C6.C2^5 - GroupNames

G = C₆.C₂⁵ order 192 = 2⁶·3

14^th non-split extension by C₆ of C₂⁵ acting via C₂⁵/C₂⁴=C₂

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C₆.₁₄C₂⁵, D₆.₈C₂⁴, C₁₂.₄₉C₂⁴, D₁₂.₄₁C₂³, Dic₃.₉C₂⁴, Dic₆.₄₁C₂³, C₄○D₄⋊₂₂D₆, (C₂×D₄)⋊₄₇D₆, (C₂×Q₈)⋊₃₉D₆, D₄○D₁₂⋊₁₃C₂, Q₈○D₁₂⋊₁₃C₂, (C₂²×C₄)⋊₃₇D₆, (C₂×C₆).₅C₂⁴, D₄⋊₆D₆⋊₁₁C₂, (S₃×D₄)⋊₁₃C₂², (C₆×D₄)⋊₅₄C₂², C₄.₄₆(S₃×C₂³), C₂.₁₅(S₃×C₂⁴), (S₃×Q₈)⋊₁₅C₂², (C₆×Q₈)⋊₄₇C₂², C₃⋊D₄.₂C₂³, C₄○D₁₂⋊₂₇C₂², (C₂×D₁₂)⋊₆₅C₂², (C₄×S₃).₃₈C₂³, Q₈.₁₅D₆⋊₉C₂, C₃⋊₁(C₂.C₂⁵), (C₃×D₄).₃₀C₂³, D₄.₃₀(C₂²×S₃), Q₈.₄₁(C₂²×S₃), (C₃×Q₈).₃₁C₂³, D₄⋊₂S₃⋊₁₄C₂², (C₂×C₁₂).₅₆₈C₂³, (C₂²×C₁₂)⋊₂₉C₂², Q₈⋊₃S₃⋊₁₄C₂², (C₂×Dic₆)⋊₇₆C₂², C₂².₁₀(S₃×C₂³), C₂³.₁₅₂(C₂²×S₃), (C₂²×C₆).₂₅₀C₂³, (C₂²×S₃).₁₄₂C₂³, (C₂×Dic₃).₁₆₈C₂³, (S₃×C₄○D₄)⋊₆C₂, (C₂×C₄○D₄)⋊₁₉S₃, (C₆×C₄○D₄)⋊₁₆C₂, (S₃×C₂×C₄)⋊₃₅C₂², (C₂×C₄○D₁₂)⋊₃₉C₂, (C₃×C₄○D₄)⋊₂₂C₂², (C₂×C₃⋊D₄)⋊₅₅C₂², (C₂×C₄).₆₄₆(C₂²×S₃), SmallGroup(192,1523)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₆ — C₆.C₂⁵

Chief series

C₁ — C₃ — C₆ — D₆ — C₂²×S₃ — S₃×C₂×C₄ — S₃×C₄○D₄ — C₆.C₂⁵

Lower central

C₃ — C₆ — C₆.C₂⁵

Upper central

C₁ — C₄ — C₂×C₄○D₄

Generators and relations for C₆.C₂⁵
G = < a,b,c,d,e,f | a⁶=b²=c²=e²=f²=1, d²=a³, bab=a^-1, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, fbf=a³b, cd=dc, ece=a³c, cf=fc, de=ed, df=fd, ef=fe >

