C3xC2^2.SD16 - GroupNames

G = C₃×C₂².SD₁₆ order 192 = 2⁶·3

Direct product of C₃ and C₂².SD₁₆

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

Aliases: C₃×C₂².SD₁₆, C₄⋊C₄⋊₁C₁₂, (C₆×D₄)⋊₃C₄, C₆.₂₂C₄≀C₂, (C₂×D₄)⋊₁C₁₂, C₂²⋊C₈⋊₂C₆, (C₂×C₆).₂₄D₈, C₄⋊D₄.₁C₆, C₂².₂(C₃×D₈), (C₂×C₁₂).₄₄₃D₄, (C₂×C₆).₃₅SD₁₆, C₂³.₃₄(C₃×D₄), C₆.₂₉(C₂³⋊C₄), C₂.C₄²⋊₉C₆, (C₂²×C₆).₁₄₉D₄, C₆.₃₃(D₄⋊C₄), C₂².₇(C₃×SD₁₆), (C₂²×C₁₂).₃₈₄C₂², (C₃×C₄⋊C₄)⋊₃C₄, C₂.₄(C₃×C₄≀C₂), (C₃×C₂²⋊C₈)⋊₄C₂, (C₂×C₄).₇(C₂×C₁₂), (C₂×C₄).₉₅(C₃×D₄), C₂.₄(C₃×C₂³⋊C₄), C₂.₃(C₃×D₄⋊C₄), (C₂×C₁₂).₁₇₄(C₂×C₄), (C₃×C₄⋊D₄).₁₁C₂, (C₂²×C₄).₁₉(C₂×C₆), C₂².₃₅(C₃×C₂²⋊C₄), (C₂×C₆).₁₂₂(C₂²⋊C₄), (C₃×C₂.C₄²)⋊₂₂C₂, SmallGroup(192,133)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂×C₄ — C₃×C₂².SD₁₆

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²×C₄ — C₂²×C₁₂ — C₃×C₂.C₄² — C₃×C₂².SD₁₆

Lower central

C₁ — C₂² — C₂×C₄ — C₃×C₂².SD₁₆

Upper central

C₁ — C₂×C₆ — C₂²×C₁₂ — C₃×C₂².SD₁₆

Generators and relations for C₃×C₂².SD₁₆
G = < a,b,c,d,e | a³=b²=c²=d⁸=e²=1, ab=ba, ac=ca, ad=da, ae=ea, dbd^-1=ebe=bc=cb, cd=dc, ce=ec, ede=bcd³ >

Subgroups: 210 in 90 conjugacy classes, 34 normal (all characteristic)
C₁, C₂, C₂, C₃, C₄, C₂², C₂², C₆, C₆, C₈, C₂×C₄, C₂×C₄, D₄, C₂³, C₂³, C₁₂, C₂×C₆, C₂×C₆, C₂²⋊C₄, C₄⋊C₄, C₂×C₈, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×D₄, C₂₄, C₂×C₁₂, C₂×C₁₂, C₃×D₄, C₂²×C₆, C₂²×C₆, C₂.C₄², C₂²⋊C₈, C₄⋊D₄, C₃×C₂²⋊C₄, C₃×C₄⋊C₄, C₂×C₂₄, C₂²×C₁₂, C₂²×C₁₂, C₆×D₄, C₆×D₄, C₂².SD₁₆, C₃×C₂.C₄², C₃×C₂²⋊C₈, C₃×C₄⋊D₄, C₃×C₂².SD₁₆
Quotients: C₁, C₂, C₃, C₄, C₂², C₆, C₂×C₄, D₄, C₁₂, C₂×C₆, C₂²⋊C₄, D₈, SD₁₆, C₂×C₁₂, C₃×D₄, C₂³⋊C₄, D₄⋊C₄, C₄≀C₂, C₃×C₂²⋊C₄, C₃×D₈, C₃×SD₁₆, C₂².SD₁₆, C₃×C₂³⋊C₄, C₃×D₄⋊C₄, C₃×C₄≀C₂, C₃×C₂².SD₁₆

Smallest permutation representation of C₃×C₂².SD₁₆
►On 48 points

Generators in S₄₈

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

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

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

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

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

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

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

57 conjugacy classes

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

57 irreducible representations

dim	1	1	1	1	1	1	1	1	1	1	1	1	2	2	2	2	2	2	2	2	2	2	4	4
type	+	+	+	+									+	+	+								+
image	C₁	C₂	C₂	C₂	C₃	C₄	C₄	C₆	C₆	C₆	C₁₂	C₁₂	D₄	D₄	D₈	SD₁₆	C₃×D₄	C₃×D₄	C₄≀C₂	C₃×D₈	C₃×SD₁₆	C₃×C₄≀C₂	C₂³⋊C₄	C₃×C₂³⋊C₄
kernel	C₃×C₂².SD₁₆	C₃×C₂.C₄²	C₃×C₂²⋊C₈	C₃×C₄⋊D₄	C₂².SD₁₆	C₃×C₄⋊C₄	C₆×D₄	C₂.C₄²	C₂²⋊C₈	C₄⋊D₄	C₄⋊C₄	C₂×D₄	C₂×C₁₂	C₂²×C₆	C₂×C₆	C₂×C₆	C₂×C₄	C₂³	C₆	C₂²	C₂²	C₂	C₆	C₂
# reps	1	1	1	1	2	2	2	2	2	2	4	4	1	1	2	2	2	2	4	4	4	8	1	2

Matrix representation of C₃×C₂².SD₁₆ ►in GL₄(𝔽₇₃) generated by

64	0	0	0
0	64	0	0
0	0	64	0
0	0	0	64

72	0	0	0
0	72	0	0
0	0	72	0
0	0	44	1

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

57	16	0	0
57	57	0	0
0	0	63	46
0	0	0	10

57	57	0	0
57	16	0	0
0	0	51	72
0	0	45	22

G:=sub<GL(4,GF(73))| [64,0,0,0,0,64,0,0,0,0,64,0,0,0,0,64],[72,0,0,0,0,72,0,0,0,0,72,44,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,72,0,0,0,0,72],[57,57,0,0,16,57,0,0,0,0,63,0,0,0,46,10],[57,57,0,0,57,16,0,0,0,0,51,45,0,0,72,22] >;

C₃×C₂².SD₁₆ in GAP, Magma, Sage, TeX

C_3\times C_2^2.{\rm SD}_{16}

% in TeX

G:=Group("C3xC2^2.SD16");

// GroupNames label

G:=SmallGroup(192,133);

// by ID

G=gap.SmallGroup(192,133);

# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-2,168,197,1683,1522,248,2951]);

// Polycyclic

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

// generators/relations

G = C3×C22.SD16 order 192 = 26·3

Direct product of C3 and C22.SD16

G = C₃×C₂².SD₁₆ order 192 = 2⁶·3

Direct product of C₃ and C₂².SD₁₆