C2^3.36C4^2 - GroupNames

G = C₂³.₃₆C₄² order 128 = 2⁷

18^th non-split extension by C₂³ of C₄² acting via C₄²/C₂×C₄=C₂

p-group, metabelian, nilpotent (class 2), monomial

Aliases: C₂³.₃₆C₄², C₂³.₂₀M₄(2), C₂²⋊C₈⋊₁₂C₄, C₈⋊₉(C₂²⋊C₄), (C₂²×C₈)⋊₁₉C₄, C₄.₁₅₉(C₄×D₄), (C₂×C₈).₃₈₁D₄, C₂.₁(C₈⋊₉D₄), C₂⁴.₉₄(C₂×C₄), (C₂×C₄).₄₆C₄², (C₂³×C₈).₂₂C₂, C₂².₈₄(C₄×D₄), C₂²⋊₂(C₈⋊C₄), (C₂×C₄²).₂C₂², C₂².₂₃(C₈○D₄), C₂.C₄².₈C₄, C₂².₅₁(C₂×C₄²), C₄.₇₁(C₄²⋊C₂), (C₂²×C₈).₃₇₈C₂², (C₂³×C₄).₆₃₁C₂², C₂³.₂₅₄(C₂²×C₄), C₂².₃₈(C₂×M₄(2)), C₂.₁₃(C₈○₂M₄(2)), C₂².₇C₄²⋊₃₇C₂, (C₂²×C₄).₁₆₀₇C₂³, C₂.₈(C₂×C₈⋊C₄), (C₂×C₈⋊C₄)⋊₁₉C₂, (C₂×C₈).₁₃₁(C₂×C₄), (C₄×C₂²⋊C₄).₂C₂, C₂.₁₀(C₄×C₂²⋊C₄), (C₂×C₄).₁₄₉₇(C₂×D₄), (C₂×C₂²⋊C₈).₄₂C₂, (C₂×C₂²⋊C₄).₂₃C₄, C₄.₁₀₇(C₂×C₂²⋊C₄), (C₂×C₄).₉₁₇(C₄○D₄), (C₂²×C₄).₁₀₇(C₂×C₄), (C₂×C₄).₅₉₇(C₂²×C₄), SmallGroup(128,484)

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

Derived series

C₁ — C₂² — C₂³.₃₆C₄²

Chief series

C₁ — C₂ — C₂² — C₂×C₄ — C₂²×C₄ — C₂²×C₈ — C₂×C₈⋊C₄ — C₂³.₃₆C₄²

Lower central

C₁ — C₂² — C₂³.₃₆C₄²

Upper central

C₁ — C₂²×C₄ — C₂³.₃₆C₄²

Jennings

C₁ — C₂ — C₂ — C₂²×C₄ — C₂³.₃₆C₄²

Generators and relations for C₂³.₃₆C₄²
G = < a,b,c,d,e | a²=b²=c²=d⁴=1, e⁴=c, ab=ba, ac=ca, dad^-1=abc, ae=ea, bc=cb, ede^-1=bd=db, be=eb, cd=dc, ce=ec >

Subgroups: 276 in 180 conjugacy classes, 92 normal (20 characteristic)
C₁, C₂, C₂, C₂, C₄, C₄, C₄, C₂², C₂², C₂², C₈, C₈, C₂×C₄, C₂×C₄, C₂×C₄, C₂³, C₂³, C₂³, C₄², C₂²⋊C₄, C₂×C₈, C₂×C₈, C₂²×C₄, C₂²×C₄, C₂²×C₄, C₂⁴, C₂.C₄², C₈⋊C₄, C₂²⋊C₈, C₂×C₄², C₂×C₂²⋊C₄, C₂²×C₈, C₂²×C₈, C₂²×C₈, C₂³×C₄, C₂².₇C₄², C₄×C₂²⋊C₄, C₂×C₈⋊C₄, C₂×C₂²⋊C₈, C₂³×C₈, C₂³.₃₆C₄²
Quotients: C₁, C₂, C₄, C₂², C₂×C₄, D₄, C₂³, C₄², C₂²⋊C₄, M₄(2), C₂²×C₄, C₂×D₄, C₄○D₄, C₈⋊C₄, C₂×C₄², C₂×C₂²⋊C₄, C₄²⋊C₂, C₄×D₄, C₂×M₄(2), C₈○D₄, C₄×C₂²⋊C₄, C₂×C₈⋊C₄, C₈○₂M₄(2), C₈⋊₉D₄, C₂³.₃₆C₄²

Smallest permutation representation of C₂³.₃₆C₄²
►On 64 points

Generators in S₆₄

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

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

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

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

(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)

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

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

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

56 conjugacy classes

class	1	2A	···	2G	2H	2I	2J	2K	4A	···	4H	4I	4J	4K	4L	4M	···	4T	8A	···	8P	8Q	···	8X
order	1	2	···	2	2	2	2	2	4	···	4	4	4	4	4	4	···	4	8	···	8	8	···	8
size	1	1	···	1	2	2	2	2	1	···	1	2	2	2	2	4	···	4	2	···	2	4	···	4

56 irreducible representations

dim	1	1	1	1	1	1	1	1	1	1	2	2	2	2
type	+	+	+	+	+	+					+
image	C₁	C₂	C₂	C₂	C₂	C₂	C₄	C₄	C₄	C₄	D₄	C₄○D₄	M₄(2)	C₈○D₄
kernel	C₂³.₃₆C₄²	C₂².₇C₄²	C₄×C₂²⋊C₄	C₂×C₈⋊C₄	C₂×C₂²⋊C₈	C₂³×C₈	C₂.C₄²	C₂²⋊C₈	C₂×C₂²⋊C₄	C₂²×C₈	C₂×C₈	C₂×C₄	C₂³	C₂²
# reps	1	2	1	2	1	1	4	8	4	8	4	4	8	8

Matrix representation of C₂³.₃₆C₄² ►in GL₅(𝔽₁₇)

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

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

4	0	0	0	0
0	12	16	0	0
0	9	5	0	0
0	0	0	16	15
0	0	0	0	1

1	0	0	0	0
0	4	15	0	0
0	10	13	0	0
0	0	0	4	0
0	0	0	13	13

G:=sub<GL(5,GF(17))| [1,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,1,16,0,0,0,0,16],[1,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,16],[1,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,1,0,0,0,0,0,1],[4,0,0,0,0,0,12,9,0,0,0,16,5,0,0,0,0,0,16,0,0,0,0,15,1],[1,0,0,0,0,0,4,10,0,0,0,15,13,0,0,0,0,0,4,13,0,0,0,0,13] >;

C₂³.₃₆C₄² in GAP, Magma, Sage, TeX

C_2^3._{36}C_4^2

% in TeX

G:=Group("C2^3.36C4^2");

// GroupNames label

G:=SmallGroup(128,484);

// by ID

G=gap.SmallGroup(128,484);

# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,112,141,232,723,100,172]);

// Polycyclic

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

// generators/relations

G = C23.36C42 order 128 = 27

18th non-split extension by C23 of C42 acting via C42/C2×C4=C2

G = C₂³.₃₆C₄² order 128 = 2⁷

18^th non-split extension by C₂³ of C₄² acting via C₄²/C₂×C₄=C₂