Subgroups: 1672 in 810 conjugacy classes, 443 normal (17 characteristic)
C₁, C₂, C₂, C₃, C₄, C₄, C₄, C₂², C₂², C₂², S₃, C₆, C₆, C₂×C₄, C₂×C₄, C₂×C₄, D₄, D₄, Q₈, Q₈, C₂³, C₂³, Dic₃, C₁₂, C₁₂, D₆, D₆, C₂×C₆, C₂×C₆, C₂×C₆, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×D₄, C₂×Q₈, C₂×Q₈, C₄○D₄, C₄○D₄, Dic₆, C₄×S₃, D₁₂, C₂×Dic₃, C₃⋊D₄, C₂×C₁₂, C₂×C₁₂, C₃×D₄, C₃×Q₈, C₂²×S₃, C₂²×C₆, C₂×C₄○D₄, C₂×C₄○D₄, 2₊¹⁺⁴, 2_-¹⁺⁴, C₂×Dic₆, S₃×C₂×C₄, C₂×D₁₂, C₄○D₁₂, S₃×D₄, D₄⋊₂S₃, S₃×Q₈, Q₈⋊₃S₃, C₂×C₃⋊D₄, C₂²×C₁₂, C₆×D₄, C₆×Q₈, C₃×C₄○D₄, C₂.C₂⁵, C₂×C₄○D₁₂, D₄⋊₆D₆, Q₈.₁₅D₆, S₃×C₄○D₄, D₄○D₁₂, Q₈○D₁₂, C₆×C₄○D₄, C₆.C₂⁵
Quotients: C₁, C₂, C₂², S₃, C₂³, D₆, C₂⁴, C₂²×S₃, C₂⁵, S₃×C₂³, C₂.C₂⁵, S₃×C₂⁴, C₆.C₂⁵

Smallest permutation representation of C₆.C₂⁵
►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)

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

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

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

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

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

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

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

54 conjugacy classes

class	1	2A	2B	···	2H	2I	···	2P	3	4A	4B	4C	···	4I	4J	···	4Q	6A	6B	6C	6D	···	6I	12A	12B	12C	12D	12E	···	12J
order	1	2	2	···	2	2	···	2	3	4	4	4	···	4	4	···	4	6	6	6	6	···	6	12	12	12	12	12	···	12
size	1	1	2	···	2	6	···	6	2	1	1	2	···	2	6	···	6	2	2	2	4	···	4	2	2	2	2	4	···	4

54 irreducible representations

dim

type

image

C₁

C₂

S₃

D₆

C₂.C₂⁵

C₆.C₂⁵

kernel

C₆.C₂⁵

C₂×C₄○D₁₂

D₄⋊₆D₆

Q₈.₁₅D₆

S₃×C₄○D₄

D₄○D₁₂

Q₈○D₁₂

C₆×C₄○D₄

C₂×C₄○D₄

C₂²×C₄

C₂×D₄

C₂×Q₈

C₄○D₄

C₃

C₁

# reps

Matrix representation of C₆.C₂⁵ ►in GL₆(𝔽₁₃)

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

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

1	0	0	0	0	0
0	1	0	0	0	0
0	0	0	0	5	0
0	0	0	0	0	5
0	0	8	0	0	0
0	0	0	8	0	0

12	0	0	0	0	0
0	12	0	0	0	0
0	0	5	0	0	0
0	0	0	5	0	0
0	0	0	0	5	0
0	0	0	0	0	5

1	0	0	0	0	0
0	1	0	0	0	0
0	0	0	0	1	0
0	0	0	0	0	1
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	0	1	0	0
0	0	1	0	0	0
0	0	0	0	0	1
0	0	0	0	1	0

G:=sub<GL(6,GF(13))| [0,1,0,0,0,0,12,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[0,12,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,8,0,0,5,0,0,0,0,0,0,5,0,0],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,5,0,0,0,0,0,0,5,0,0,0,0,0,0,5,0,0,0,0,0,0,5],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;

C₆.C₂⁵ in GAP, Magma, Sage, TeX

C_6.C_2^5

% in TeX

G:=Group("C6.C2^5");

// GroupNames label

G:=SmallGroup(192,1523);

// by ID

G=gap.SmallGroup(192,1523);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,184,570,1684,6278]);

// Polycyclic

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

// generators/relations

G = C6.C25 order 192 = 26·3

14th non-split extension by C6 of C25 acting via C25/C24=C2

G = C₆.C₂⁵ order 192 = 2⁶·3

14^th non-split extension by C₆ of C₂⁵ acting via C₂⁵/C₂⁴=C₂