C2^2xC17:C4 - GroupNames

G = C₂²×C₁₇⋊C₄ order 272 = 2⁴·17

Direct product of C₂² and C₁₇⋊C₄

direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary

Aliases: C₂²×C₁₇⋊C₄, D₃₄⋊₃C₄, D₁₇.C₂³, D₃₄.₇C₂², C₃₄⋊(C₂×C₄), D₁₇⋊(C₂×C₄), C₁₇⋊(C₂²×C₄), (C₂×C₃₄)⋊₂C₄, (C₂²×D₁₇).₃C₂, SmallGroup(272,52)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₁₇ — C₂²×C₁₇⋊C₄

Chief series

C₁ — C₁₇ — D₁₇ — C₁₇⋊C₄ — C₂×C₁₇⋊C₄ — C₂²×C₁₇⋊C₄

Lower central

C₁₇ — C₂²×C₁₇⋊C₄

Upper central

C₁ — C₂²

Generators and relations for C₂²×C₁₇⋊C₄
G = < a,b,c,d | a²=b²=c¹⁷=d⁴=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd^-1=c⁴ >

Subgroups: 406 in 54 conjugacy classes, 32 normal (7 characteristic)
C₁, C₂, C₂, C₄, C₂², C₂², C₂×C₄, C₂³, C₂²×C₄, C₁₇, D₁₇, D₁₇, C₃₄, C₁₇⋊C₄, D₃₄, C₂×C₃₄, C₂×C₁₇⋊C₄, C₂²×D₁₇, C₂²×C₁₇⋊C₄
Quotients: C₁, C₂, C₄, C₂², C₂×C₄, C₂³, C₂²×C₄, C₁₇⋊C₄, C₂×C₁₇⋊C₄, C₂²×C₁₇⋊C₄

Smallest permutation representation of C₂²×C₁₇⋊C₄
►On 68 points

Generators in S₆₈

(1 52)(2 53)(3 54)(4 55)(5 56)(6 57)(7 58)(8 59)(9 60)(10 61)(11 62)(12 63)(13 64)(14 65)(15 66)(16 67)(17 68)(18 35)(19 36)(20 37)(21 38)(22 39)(23 40)(24 41)(25 42)(26 43)(27 44)(28 45)(29 46)(30 47)(31 48)(32 49)(33 50)(34 51)

(1 18)(2 19)(3 20)(4 21)(5 22)(6 23)(7 24)(8 25)(9 26)(10 27)(11 28)(12 29)(13 30)(14 31)(15 32)(16 33)(17 34)(35 52)(36 53)(37 54)(38 55)(39 56)(40 57)(41 58)(42 59)(43 60)(44 61)(45 62)(46 63)(47 64)(48 65)(49 66)(50 67)(51 68)

(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 49 50 51)(52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68)

(1 18)(2 31 17 22)(3 27 16 26)(4 23 15 30)(5 19 14 34)(6 32 13 21)(7 28 12 25)(8 24 11 29)(9 20 10 33)(35 52)(36 65 51 56)(37 61 50 60)(38 57 49 64)(39 53 48 68)(40 66 47 55)(41 62 46 59)(42 58 45 63)(43 54 44 67)

G:=sub<Sym(68)| (1,52)(2,53)(3,54)(4,55)(5,56)(6,57)(7,58)(8,59)(9,60)(10,61)(11,62)(12,63)(13,64)(14,65)(15,66)(16,67)(17,68)(18,35)(19,36)(20,37)(21,38)(22,39)(23,40)(24,41)(25,42)(26,43)(27,44)(28,45)(29,46)(30,47)(31,48)(32,49)(33,50)(34,51), (1,18)(2,19)(3,20)(4,21)(5,22)(6,23)(7,24)(8,25)(9,26)(10,27)(11,28)(12,29)(13,30)(14,31)(15,32)(16,33)(17,34)(35,52)(36,53)(37,54)(38,55)(39,56)(40,57)(41,58)(42,59)(43,60)(44,61)(45,62)(46,63)(47,64)(48,65)(49,66)(50,67)(51,68), (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)(35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51)(52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68), (1,18)(2,31,17,22)(3,27,16,26)(4,23,15,30)(5,19,14,34)(6,32,13,21)(7,28,12,25)(8,24,11,29)(9,20,10,33)(35,52)(36,65,51,56)(37,61,50,60)(38,57,49,64)(39,53,48,68)(40,66,47,55)(41,62,46,59)(42,58,45,63)(43,54,44,67)>;

