C3xC4^2:C2^2 - GroupNames

G = C₃×C₄²⋊C₂² order 192 = 2⁶·3

Direct product of C₃ and C₄²⋊C₂²

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

Aliases: C₃×C₄²⋊C₂², C₄≀C₂⋊₅C₆, C₄○D₄⋊₆C₁₂, (C₆×D₄)⋊₂₂C₄, C₄²⋊₄(C₂×C₆), (C₆×Q₈)⋊₁₈C₄, C₄.₇₃(C₆×D₄), (C₂×D₄)⋊₁₀C₁₂, (C₂×Q₈)⋊₁₀C₁₂, D₄.₇(C₂×C₁₂), C₄²⋊C₂⋊₄C₆, (C₄×C₁₂)⋊₃₄C₂², (C₂×C₁₂).₅₂₀D₄, C₁₂.₄₇₈(C₂×D₄), C₄.₈(C₂²×C₁₂), Q₈.₁₂(C₂×C₁₂), C₂².₁₃(C₆×D₄), C₂³.₁₂(C₃×D₄), (C₂²×C₆).₃₀D₄, (C₆×M₄(2))⋊₃₁C₂, M₄(2)⋊₁₀(C₂×C₆), (C₂×M₄(2))⋊₁₃C₆, C₁₂.₈₄(C₂²⋊C₄), (C₂×C₁₂).₈₉₇C₂³, C₁₂.₁₅₃(C₂²×C₄), (C₃×M₄(2))⋊₃₉C₂², (C₂²×C₁₂).₄₁₃C₂², (C₃×C₄≀C₂)⋊₁₃C₂, (C₃×C₄○D₄)⋊₁₀C₄, (C₂×C₄).₂₅(C₃×D₄), (C₂×C₄).₂₃(C₂×C₁₂), C₄○D₄.₁₄(C₂×C₆), (C₆×C₄○D₄).₂₁C₂, (C₂×C₄○D₄).₁₃C₆, (C₃×D₄).₂₉(C₂×C₄), (C₂×C₆).₄₀₈(C₂×D₄), C₄.₁₆(C₃×C₂²⋊C₄), C₂.₂₄(C₆×C₂²⋊C₄), (C₃×Q₈).₃₁(C₂×C₄), (C₂×C₁₂).₁₉₆(C₂×C₄), C₆.₁₁₂(C₂×C₂²⋊C₄), (C₂×C₄).₇₂(C₂²×C₆), (C₂²×C₄).₃₇(C₂×C₆), (C₃×C₄²⋊C₂)⋊₂₅C₂, (C₂×C₆).₈₃(C₂²⋊C₄), (C₃×C₄○D₄).₅₂C₂², C₂².₂₂(C₃×C₂²⋊C₄), SmallGroup(192,854)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₄ — C₃×C₄²⋊C₂²

Chief series

C₁ — C₂ — C₄ — C₂×C₄ — C₂×C₁₂ — C₃×M₄(2) — C₃×C₄≀C₂ — C₃×C₄²⋊C₂²

Lower central

C₁ — C₂ — C₄ — C₃×C₄²⋊C₂²

Upper central

C₁ — C₁₂ — C₂²×C₁₂ — C₃×C₄²⋊C₂²

Generators and relations for C₃×C₄²⋊C₂²
G = < a,b,c,d,e | a³=b⁴=c⁴=d²=e²=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd=b^-1c, ebe=bc², cd=dc, ce=ec, de=ed >

