C2^3.7M4(2) - GroupNames

G = C₂³.₇M₄(2) order 128 = 2⁷

3^rd non-split extension by C₂³ of M₄(2) acting via M₄(2)/C₄=C₂²

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

Aliases: C₂³.₇M₄(2), C₄⋊C₄.₁C₈, (C₂×C₈).₂₉₃D₄, C₂²⋊C₁₆.₂C₂, C₂.₅(D₄.C₈), C₄.₄₀(C₂³⋊C₄), C₄²⋊C₂.₁C₄, (C₂×M₄(2)).₇C₄, (C₂²×C₈).₄C₂², C₄.₂₁(C₄.₁₀D₄), C₂².₄₉(C₂²⋊C₈), C₄².₆C₂².₈C₂, C₂.₄(C₂².M₄(2)), (C₂×C₄).₁₁(C₂×C₈), (C₂²×C₄).₁₆₅(C₂×C₄), (C₂×C₄).₃₇₉(C₂²⋊C₄), SmallGroup(128,55)

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

Derived series

C₁ — C₂×C₄ — C₂³.₇M₄(2)

Chief series

C₁ — C₂ — C₄ — C₂×C₄ — C₂×C₈ — C₂²×C₈ — C₄².₆C₂² — C₂³.₇M₄(2)

Lower central

C₁ — C₂² — C₂×C₄ — C₂³.₇M₄(2)

Upper central

C₁ — C₂×C₄ — C₂²×C₈ — C₂³.₇M₄(2)

Jennings

C₁ — C₂ — C₂ — C₂ — C₂ — C₄ — C₂×C₄ — C₂²×C₈ — C₂³.₇M₄(2)

Generators and relations for C₂³.₇M₄(2)
G = < a,b,c,d,e | a²=b²=c²=1, d⁸=c, e²=b, ab=ba, eae^-1=ac=ca, dad^-1=abc, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede^-1=abcd⁵ >

C₁

C₂

₄C₂

C₂²

C₄

₂C₄

₂C₂²

₄C₄

C₂×C₄

C₂³

C₂×C₄

₂C₈

₂C₂×C₄

₂C₈

₂C₂×C₄

₄C₈

C₄⋊C₄

C₂×C₈

C₂²×C₄

C₂×C₈

C₄⋊C₄

₂C₄²

₂C₂×C₈

₂C₂²⋊C₄

₄M₄(2)

₄C₁₆

₄C₂×C₈

₄C₁₆

C₂²×C₈

C₂×M₄(2)

C₄²⋊C₂

₂C₄⋊C₈

₂C₂×C₁₆

₂C₄⋊C₈

₂C₂×C₁₆

C₂²⋊C₁₆

C₄².₆C₂²

C₂³.₇M₄(2)

Smallest permutation representation of C₂³.₇M₄(2)
►On 64 points

Generators in S₆₄

(2 52)(4 54)(6 56)(8 58)(10 60)(12 62)(14 64)(16 50)(17 45)(18 26)(19 47)(20 28)(21 33)(22 30)(23 35)(24 32)(25 37)(27 39)(29 41)(31 43)(34 42)(36 44)(38 46)(40 48)

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

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

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

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

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

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

38 conjugacy classes

class	1	2A	2B	2C	2D	4A	4B	4C	4D	4E	4F	4G	8A	···	8H	8I	8J	16A	···	16P
order	1	2	2	2	2	4	4	4	4	4	4	4	8	···	8	8	8	16	···	16
size	1	1	1	1	4	1	1	1	1	4	8	8	2	···	2	8	8	4	···	4

38 irreducible representations

dim	1	1	1	1	1	1	2	2	2	4	4
type	+	+	+				+			+	-
image	C₁	C₂	C₂	C₄	C₄	C₈	D₄	M₄(2)	D₄.C₈	C₂³⋊C₄	C₄.₁₀D₄
kernel	C₂³.₇M₄(2)	C₂²⋊C₁₆	C₄².₆C₂²	C₄²⋊C₂	C₂×M₄(2)	C₄⋊C₄	C₂×C₈	C₂³	C₂	C₄	C₄
# reps	1	2	1	2	2	8	2	2	16	1	1

Matrix representation of C₂³.₇M₄(2) ►in GL₄(𝔽₁₇) generated by

1	0	0	0
13	16	0	0
0	0	1	0
0	0	2	16

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

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

5	0	0	0
1	14	0	0
0	0	12	5
0	0	15	5

13	15	0	0
0	4	0	0
0	0	16	1
0	0	0	1

G:=sub<GL(4,GF(17))| [1,13,0,0,0,16,0,0,0,0,1,2,0,0,0,16],[16,0,0,0,0,16,0,0,0,0,1,0,0,0,0,1],[16,0,0,0,0,16,0,0,0,0,16,0,0,0,0,16],[5,1,0,0,0,14,0,0,0,0,12,15,0,0,5,5],[13,0,0,0,15,4,0,0,0,0,16,0,0,0,1,1] >;

C₂³.₇M₄(2) in GAP, Magma, Sage, TeX

C_2^3._7M_4(2)

% in TeX

G:=Group("C2^3.7M4(2)");

// GroupNames label

G:=SmallGroup(128,55);

// by ID

G=gap.SmallGroup(128,55);

# by ID

G:=PCGroup([7,-2,2,-2,2,-2,2,-2,56,85,120,422,723,352,1242,136,124]);

// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^2=1,d^8=c,e^2=b,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=a*b*c*d^5>;

// generators/relations

Export

Subgroup lattice of C₂³.₇M₄(2) in TeX

G = C23.7M4(2) order 128 = 27

3rd non-split extension by C23 of M4(2) acting via M4(2)/C4=C22

G = C₂³.₇M₄(2) order 128 = 2⁷

3^rd non-split extension by C₂³ of M₄(2) acting via M₄(2)/C₄=C₂²