G:=Group( (1,52)(2,53)(3,54)(4,55)(5,56)(6,57)(7,58)(8,59)(9,60)(10,61)(11,62)(12,63)(13,64)(14,65)(15,66)(16,67)(17,68)(18,35)(19,36)(20,37)(21,38)(22,39)(23,40)(24,41)(25,42)(26,43)(27,44)(28,45)(29,46)(30,47)(31,48)(32,49)(33,50)(34,51), (1,18)(2,19)(3,20)(4,21)(5,22)(6,23)(7,24)(8,25)(9,26)(10,27)(11,28)(12,29)(13,30)(14,31)(15,32)(16,33)(17,34)(35,52)(36,53)(37,54)(38,55)(39,56)(40,57)(41,58)(42,59)(43,60)(44,61)(45,62)(46,63)(47,64)(48,65)(49,66)(50,67)(51,68), (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)(35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51)(52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68), (1,18)(2,31,17,22)(3,27,16,26)(4,23,15,30)(5,19,14,34)(6,32,13,21)(7,28,12,25)(8,24,11,29)(9,20,10,33)(35,52)(36,65,51,56)(37,61,50,60)(38,57,49,64)(39,53,48,68)(40,66,47,55)(41,62,46,59)(42,58,45,63)(43,54,44,67) );

G=PermutationGroup([[(1,52),(2,53),(3,54),(4,55),(5,56),(6,57),(7,58),(8,59),(9,60),(10,61),(11,62),(12,63),(13,64),(14,65),(15,66),(16,67),(17,68),(18,35),(19,36),(20,37),(21,38),(22,39),(23,40),(24,41),(25,42),(26,43),(27,44),(28,45),(29,46),(30,47),(31,48),(32,49),(33,50),(34,51)], [(1,18),(2,19),(3,20),(4,21),(5,22),(6,23),(7,24),(8,25),(9,26),(10,27),(11,28),(12,29),(13,30),(14,31),(15,32),(16,33),(17,34),(35,52),(36,53),(37,54),(38,55),(39,56),(40,57),(41,58),(42,59),(43,60),(44,61),(45,62),(46,63),(47,64),(48,65),(49,66),(50,67),(51,68)], [(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),(35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51),(52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68)], [(1,18),(2,31,17,22),(3,27,16,26),(4,23,15,30),(5,19,14,34),(6,32,13,21),(7,28,12,25),(8,24,11,29),(9,20,10,33),(35,52),(36,65,51,56),(37,61,50,60),(38,57,49,64),(39,53,48,68),(40,66,47,55),(41,62,46,59),(42,58,45,63),(43,54,44,67)]])

32 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	2G	4A	···	4H	17A	17B	17C	17D	34A	···	34L
order	1	2	2	2	2	2	2	2	4	···	4	17	17	17	17	34	···	34
size	1	1	1	1	17	17	17	17	17	···	17	4	4	4	4	4	···	4

32 irreducible representations

dim	1	1	1	1	1	4	4
type	+	+	+			+	+
image	C₁	C₂	C₂	C₄	C₄	C₁₇⋊C₄	C₂×C₁₇⋊C₄
kernel	C₂²×C₁₇⋊C₄	C₂×C₁₇⋊C₄	C₂²×D₁₇	D₃₄	C₂×C₃₄	C₂²	C₂
# reps	1	6	1	6	2	4	12

Matrix representation of C₂²×C₁₇⋊C₄ ►in GL₆(𝔽₁₃₇)

136	0	0	0	0	0
0	136	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

1	0	0	0	0	0
0	136	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

1	0	0	0	0	0
0	1	0	0	0	0
0	0	36	101	64	136
0	0	37	101	64	136
0	0	36	102	64	136
0	0	36	101	65	136

100	0	0	0	0	0
0	37	0	0	0	0
0	0	0	0	0	1
0	0	4	134	112	13
0	0	77	61	2	13
0	0	101	36	73	1

G:=sub<GL(6,GF(137))| [136,0,0,0,0,0,0,136,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,136,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,36,37,36,36,0,0,101,101,102,101,0,0,64,64,64,65,0,0,136,136,136,136],[100,0,0,0,0,0,0,37,0,0,0,0,0,0,0,4,77,101,0,0,0,134,61,36,0,0,0,112,2,73,0,0,1,13,13,1] >;

C₂²×C₁₇⋊C₄ in GAP, Magma, Sage, TeX

C_2^2\times C_{17}\rtimes C_4

% in TeX

G:=Group("C2^2xC17:C4");

// GroupNames label

G:=SmallGroup(272,52);

// by ID

G=gap.SmallGroup(272,52);

# by ID

G:=PCGroup([5,-2,-2,-2,-2,-17,40,5204,819]);

// Polycyclic

G:=Group<a,b,c,d|a^2=b^2=c^17=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^4>;

// generators/relations

1	0	0	0	0	0
0	1	0	0	0	0
0	0	36	101	64	136
0	0	37	101	64	136
0	0	36	102	64	136
0	0	36	101	65	136

1	0	0	0	0	0
0	1	0	0	0	0
0	0	36	101	64	136
0	0	37	101	64	136
0	0	36	102	64	136
0	0	36	101	65	136

G = C22×C17⋊C4 order 272 = 24·17

Direct product of C22 and C17⋊C4

G = C₂²×C₁₇⋊C₄ order 272 = 2⁴·17

Direct product of C₂² and C₁₇⋊C₄

1	0	0	0	0	0
0	1	0	0	0	0
0	0	36	101	64	136
0	0	37	101	64	136
0	0	36	102	64	136
0	0	36	101	65	136