C2^3.36D8 - GroupNames

G = C₂³.₃₆D₈ order 128 = 2⁷

7^th non-split extension by C₂³ of D₈ acting via D₈/D₄=C₂

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

Aliases: C₂³.₃₆D₈, C₂⁴.₁₅₆D₄, C₂³.₃₃SD₁₆, C₂².₃₃(C₂×D₈), C₂².₄Q₁₆⋊₅C₂, C₂³.₇₄₇(C₂×D₄), (C₂²×C₄).₂₇₇D₄, (C₂²×C₈).₁₆C₂², C₂².₄₉(C₂×SD₁₆), C₄.₁₈(C₄²⋊C₂), C₂³.₇Q₈.₉C₂, (C₂³×C₄).₂₄₃C₂², C₂.₁(C₂².D₈), C₂².₂₈(D₄⋊C₄), C₂³.₁₉₉(C₂²⋊C₄), (C₂²×C₄).₁₃₃₄C₂³, C₂.₁(C₂³.₄₇D₄), C₂².₅₂(C₈.C₂²), C₂.₁₁(C₂³.₃₄D₄), C₂.₁₉(C₂³.₃₈D₄), C₄.₁₀₃(C₂².D₄), C₂².₇₈(C₂².D₄), (C₂×C₄⋊C₄)⋊₂₈C₄, C₄⋊C₄.₁₉₄(C₂×C₄), (C₂×C₄).₁₃₂₄(C₂×D₄), C₂.₁₉(C₂×D₄⋊C₄), (C₂²×C₄⋊C₄).₁₃C₂, (C₂×C₂²⋊C₈).₁₇C₂, (C₂×C₄⋊C₄).₄₁C₂², (C₂×C₄).₇₄₀(C₄○D₄), (C₂×C₄).₃₆₇(C₂²×C₄), (C₂²×C₄).₂₆₆(C₂×C₄), (C₂×C₄).₁₂₆(C₂²⋊C₄), C₂².₂₅₇(C₂×C₂²⋊C₄), SmallGroup(128,555)

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

Derived series

C₁ — C₂×C₄ — C₂³.₃₆D₈

Chief series

C₁ — C₂ — C₂² — C₂×C₄ — C₂²×C₄ — C₂×C₄⋊C₄ — C₂²×C₄⋊C₄ — C₂³.₃₆D₈

Lower central

C₁ — C₂ — C₂×C₄ — C₂³.₃₆D₈

Upper central

C₁ — C₂³ — C₂³×C₄ — C₂³.₃₆D₈

Jennings

C₁ — C₂ — C₂ — C₂²×C₄ — C₂³.₃₆D₈

Generators and relations for C₂³.₃₆D₈
G = < a,b,c,d,e | a²=b²=c²=d⁸=1, e²=c, dad^-1=eae^-1=ab=ba, ac=ca, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede^-1=bcd^-1 >

Subgroups: 348 in 172 conjugacy classes, 68 normal (18 characteristic)
C₁, C₂, C₂, C₂, C₄, C₄, C₂², C₂², C₂², C₈, C₂×C₄, C₂×C₄, C₂×C₄, C₂³, C₂³, C₂³, C₂²⋊C₄, C₄⋊C₄, C₄⋊C₄, C₂×C₈, C₂²×C₄, C₂²×C₄, C₂²×C₄, C₂⁴, C₂.C₄², C₂²⋊C₈, C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₂×C₄⋊C₄, C₂²×C₈, C₂³×C₄, C₂³×C₄, C₂².₄Q₁₆, C₂³.₇Q₈, C₂×C₂²⋊C₈, C₂²×C₄⋊C₄, C₂³.₃₆D₈
Quotients: C₁, C₂, C₄, C₂², C₂×C₄, D₄, C₂³, C₂²⋊C₄, D₈, SD₁₆, C₂²×C₄, C₂×D₄, C₄○D₄, D₄⋊C₄, C₂×C₂²⋊C₄, C₄²⋊C₂, C₂².D₄, C₂×D₈, C₂×SD₁₆, C₈.C₂², C₂³.₃₄D₄, C₂×D₄⋊C₄, C₂³.₃₈D₄, C₂².D₈, C₂³.₄₇D₄, C₂³.₃₆D₈

Smallest permutation representation of C₂³.₃₆D₈
►On 64 points

Generators in S₆₄

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

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

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

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

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

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

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

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	2	2	2	2	2	4
type	+	+	+	+	+		+	+		+		-
image	C₁	C₂	C₂	C₂	C₂	C₄	D₄	D₄	C₄○D₄	D₈	SD₁₆	C₈.C₂²
kernel	C₂³.₃₆D₈	C₂².₄Q₁₆	C₂³.₇Q₈	C₂×C₂²⋊C₈	C₂²×C₄⋊C₄	C₂×C₄⋊C₄	C₂²×C₄	C₂⁴	C₂×C₄	C₂³	C₂³	C₂²
# reps	1	4	1	1	1	8	3	1	8	4	4	2

Matrix representation of C₂³.₃₆D₈ ►in GL₆(𝔽₁₇)

16	0	0	0	0	0
0	16	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	10	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

16	0	0	0	0	0
0	16	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

10	10	0	0	0	0
12	0	0	0	0	0
0	0	6	6	0	0
0	0	14	0	0	0
0	0	0	0	6	9
0	0	0	0	15	11

11	2	0	0	0	0
7	6	0	0	0	0
0	0	16	15	0	0
0	0	0	1	0	0
0	0	0	0	10	15
0	0	0	0	7	7

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,10,0,0,0,0,0,16],[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],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[10,12,0,0,0,0,10,0,0,0,0,0,0,0,6,14,0,0,0,0,6,0,0,0,0,0,0,0,6,15,0,0,0,0,9,11],[11,7,0,0,0,0,2,6,0,0,0,0,0,0,16,0,0,0,0,0,15,1,0,0,0,0,0,0,10,7,0,0,0,0,15,7] >;

C₂³.₃₆D₈ in GAP, Magma, Sage, TeX

C_2^3._{36}D_8

% in TeX

G:=Group("C2^3.36D8");

// GroupNames label

G:=SmallGroup(128,555);

// by ID

G=gap.SmallGroup(128,555);

# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,422,58,2804,718,172]);

// Polycyclic

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

// generators/relations

G = C23.36D8 order 128 = 27

7th non-split extension by C23 of D8 acting via D8/D4=C2

G = C₂³.₃₆D₈ order 128 = 2⁷

7^th non-split extension by C₂³ of D₈ acting via D₈/D₄=C₂