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 19)(4 21)(6 23)(8 25)(10 27)(12 29)(14 31)(16 17)(33 41)(34 54)(35 43)(36 56)(37 45)(38 58)(39 47)(40 60)(42 62)(44 64)(46 50)(48 52)(49 57)(51 59)(53 61)(55 63)

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

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

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

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

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

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