C2xC2^2:Q16 - GroupNames

G = C₂×C₂²⋊Q₁₆ order 128 = 2⁷

Direct product of C₂ and C₂²⋊Q₁₆

direct product, p-group, metabelian, nilpotent (class 3), monomial

Aliases: C₂×C₂²⋊Q₁₆, C₂³⋊₅Q₁₆, C₂⁴.₁₇₆D₄, C₄⋊C₄.₂C₂³, Q₈.₃₇(C₂×D₄), C₂²⋊₃(C₂×Q₁₆), C₄.₃₆C₂²≀C₂, (C₂×Q₈).₂₂₇D₄, (C₂²×Q₁₆)⋊₅C₂, C₄.₃₆(C₂²×D₄), C₂.₄(C₂²×Q₁₆), (C₂×C₄).₂₁₈C₂⁴, (C₂×C₈).₁₂₆C₂³, (C₂×Q₁₆)⋊₃₆C₂², (C₂²×C₄).₄₁₈D₄, C₂³.₈₄₉(C₂×D₄), (C₂×Q₈).₁₆C₂³, (Q₈×C₂³).₁₂C₂, Q₈⋊C₄⋊₆₃C₂², C₂².₁₁₅C₂²≀C₂, C₂²⋊C₈.₁₇₀C₂², (C₂²×C₄).₉₅₆C₂³, (C₂³×C₄).₅₃₈C₂², (C₂²×C₈).₁₃₃C₂², C₂².₄₇₈(C₂²×D₄), C₂²⋊Q₈.₁₄₉C₂², C₂².₉₉(C₈.C₂²), (C₂²×Q₈).₄₆₄C₂², C₂.₈(C₂×C₈.C₂²), (C₂×Q₈⋊C₄)⋊₂₁C₂, C₂.₃₆(C₂×C₂²≀C₂), (C₂×C₄).₁₀₉₂(C₂×D₄), (C₂×C₂²⋊C₈).₃₀C₂, (C₂×C₂²⋊Q₈).₅₁C₂, (C₂×C₄⋊C₄).₅₇₉C₂², SmallGroup(128,1731)

Series: Derived ►Chief ►Lower central ►Upper central ►Jennings

Derived series

C₁ — C₂×C₄ — C₂×C₂²⋊Q₁₆

Chief series

C₁ — C₂ — C₂² — C₂×C₄ — C₂²×C₄ — C₂³×C₄ — Q₈×C₂³ — C₂×C₂²⋊Q₁₆

Lower central

C₁ — C₂ — C₂×C₄ — C₂×C₂²⋊Q₁₆

Upper central

C₁ — C₂³ — C₂³×C₄ — C₂×C₂²⋊Q₁₆

Jennings

C₁ — C₂ — C₂ — C₂×C₄ — C₂×C₂²⋊Q₁₆

Generators and relations for C₂×C₂²⋊Q₁₆
G = < a,b,c,d,e | a²=b²=c²=d⁸=1, e²=d⁴, ab=ba, ac=ca, ad=da, ae=ea, dbd^-1=bc=cb, be=eb, cd=dc, ce=ec, ede^-1=d^-1 >

