C6xC8:C2^2 - GroupNames

G = C₆xC₈:C₂² order 192 = 2⁶·3

Direct product of C₆ and C₈:C₂²

direct product, metabelian, nilpotent (class 3), monomial, 2-elementary

Aliases: C₆xC₈:C₂², C₂₄:₇C₂³, C₁₂.₈₂C₂⁴, C₈:(C₂²xC₆), D₈:₃(C₂xC₆), (C₂xD₈):₁₁C₆, (C₆xD₈):₂₅C₂, C₄.₆₆(C₆xD₄), (C₂xSD₁₆):₄C₆, SD₁₆:₁(C₂xC₆), D₄:₂(C₂²xC₆), C₄.₅(C₂³xC₆), Q₈:₃(C₂²xC₆), (C₂xC₂₄):₂₁C₂², (C₆xSD₁₆):₁₅C₂, C₁₂.₃₂₉(C₂xD₄), (C₂xC₁₂).₅₂₅D₄, (C₃xD₈):₁₉C₂², (C₆xD₄):₆₆C₂², (C₂²xD₄):₁₄C₆, (C₃xD₄):₁₃C₂³, (C₆xM₄(2)):₈C₂, (C₂xM₄(2)):₃C₆, M₄(2):₃(C₂xC₆), (C₆xQ₈):₅₄C₂², (C₃xQ₈):₁₂C₂³, C₂³.₅₅(C₃xD₄), C₂².₂₃(C₆xD₄), (C₂²xC₆).₁₇₂D₄, C₆.₂₀₃(C₂²xD₄), (C₂xC₁₂).₉₇₅C₂³, (C₃xSD₁₆):₁₇C₂², (C₃xM₄(2)):₂₄C₂², (C₂²xC₁₂).₄₆₅C₂², (C₂xC₈):₂(C₂xC₆), (D₄xC₂xC₆):₂₆C₂, C₂.₂₇(D₄xC₂xC₆), C₄oD₄:₆(C₂xC₆), (C₆xC₄oD₄):₂₇C₂, (C₂xC₄oD₄):₁₅C₆, (C₂xD₄):₁₅(C₂xC₆), (C₂xQ₈):₁₆(C₂xC₆), (C₂xC₄).₁₃₆(C₃xD₄), (C₂xC₆).₄₁₉(C₂xD₄), (C₃xC₄oD₄):₂₄C₂², (C₂²xC₄).₈₁(C₂xC₆), (C₂xC₄).₄₅(C₂²xC₆), SmallGroup(192,1462)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₄ — C₆xC₈:C₂²

Chief series

C₁ — C₂ — C₄ — C₁₂ — C₃xD₄ — C₃xD₈ — C₃xC₈:C₂² — C₆xC₈:C₂²

Lower central

C₁ — C₂ — C₄ — C₆xC₈:C₂²

Upper central

C₁ — C₂xC₆ — C₂²xC₁₂ — C₆xC₈:C₂²

Generators and relations for C₆xC₈:C₂²
G = < a,b,c,d | a⁶=b⁸=c²=d²=1, ab=ba, ac=ca, ad=da, cbc=b³, dbd=b⁵, cd=dc >

Subgroups: 530 in 298 conjugacy classes, 162 normal (30 characteristic)
C₁, C₂, C₂, C₂, C₃, C₄, C₄, C₄, C₂², C₂², C₂², C₆, C₆, C₆, C₈, C₂xC₄, C₂xC₄, C₂xC₄, D₄, D₄, Q₈, Q₈, C₂³, C₂³, C₁₂, C₁₂, C₁₂, C₂xC₆, C₂xC₆, C₂xC₆, C₂xC₈, M₄(2), D₈, SD₁₆, C₂²xC₄, C₂²xC₄, C₂xD₄, C₂xD₄, C₂xD₄, C₂xQ₈, C₄oD₄, C₄oD₄, C₂⁴, C₂₄, C₂xC₁₂, C₂xC₁₂, C₂xC₁₂, C₃xD₄, C₃xD₄, C₃xQ₈, C₃xQ₈, C₂²xC₆, C₂²xC₆, C₂xM₄(2), C₂xD₈, C₂xSD₁₆, C₈:C₂², C₂²xD₄, C₂xC₄oD₄, C₂xC₂₄, C₃xM₄(2), C₃xD₈, C₃xSD₁₆, C₂²xC₁₂, C₂²xC₁₂, C₆xD₄, C₆xD₄, C₆xD₄, C₆xQ₈, C₃xC₄oD₄, C₃xC₄oD₄, C₂³xC₆, C₂xC₈:C₂², C₆xM₄(2), C₆xD₈, C₆xSD₁₆, C₃xC₈:C₂², D₄xC₂xC₆, C₆xC₄oD₄, C₆xC₈:C₂²
Quotients: C₁, C₂, C₃, C₂², C₆, D₄, C₂³, C₂xC₆, C₂xD₄, C₂⁴, C₃xD₄, C₂²xC₆, C₈:C₂², C₂²xD₄, C₆xD₄, C₂³xC₆, C₂xC₈:C₂², C₃xC₈:C₂², D₄xC₂xC₆, C₆xC₈:C₂²

