C2^3:2Q16 - GroupNames

G = C₂³⋊₂Q₁₆ order 128 = 2⁷

1^st semidirect product of C₂³ and Q₁₆ acting via Q₁₆/C₄=C₂²

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

Aliases: C₂³⋊₂Q₁₆, C₂⁴.₈₁D₄, C₄⋊C₄.₇₈D₄, (C₂×C₈).₄₀D₄, (C₂×Q₈).₇₇D₄, C₄.₆₂C₂²≀C₂, (C₂²×Q₁₆)⋊₁C₂, C₄.₁₅(C₄⋊D₄), C₄.₂₈(C₄⋊₁D₄), C₂³.₈₉₈(C₂×D₄), (C₂²×C₄).₁₃₉D₄, C₂².₅₂(C₂×Q₁₆), C₂².₄Q₁₆⋊₃₇C₂, C₂.₁₂(C₈.D₄), C₂.₆(C₂³⋊₂D₄), C₂².₁₉₄C₂²≀C₂, C₂.₂₈(D₄.₇D₄), C₂.₁₂(C₈.₁₈D₄), C₂².₁₀₂(C₄○D₈), (C₂²×C₈).₁₀₆C₂², (C₂³×C₄).₂₆₉C₂², C₂.₁₈(C₂²⋊Q₁₆), (C₂²×Q₈).₄₇C₂², C₂².₂₁₉(C₄⋊D₄), (C₂²×C₄).₁₄₃₂C₂³, C₂².₁₁₇(C₈.C₂²), (C₂×Q₈⋊C₄)⋊₁₂C₂, (C₂×C₄).₁₀₂₂(C₂×D₄), (C₂×C₂²⋊C₈).₂₆C₂, (C₂×C₂²⋊Q₈).₁₂C₂, (C₂×C₄).₆₁₅(C₄○D₄), (C₂×C₄⋊C₄).₁₀₃C₂², SmallGroup(128,733)

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

Derived series

C₁ — C₂²×C₄ — C₂³⋊₂Q₁₆

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²×C₄ — C₂³×C₄ — C₂×C₂²⋊Q₈ — C₂³⋊₂Q₁₆

Lower central

C₁ — C₂ — C₂²×C₄ — C₂³⋊₂Q₁₆

Upper central

C₁ — C₂³ — C₂³×C₄ — C₂³⋊₂Q₁₆

Jennings

C₁ — C₂ — C₂ — C₂²×C₄ — C₂³⋊₂Q₁₆

Generators and relations for C₂³⋊₂Q₁₆
G = < a,b,c,d,e | a²=b²=c²=d⁸=1, e²=d⁴, ab=ba, eae^-1=ac=ca, dad^-1=abc, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede^-1=d^-1 >

Subgroups: 400 in 196 conjugacy classes, 56 normal (22 characteristic)
C₁, C₂, C₂, C₂, C₄, C₄, C₄, C₂², C₂², C₂², C₈, C₂×C₄, C₂×C₄, C₂×C₄, Q₈, C₂³, C₂³, C₂³, C₂²⋊C₄, C₄⋊C₄, C₄⋊C₄, C₂×C₈, C₂×C₈, Q₁₆, 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×C₈, C₂×Q₁₆, C₂³×C₄, C₂²×Q₈, C₂².₄Q₁₆, C₂×C₂²⋊C₈, C₂×Q₈⋊C₄, C₂×C₂²⋊Q₈, C₂²×Q₁₆, C₂³⋊₂Q₁₆
Quotients: C₁, C₂, C₂², D₄, C₂³, Q₁₆, C₂×D₄, C₄○D₄, C₂²≀C₂, C₄⋊D₄, C₄⋊₁D₄, C₂×Q₁₆, C₄○D₈, C₈.C₂², C₂³⋊₂D₄, C₂²⋊Q₁₆, D₄.₇D₄, C₈.₁₈D₄, C₈.D₄, C₂³⋊₂Q₁₆

Smallest permutation representation of C₂³⋊₂Q₁₆
►On 64 points

Generators in S₆₄

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

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

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

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

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

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

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

32 conjugacy classes

class	1	2A	···	2G	2H	2I	4A	4B	4C	4D	4E	4F	4G	···	4N	8A	···	8H
order	1	2	···	2	2	2	4	4	4	4	4	4	4	···	4	8	···	8
size	1	1	···	1	4	4	2	2	2	2	4	4	8	···	8	4	···	4

32 irreducible representations

dim	1	1	1	1	1	1	2	2	2	2	2	2	2	2	4
type	+	+	+	+	+	+	+	+	+	+	+		-		-
image	C₁	C₂	C₂	C₂	C₂	C₂	D₄	D₄	D₄	D₄	D₄	C₄○D₄	Q₁₆	C₄○D₈	C₈.C₂²
kernel	C₂³⋊₂Q₁₆	C₂².₄Q₁₆	C₂×C₂²⋊C₈	C₂×Q₈⋊C₄	C₂×C₂²⋊Q₈	C₂²×Q₁₆	C₄⋊C₄	C₂×C₈	C₂²×C₄	C₂×Q₈	C₂⁴	C₂×C₄	C₂³	C₂²	C₂²
# reps	1	1	1	2	2	1	4	2	1	4	1	2	4	4	2

Matrix representation of C₂³⋊₂Q₁₆ ►in GL₆(𝔽₁₇)

16	0	0	0	0	0
0	16	0	0	0	0
0	0	1	0	0	0
0	0	8	16	0	0
0	0	0	0	16	0
0	0	0	0	14	1

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

1	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	16	0
0	0	0	0	0	16

14	3	0	0	0	0
14	14	0	0	0	0
0	0	9	2	0	0
0	0	11	8	0	0
0	0	0	0	8	0
0	0	0	0	15	15

7	16	0	0	0	0
16	10	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	0	0	0	3	15
0	0	0	0	5	14

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,8,0,0,0,0,0,16,0,0,0,0,0,0,16,14,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[14,14,0,0,0,0,3,14,0,0,0,0,0,0,9,11,0,0,0,0,2,8,0,0,0,0,0,0,8,15,0,0,0,0,0,15],[7,16,0,0,0,0,16,10,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,3,5,0,0,0,0,15,14] >;

C₂³⋊₂Q₁₆ in GAP, Magma, Sage, TeX

C_2^3\rtimes_2Q_{16}

% in TeX

G:=Group("C2^3:2Q16");

// GroupNames label

G:=SmallGroup(128,733);

// by ID

G=gap.SmallGroup(128,733);

# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,448,141,456,422,387,2019,1018,248]);

// 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,e*a*e^-1=a*c=c*a,d*a*d^-1=a*b*c,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^-1>;

// generators/relations

G = C23⋊2Q16 order 128 = 27

1st semidirect product of C23 and Q16 acting via Q16/C4=C22

G = C₂³⋊₂Q₁₆ order 128 = 2⁷

1^st semidirect product of C₂³ and Q₁₆ acting via Q₁₆/C₄=C₂²