Subgroups: 258 in 154 conjugacy classes, 78 normal (46 characteristic)
C₁, C₂, C₂, C₃, C₄, C₄, C₂², C₂², C₆, C₆, C₈, C₂×C₄, C₂×C₄, D₄, D₄, Q₈, Q₈, C₂³, C₂³, C₁₂, C₁₂, C₂×C₆, C₂×C₆, C₄², C₂²⋊C₄, C₄⋊C₄, C₂×C₈, M₄(2), M₄(2), C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×D₄, C₂×Q₈, C₄○D₄, C₄○D₄, C₂₄, C₂×C₁₂, C₂×C₁₂, C₃×D₄, C₃×D₄, C₃×Q₈, C₃×Q₈, C₂²×C₆, C₂²×C₆, C₄≀C₂, C₄²⋊C₂, C₂×M₄(2), C₂×C₄○D₄, C₄×C₁₂, C₃×C₂²⋊C₄, C₃×C₄⋊C₄, C₂×C₂₄, C₃×M₄(2), C₃×M₄(2), C₂²×C₁₂, C₂²×C₁₂, C₆×D₄, C₆×D₄, C₆×Q₈, C₃×C₄○D₄, C₃×C₄○D₄, C₄²⋊C₂², C₃×C₄≀C₂, C₃×C₄²⋊C₂, C₆×M₄(2), C₆×C₄○D₄, C₃×C₄²⋊C₂²
Quotients: C₁, C₂, C₃, C₄, C₂², C₆, C₂×C₄, D₄, C₂³, C₁₂, C₂×C₆, C₂²⋊C₄, C₂²×C₄, C₂×D₄, C₂×C₁₂, C₃×D₄, C₂²×C₆, C₂×C₂²⋊C₄, C₃×C₂²⋊C₄, C₂²×C₁₂, C₆×D₄, C₄²⋊C₂², C₆×C₂²⋊C₄, C₃×C₄²⋊C₂²

Smallest permutation representation of C₃×C₄²⋊C₂²
►On 48 points

Generators in S₄₈

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

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

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

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

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

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

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

66 conjugacy classes

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

66 irreducible representations

dim	1	1	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	C₃	C₄	C₄	C₄	C₆	C₆	C₆	C₆	C₁₂	C₁₂	C₁₂	D₄	D₄	C₃×D₄	C₃×D₄	C₄²⋊C₂²	C₃×C₄²⋊C₂²
kernel	C₃×C₄²⋊C₂²	C₃×C₄≀C₂	C₃×C₄²⋊C₂	C₆×M₄(2)	C₆×C₄○D₄	C₄²⋊C₂²	C₆×D₄	C₆×Q₈	C₃×C₄○D₄	C₄≀C₂	C₄²⋊C₂	C₂×M₄(2)	C₂×C₄○D₄	C₂×D₄	C₂×Q₈	C₄○D₄	C₂×C₁₂	C₂²×C₆	C₂×C₄	C₂³	C₃	C₁
# reps	1	4	1	1	1	2	2	2	4	8	2	2	2	4	4	8	3	1	6	2	2	4

Matrix representation of C₃×C₄²⋊C₂² ►in GL₆(𝔽₇₃)

64	0	0	0	0	0
0	64	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	1

72	0	0	0	0	0
1	1	0	0	0	0
0	0	0	72	32	63
0	0	1	0	27	63
0	0	0	0	41	23
0	0	0	0	19	32

72	0	0	0	0	0
0	72	0	0	0	0
0	0	46	0	0	0
0	0	0	46	0	0
0	0	0	0	46	0
0	0	0	0	0	46

72	71	0	0	0	0
0	1	0	0	0	0
0	0	0	12	1	0
0	0	0	1	0	0
0	0	1	61	0	0
0	0	0	71	0	72

72	0	0	0	0	0
0	72	0	0	0	0
0	0	0	72	61	35
0	0	72	0	72	0
0	0	0	0	12	38
0	0	0	0	2	61

G:=sub<GL(6,GF(73))| [64,0,0,0,0,0,0,64,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[72,1,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,72,0,0,0,0,0,32,27,41,19,0,0,63,63,23,32],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,46,0,0,0,0,0,0,46,0,0,0,0,0,0,46,0,0,0,0,0,0,46],[72,0,0,0,0,0,71,1,0,0,0,0,0,0,0,0,1,0,0,0,12,1,61,71,0,0,1,0,0,0,0,0,0,0,0,72],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,72,0,0,0,0,72,0,0,0,0,0,61,72,12,2,0,0,35,0,38,61] >;

C₃×C₄²⋊C₂² in GAP, Magma, Sage, TeX

C_3\times C_4^2\rtimes C_2^2

% in TeX

G:=Group("C3xC4^2:C2^2");

// GroupNames label

G:=SmallGroup(192,854);

// by ID

G=gap.SmallGroup(192,854);

# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-2,336,365,1059,4204,2111,172,124]);

// Polycyclic

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

// generators/relations

G = C3×C42⋊C22 order 192 = 26·3

Direct product of C3 and C42⋊C22

G = C₃×C₄²⋊C₂² order 192 = 2⁶·3

Direct product of C₃ and C₄²⋊C₂²