Subgroups: 636 in 370 conjugacy classes, 124 normal (18 characteristic)
C₁, C₂, C₂, C₂, C₄, C₄, C₂², C₂², C₂², C₈, C₂×C₄, C₂×C₄, C₂×C₄, Q₈, Q₈, C₂³, C₂³, C₂³, C₂²⋊C₄, C₄⋊C₄, C₄⋊C₄, C₂×C₈, C₂×C₈, Q₁₆, C₂²×C₄, C₂²×C₄, C₂²×C₄, C₂×Q₈, C₂×Q₈, C₂⁴, C₂²⋊C₈, Q₈⋊C₄, C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₂×C₄⋊C₄, C₂²⋊Q₈, C₂²⋊Q₈, C₂²×C₈, C₂×Q₁₆, C₂×Q₁₆, C₂³×C₄, C₂³×C₄, C₂²×Q₈, C₂²×Q₈, C₂²×Q₈, C₂×C₂²⋊C₈, C₂×Q₈⋊C₄, C₂²⋊Q₁₆, C₂×C₂²⋊Q₈, C₂²×Q₁₆, Q₈×C₂³, C₂×C₂²⋊Q₁₆
Quotients: C₁, C₂, C₂², D₄, C₂³, Q₁₆, C₂×D₄, C₂⁴, C₂²≀C₂, C₂×Q₁₆, C₈.C₂², C₂²×D₄, C₂²⋊Q₁₆, C₂×C₂²≀C₂, C₂²×Q₁₆, C₂×C₈.C₂², C₂×C₂²⋊Q₁₆

Smallest permutation representation of C₂×C₂²⋊Q₁₆
►On 64 points

Generators in S₆₄

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

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

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

(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)

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

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

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

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

38 conjugacy classes

class	1	2A	···	2G	2H	2I	2J	2K	4A	4B	4C	4D	4E	···	4N	4O	4P	4Q	4R	8A	···	8H
order	1	2	···	2	2	2	2	2	4	4	4	4	4	···	4	4	4	4	4	8	···	8
size	1	1	···	1	2	2	2	2	2	2	2	2	4	···	4	8	8	8	8	4	···	4

38 irreducible representations

dim	1	1	1	1	1	1	1	2	2	2	2	4
type	+	+	+	+	+	+	+	+	+	+	-	-
image	C₁	C₂	C₂	C₂	C₂	C₂	C₂	D₄	D₄	D₄	Q₁₆	C₈.C₂²
kernel	C₂×C₂²⋊Q₁₆	C₂×C₂²⋊C₈	C₂×Q₈⋊C₄	C₂²⋊Q₁₆	C₂×C₂²⋊Q₈	C₂²×Q₁₆	Q₈×C₂³	C₂²×C₄	C₂×Q₈	C₂⁴	C₂³	C₂²
# reps	1	1	2	8	1	2	1	3	8	1	8	2

Matrix representation of C₂×C₂²⋊Q₁₆ ►in GL₅(𝔽₁₇)

16	0	0	0	0
0	16	0	0	0
0	0	16	0	0
0	0	0	16	0
0	0	0	0	16

16	0	0	0	0
0	16	0	0	0
0	0	16	0	0
0	0	0	1	0
0	0	0	16	16

1	0	0	0	0
0	1	0	0	0
0	0	1	0	0
0	0	0	16	0
0	0	0	0	16

1	0	0	0	0
0	2	0	0	0
0	0	9	0	0
0	0	0	16	15
0	0	0	0	1

1	0	0	0	0
0	0	1	0	0
0	16	0	0	0
0	0	0	1	0
0	0	0	0	1

G:=sub<GL(5,GF(17))| [16,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,16],[16,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,1,16,0,0,0,0,16],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,16,0,0,0,0,0,16],[1,0,0,0,0,0,2,0,0,0,0,0,9,0,0,0,0,0,16,0,0,0,0,15,1],[1,0,0,0,0,0,0,16,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1] >;

C₂×C₂²⋊Q₁₆ in GAP, Magma, Sage, TeX

C_2\times C_2^2\rtimes Q_{16}

% in TeX

G:=Group("C2xC2^2:Q16");

// GroupNames label

G:=SmallGroup(128,1731);

// by ID

G=gap.SmallGroup(128,1731);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,456,758,2804,1411,172]);

// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^8=1,e^2=d^4,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,d*b*d^-1=b*c=c*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^-1>;

// generators/relations

G = C2×C22⋊Q16 order 128 = 27

Direct product of C2 and C22⋊Q16

G = C₂×C₂²⋊Q₁₆ order 128 = 2⁷

Direct product of C₂ and C₂²⋊Q₁₆