C6.51ES+(2,2) - GroupNames

G = C₆.₅₁2₊¹⁺⁴ order 192 = 2⁶·3

51^st non-split extension by C₆ of 2₊¹⁺⁴ acting via 2₊¹⁺⁴/C₂×D₄=C₂

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C₆.₅₁2₊¹⁺⁴, C₄⋊C₄⋊₁₁D₆, (C₂×Q₈)⋊₈D₆, D₆.₂(C₂×Q₈), C₂²⋊Q₈⋊₁₁S₃, (C₆×Q₈)⋊₈C₂², (C₂²×S₃)⋊₃Q₈, D₆⋊Q₈⋊₂₁C₂, D₆⋊₃Q₈⋊₁₆C₂, C₄.D₁₂⋊₂₇C₂, C₂².₇(S₃×Q₈), C₂²⋊C₄.₅₉D₆, C₃⋊₂(C₂³⋊₂Q₈), C₂.₃₅(D₄○D₁₂), C₆.₃₆(C₂²×Q₈), (C₂×C₁₂).₅₇C₂³, (C₂×C₆).₁₇₈C₂⁴, C₄⋊Dic₃⋊₃₆C₂², (C₂×Dic₆)⋊₉C₂², (C₂²×C₄).₂₅₆D₆, C₁₂.₄₈D₄⋊₄₆C₂, C₂.₅₃(D₄⋊₆D₆), Dic₃⋊C₄⋊₁₈C₂², D₆⋊C₄.₁₄₇C₂², Dic₃.D₄⋊₂₄C₂, (S₃×C₂³).₅₃C₂², (C₂²×C₆).₂₀₆C₂³, C₂³.₂₀₁(C₂²×S₃), C₂².₁₉₉(S₃×C₂³), (C₂×Dic₃).₈₉C₂³, (C₂²×S₃).₂₀₀C₂³, (C₂²×C₁₂).₃₁₅C₂², C₆.D₄.₃₄C₂², (C₂²×Dic₃).₁₁₉C₂², C₂.₁₉(C₂×S₃×Q₈), (C₂×C₆).₇(C₂×Q₈), (C₃×C₄⋊C₄)⋊₂₀C₂², (C₂×D₆⋊C₄).₂₀C₂, (S₃×C₂²⋊C₄).₂C₂, (S₃×C₂×C₄).₉₈C₂², (C₃×C₂²⋊Q₈)⋊₁₄C₂, (C₂×C₄).₁₈₃(C₂²×S₃), (C₃×C₂²⋊C₄).₃₃C₂², SmallGroup(192,1193)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂×C₆ — C₆.₅₁2₊¹⁺⁴

Chief series

C₁ — C₃ — C₆ — C₂×C₆ — C₂²×S₃ — S₃×C₂³ — S₃×C₂²⋊C₄ — C₆.₅₁2₊¹⁺⁴

Lower central

C₃ — C₂×C₆ — C₆.₅₁2₊¹⁺⁴

Upper central

C₁ — C₂² — C₂²⋊Q₈

Generators and relations for C₆.₅₁2₊¹⁺⁴
G = < a,b,c,d,e | a⁶=b⁴=c²=1, d²=b², e²=a³, ab=ba, ac=ca, dad^-1=a^-1, ae=ea, cbc=a³b^-1, dbd^-1=ebe^-1=a³b, cd=dc, ce=ec, ede^-1=a³b²d >

