Copied to
clipboard

G = C23.7M4(2)  order 128 = 27

3rd 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

C1C2×C4 — C23.7M4(2)
C1C2C4C2×C4C2×C8C22×C8C42.6C22 — C23.7M4(2)
C1C22C2×C4 — C23.7M4(2)
C1C2×C4C22×C8 — C23.7M4(2)
C1C2C2C2C2C4C2×C4C22×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 >

4C2
2C4
2C22
2C22
2C22
4C4
4C4
2C8
2C2×C4
2C2×C4
2C8
2C2×C4
2C2×C4
4C8
2C42
2C2×C8
2C22⋊C4
4M4(2)
4C16
4C2×C8
4C16
2C4⋊C8
2C2×C16
2C4⋊C8
2C2×C16

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 2A2B2C2D4A4B4C4D4E4F4G8A···8H8I8J16A···16P
order1222244444448···88816···16
size1111411114882···2884···4

38 irreducible representations

dim11111122244
type+++++-
imageC1C2C2C4C4C8D4M4(2)D4.C8C23⋊C4C4.10D4
kernelC23.7M4(2)C22⋊C16C42.6C22C42⋊C2C2×M4(2)C4⋊C4C2×C8C23C2C4C4
# reps121228221611

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

1000
131600
0010
00216
,
16000
01600
0010
0001
,
16000
01600
00160
00016
,
5000
11400
00125
00155
,
131500
0400
00161
0001
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

׿
×
𝔽