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G = C23.27C42order 128 = 27

9th non-split extension by C23 of C42 acting via C42/C2×C4=C2

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

Aliases: C23.27C42, C42.744C23, C4⋊C8.19C4, (C4×C8).17C4, C8⋊C820C2, C4.4(C8⋊C4), C4.54(C8○D4), C22⋊C8.18C4, (C22×C8).27C4, (C2×C4).61C42, C42.293(C2×C4), (C4×C8).305C22, C4.58(C2×M4(2)), (C2×C4).71M4(2), C22.4(C8⋊C4), C22.41(C2×C42), C2.7(C82M4(2)), C42.12C4.42C2, (C2×C42).1031C22, (C2×C4×C8).8C2, C2.6(C2×C8⋊C4), (C2×C8).118(C2×C4), (C2×C4).581(C22×C4), (C22×C4).469(C2×C4), SmallGroup(128,184)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C23.27C42
C1C2C22C2×C4C42C2×C42C42.12C4 — C23.27C42
C1C22 — C23.27C42
C1C42 — C23.27C42
C1C22C22C42 — C23.27C42

Generators and relations for C23.27C42
 G = < a,b,c,d,e | a2=b2=c2=1, d4=b, e4=c, eae-1=ab=ba, ac=ca, ad=da, bc=cb, bd=db, be=eb, ede-1=cd=dc, ce=ec >

Subgroups: 124 in 98 conjugacy classes, 72 normal (12 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×8], C4 [×2], C22, C22 [×2], C22 [×2], C8 [×12], C2×C4 [×2], C2×C4 [×8], C2×C4 [×4], C23, C42 [×2], C42 [×2], C2×C8 [×12], C2×C8 [×4], C22×C4, C22×C4 [×2], C4×C8 [×8], C22⋊C8 [×4], C4⋊C8 [×4], C2×C42, C22×C8 [×2], C8⋊C8 [×4], C2×C4×C8, C42.12C4 [×2], C23.27C42
Quotients: C1, C2 [×7], C4 [×12], C22 [×7], C2×C4 [×18], C23, C42 [×4], M4(2) [×4], C22×C4 [×3], C8⋊C4 [×4], C2×C42, C2×M4(2) [×2], C8○D4 [×4], C2×C8⋊C4, C82M4(2) [×2], C23.27C42

Smallest permutation representation of C23.27C42
On 64 points
Generators in S64
(9 13)(10 14)(11 15)(12 16)(25 29)(26 30)(27 31)(28 32)(41 45)(42 46)(43 47)(44 48)(49 53)(50 54)(51 55)(52 56)
(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 20)(2 21)(3 22)(4 23)(5 24)(6 17)(7 18)(8 19)(9 41)(10 42)(11 43)(12 44)(13 45)(14 46)(15 47)(16 48)(25 55)(26 56)(27 49)(28 50)(29 51)(30 52)(31 53)(32 54)(33 58)(34 59)(35 60)(36 61)(37 62)(38 63)(39 64)(40 57)
(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 42 40 55 20 10 57 25)(2 11 33 26 21 43 58 56)(3 44 34 49 22 12 59 27)(4 13 35 28 23 45 60 50)(5 46 36 51 24 14 61 29)(6 15 37 30 17 47 62 52)(7 48 38 53 18 16 63 31)(8 9 39 32 19 41 64 54)

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

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

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

56 conjugacy classes

class 1 2A2B2C2D2E4A···4L4M···4R8A···8P8Q···8AF
order1222224···44···48···88···8
size1111221···12···22···24···4

56 irreducible representations

dim1111111122
type++++
imageC1C2C2C2C4C4C4C4M4(2)C8○D4
kernelC23.27C42C8⋊C8C2×C4×C8C42.12C4C4×C8C22⋊C8C4⋊C8C22×C8C2×C4C4
# reps14124884816

Matrix representation of C23.27C42 in GL4(𝔽17) generated by

1000
01600
0010
00016
,
16000
01600
00160
00016
,
16000
01600
0010
0001
,
15000
0200
00150
00015
,
0100
13000
0001
0010
G:=sub<GL(4,GF(17))| [1,0,0,0,0,16,0,0,0,0,1,0,0,0,0,16],[16,0,0,0,0,16,0,0,0,0,16,0,0,0,0,16],[16,0,0,0,0,16,0,0,0,0,1,0,0,0,0,1],[15,0,0,0,0,2,0,0,0,0,15,0,0,0,0,15],[0,13,0,0,1,0,0,0,0,0,0,1,0,0,1,0] >;

C23.27C42 in GAP, Magma, Sage, TeX

C_2^3._{27}C_4^2
% in TeX

G:=Group("C2^3.27C4^2");
// GroupNames label

G:=SmallGroup(128,184);
// by ID

G=gap.SmallGroup(128,184);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,56,925,120,387,136,172]);
// Polycyclic

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

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