Smallest permutation representation of C₆xC₈:C₂²
►On 48 points

Generators in S₄₈

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

(2 4)(3 7)(6 8)(9 13)(10 16)(12 14)(18 20)(19 23)(22 24)(26 28)(27 31)(30 32)(33 37)(34 40)(36 38)(42 44)(43 47)(46 48)

(1 5)(3 7)(9 13)(11 15)(17 21)(19 23)(25 29)(27 31)(33 37)(35 39)(41 45)(43 47)

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

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

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

66 conjugacy classes

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

66 irreducible representations

dim	1	1	1	1	1	1	1	1	1	1	1	1	1	1	2	2	2	2	4	4
type	+	+	+	+	+	+	+								+	+			+
image	C₁	C₂	C₂	C₂	C₂	C₂	C₂	C₃	C₆	C₆	C₆	C₆	C₆	C₆	D₄	D₄	C₃xD₄	C₃xD₄	C₈:C₂²	C₃xC₈:C₂²
kernel	C₆xC₈:C₂²	C₆xM₄(2)	C₆xD₈	C₆xSD₁₆	C₃xC₈:C₂²	D₄xC₂xC₆	C₆xC₄oD₄	C₂xC₈:C₂²	C₂xM₄(2)	C₂xD₈	C₂xSD₁₆	C₈:C₂²	C₂²xD₄	C₂xC₄oD₄	C₂xC₁₂	C₂²xC₆	C₂xC₄	C₂³	C₆	C₂
# reps	1	1	2	2	8	1	1	2	2	4	4	16	2	2	3	1	6	2	2	4

Matrix representation of C₆xC₈:C₂² ►in GL₆(F₇₃)

65	0	0	0	0	0
0	65	0	0	0	0
0	0	72	0	0	0
0	0	0	72	0	0
0	0	0	0	72	0
0	0	0	0	0	72

72	71	0	0	0	0
1	1	0	0	0	0
0	0	72	0	2	0
0	0	72	0	1	1
0	0	0	72	1	0
0	0	0	0	1	0

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

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

G:=sub<GL(6,GF(73))| [65,0,0,0,0,0,0,65,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[72,1,0,0,0,0,71,1,0,0,0,0,0,0,72,72,0,0,0,0,0,0,72,0,0,0,2,1,1,1,0,0,0,1,0,0],[72,1,0,0,0,0,0,1,0,0,0,0,0,0,1,1,1,1,0,0,0,72,0,0,0,0,0,0,0,72,0,0,0,0,72,0],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,72,72,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

C₆xC₈:C₂² in GAP, Magma, Sage, TeX

C_6\times C_8\rtimes C_2^2

% in TeX

G:=Group("C6xC8:C2^2");

// GroupNames label

G:=SmallGroup(192,1462);

// by ID

G=gap.SmallGroup(192,1462);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-2,-2,701,2102,6053,3036,124]);

// Polycyclic

G:=Group<a,b,c,d|a^6=b^8=c^2=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^3,d*b*d=b^5,c*d=d*c>;

// generators/relations

G = C6xC8:C22 order 192 = 26·3

Direct product of C6 and C8:C22

G = C₆xC₈:C₂² order 192 = 2⁶·3

Direct product of C₆ and C₈:C₂²