central

C₃² — C₃xC₆ — C₁₂.₇₈D₁₂

Upper central

C₁ — C₂xC₄

Generators and relations for C₁₂.₇₈D₁₂
G = < a,b,c | a¹²=1, b¹²=c²=a⁶, bab^-1=cac^-1=a⁵, cbc^-1=a³b¹¹ >

Subgroups: 530 in 127 conjugacy classes, 42 normal (16 characteristic)
C₁, C₂, C₂, C₃, C₃, C₄, C₄, C₂², C₂², S₃, C₆, C₆, C₈, C₂xC₄, C₂xC₄, C₂³, C₃², Dic₃, C₁₂, C₁₂, D₆, C₂xC₆, C₂xC₆, C₂xC₈, C₂²xC₄, C₃:S₃, C₃xC₆, C₃:C₈, C₂₄, C₄xS₃, C₂xDic₃, C₂xC₁₂, C₂xC₁₂, C₂²xS₃, C₂²:C₈, C₃:Dic₃, C₃xC₁₂, C₂xC₃:S₃, C₂xC₃:S₃, C₆², C₂xC₃:C₈, C₂xC₂₄, S₃xC₂xC₄, C₃xC₃:C₈, C₄xC₃:S₃, C₂xC₃:Dic₃, C₆xC₁₂, C₂²xC₃:S₃, D₆:C₈, C₆xC₃:C₈, C₂xC₄xC₃:S₃, C₁₂.₇₈D₁₂
Quotients: C₁, C₂, C₄, C₂², S₃, C₈, C₂xC₄, D₄, D₆, C₂²:C₄, C₂xC₈, M₄(2), C₄xS₃, D₁₂, C₃:D₄, C₂²:C₈, S₃², S₃xC₈, C₈:S₃, D₆:C₄, C₆.D₆, C₃:D₁₂, D₆:C₈, C₁₂.₂₉D₆, C₁₂.₃₁D₆, C₆.D₁₂, C₁₂.₇₈D₁₂

Smallest permutation representation of C₁₂.₇₈D₁₂
►On 48 points

Generators in S₄₈

(1 27 5 31 9 35 13 39 17 43 21 47)(2 36 22 32 18 28 14 48 10 44 6 40)(3 29 7 33 11 37 15 41 19 45 23 25)(4 38 24 34 20 30 16 26 12 46 8 42)

(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)

(1 3 13 15)(2 44 14 32)(4 30 16 42)(5 23 17 11)(6 40 18 28)(7 9 19 21)(8 26 20 38)(10 36 22 48)(12 46 24 34)(25 27 37 39)(29 47 41 35)(31 33 43 45)

G:=sub<Sym(48)| (1,27,5,31,9,35,13,39,17,43,21,47)(2,36,22,32,18,28,14,48,10,44,6,40)(3,29,7,33,11,37,15,41,19,45,23,25)(4,38,24,34,20,30,16,26,12,46,8,42), (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), (1,3,13,15)(2,44,14,32)(4,30,16,42)(5,23,17,11)(6,40,18,28)(7,9,19,21)(8,26,20,38)(10,36,22,48)(12,46,24,34)(25,27,37,39)(29,47,41,35)(31,33,43,45)>;

G:=Group( (1,27,5,31,9,35,13,39,17,43,21,47)(2,36,22,32,18,28,14,48,10,44,6,40)(3,29,7,33,11,37,15,41,19,45,23,25)(4,38,24,34,20,30,16,26,12,46,8,42), (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), (1,3,13,15)(2,44,14,32)(4,30,16,42)(5,23,17,11)(6,40,18,28)(7,9,19,21)(8,26,20,38)(10,36,22,48)(12,46,24,34)(25,27,37,39)(29,47,41,35)(31,33,43,45) );

G=PermutationGroup([[(1,27,5,31,9,35,13,39,17,43,21,47),(2,36,22,32,18,28,14,48,10,44,6,40),(3,29,7,33,11,37,15,41,19,45,23,25),(4,38,24,34,20,30,16,26,12,46,8,42)], [(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)], [(1,3,13,15),(2,44,14,32),(4,30,16,42),(5,23,17,11),(6,40,18,28),(7,9,19,21),(8,26,20,38),(10,36,22,48),(12,46,24,34),(25,27,37,39),(29,47,41,35),(31,33,43,45)]])

60 conjugacy classes

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

60 irreducible representations

dim	1	1	1	1	1	1	2	2	2	2	2	2	2	2	2	4	4	4	4	4
type	+	+	+				+	+	+		+					+	+	+
image	C₁	C₂	C₂	C₄	C₄	C₈	S₃	D₄	D₆	M₄(2)	D₁₂	C₃:D₄	C₄xS₃	S₃xC₈	C₈:S₃	S₃²	C₃:D₁₂	C₆.D₆	C₁₂.₂₉D₆	C₁₂.₃₁D₆
kernel	C₁₂.₇₈D₁₂	C₆xC₃:C₈	C₂xC₄xC₃:S₃	C₂xC₃:Dic₃	C₂²xC₃:S₃	C₂xC₃:S₃	C₂xC₃:C₈	C₃xC₁₂	C₂xC₁₂	C₃xC₆	C₁₂	C₁₂	C₂xC₆	C₆	C₆	C₂xC₄	C₄	C₂²	C₂	C₂
# reps	1	2	1	2	2	8	2	2	2	2	4	4	4	8	8	1	2	1	2	2

Matrix representation of C₁₂.₇₈D₁₂ ►in GL₆(F₇₃)

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

46	0	0	0	0	0
27	27	0	0	0	0
0	0	46	71	0	0
0	0	72	27	0	0
0	0	0	0	63	10
0	0	0	0	63	0

72	0	0	0	0	0
1	1	0	0	0	0
0	0	1	0	0	0
0	0	46	72	0	0
0	0	0	0	0	27
0	0	0	0	27	0

G:=sub<GL(6,GF(73))| [1,72,0,0,0,0,1,0,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,27,0,0,0,0,0,0,27],[46,27,0,0,0,0,0,27,0,0,0,0,0,0,46,72,0,0,0,0,71,27,0,0,0,0,0,0,63,63,0,0,0,0,10,0],[72,1,0,0,0,0,0,1,0,0,0,0,0,0,1,46,0,0,0,0,0,72,0,0,0,0,0,0,0,27,0,0,0,0,27,0] >;

C₁₂.₇₈D₁₂ in GAP, Magma, Sage, TeX

C_{12}._{78}D_{12}

% in TeX

G:=Group("C12.78D12");

// GroupNames label

G:=SmallGroup(288,205);

// by ID

G=gap.SmallGroup(288,205);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,28,141,92,100,1356,9414]);

// Polycyclic

G:=Group<a,b,c|a^12=1,b^12=c^2=a^6,b*a*b^-1=c*a*c^-1=a^5,c*b*c^-1=a^3*b^11>;

// generators/relations

G = C12.78D12 order 288 = 25·32

9th non-split extension by C12 of D12 acting via D12/D6=C2

G = C₁₂.₇₈D₁₂ order 288 = 2⁵·3²

9^th non-split extension by C₁₂ of D₁₂ acting via D₁₂/D₆=C₂