Subgroups: 672 in 242 conjugacy classes, 103 normal (31 characteristic)
C₁, C₂, C₂, C₃, C₄, C₂², C₂², C₂², S₃, C₆, C₆, C₂×C₄, C₂×C₄, C₂×C₄, Q₈, C₂³, C₂³, Dic₃, C₁₂, D₆, D₆, C₂×C₆, C₂×C₆, C₂×C₆, C₂²⋊C₄, C₂²⋊C₄, C₄⋊C₄, C₄⋊C₄, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂×Q₈, C₂×Q₈, C₂⁴, Dic₆, C₄×S₃, C₂×Dic₃, C₂×Dic₃, C₂×C₁₂, C₂×C₁₂, C₂×C₁₂, C₃×Q₈, C₂²×S₃, C₂²×S₃, C₂²×C₆, C₂×C₂²⋊C₄, C₂²⋊Q₈, C₂²⋊Q₈, Dic₃⋊C₄, C₄⋊Dic₃, C₄⋊Dic₃, D₆⋊C₄, C₆.D₄, C₃×C₂²⋊C₄, C₃×C₄⋊C₄, C₃×C₄⋊C₄, C₂×Dic₆, C₂×Dic₆, S₃×C₂×C₄, C₂²×Dic₃, C₂²×C₁₂, C₆×Q₈, S₃×C₂³, C₂³⋊₂Q₈, Dic₃.D₄, S₃×C₂²⋊C₄, D₆⋊Q₈, C₄.D₁₂, C₁₂.₄₈D₄, C₂×D₆⋊C₄, D₆⋊₃Q₈, C₃×C₂²⋊Q₈, C₆.₅₁2₊¹⁺⁴
Quotients: C₁, C₂, C₂², S₃, Q₈, C₂³, D₆, C₂×Q₈, C₂⁴, C₂²×S₃, C₂²×Q₈, 2₊¹⁺⁴, S₃×Q₈, S₃×C₂³, C₂³⋊₂Q₈, D₄⋊₆D₆, C₂×S₃×Q₈, D₄○D₁₂, C₆.₅₁2₊¹⁺⁴

Smallest permutation representation of C₆.₅₁2₊¹⁺⁴
►On 48 points

Generators in S₄₈

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

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

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

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

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

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

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

36 conjugacy classes

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

36 irreducible representations

dim	1	1	1	1	1	1	1	1	1	2	2	2	2	2	2	4	4	4	4
type	+	+	+	+	+	+	+	+	+	+	-	+	+	+	+	+	-		+
image	C₁	C₂	C₂	C₂	C₂	C₂	C₂	C₂	C₂	S₃	Q₈	D₆	D₆	D₆	D₆	2₊¹⁺⁴	S₃×Q₈	D₄⋊₆D₆	D₄○D₁₂
kernel	C₆.₅₁2₊¹⁺⁴	Dic₃.D₄	S₃×C₂²⋊C₄	D₆⋊Q₈	C₄.D₁₂	C₁₂.₄₈D₄	C₂×D₆⋊C₄	D₆⋊₃Q₈	C₃×C₂²⋊Q₈	C₂²⋊Q₈	C₂²×S₃	C₂²⋊C₄	C₄⋊C₄	C₂²×C₄	C₂×Q₈	C₆	C₂²	C₂	C₂
# reps	1	2	2	4	2	1	1	2	1	1	4	2	3	1	1	2	2	2	2

Matrix representation of C₆.₅₁2₊¹⁺⁴ ►in GL₆(𝔽₁₃)

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

4	10	0	0	0	0
10	9	0	0	0	0
0	0	0	0	12	0
0	0	0	0	0	12
0	0	1	0	0	0
0	0	0	1	0	0

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

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

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

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

C₆.₅₁2₊¹⁺⁴ in GAP, Magma, Sage, TeX

C_6._{51}2_+^{1+4}

% in TeX

G:=Group("C6.51ES+(2,2)");

// GroupNames label

G:=SmallGroup(192,1193);

// by ID

G=gap.SmallGroup(192,1193);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,219,184,675,297,136,6278]);

// Polycyclic

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

// generators/relations

G = C6.512+ 1+4 order 192 = 26·3

51st non-split extension by C6 of 2+ 1+4 acting via 2+ 1+4/C2×D4=C2

G = C₆.₅₁2₊¹⁺⁴ order 192 = 2⁶·3

51^st non-split extension by C₆ of 2₊¹⁺⁴ acting via 2₊¹⁺⁴/C₂×D₄=C₂