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G = C23.19C42  order 128 = 27

1st non-split extension by C23 of C42 acting via C42/C2×C4=C2

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

Aliases: C23.19C42, C23.24M4(2), C22⋊C8⋊4C4, (C22×C4)⋊1C8, C22.1(C4×C8), (C23×C4).1C4, (C22×C4).3Q8, C22.2(C4⋊C8), C2.1(C23⋊C8), C23.25(C2×C8), C23.36(C4⋊C4), C24.103(C2×C4), (C22×C4).636D4, (C23×C4).1C22, C22.1(C8⋊C4), C22.33(C23⋊C4), C22.32(C22⋊C8), C2.1(C23.9D4), C2.1(C22.C42), C23.213(C22⋊C4), C22.23(C4.D4), C22.14(C4.10D4), C2.9(C22.7C42), C22.21(C2.C42), C2.1(C22.M4(2)), (C2×C4).66(C4⋊C4), (C2×C22⋊C8).1C2, (C22×C4).89(C2×C4), (C2×C4).290(C22⋊C4), (C2×C2.C42).1C2, SmallGroup(128,12)

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

Derived series
C1 — C22 — C23.19C42
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C23×C4 — C2×C2.C42 — C23.19C42
Lower central
C1 — C2 — C22 — C23.19C42
Upper central
C1 — C23 — C23×C4 — C23.19C42
Jennings
C1 — C2 — C22 — C23×C4 — C23.19C42

Generators and relations for C23.19C42
 G = < a,b,c,d,e | a2=b2=c2=e4=1, d4=c, dad-1=ab=ba, ac=ca, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede-1=abd >

Subgroups: 280 in 132 conjugacy classes, 52 normal (18 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C8, C2×C4, C2×C4, C23, C23, C23, C2×C8, C22×C4, C22×C4, C22×C4, C24, C2.C42, C22⋊C8, C22⋊C8, C22×C8, C23×C4, C23×C4, C2×C2.C42, C2×C22⋊C8, C23.19C42
Quotients: C1, C2, C4, C22, C8, C2×C4, D4, Q8, C42, C22⋊C4, C4⋊C4, C2×C8, M4(2), C2.C42, C4×C8, C8⋊C4, C22⋊C8, C23⋊C4, C4.D4, C4.10D4, C4⋊C8, C23⋊C8, C22.M4(2), C22.7C42, C23.9D4, C22.C42, C23.19C42

Smallest permutation representation of C23.19C42
►On 64 points
Generators in S64
(1 57)(2 33)(3 59)(4 35)(5 61)(6 37)(7 63)(8 39)(9 18)(10 54)(11 20)(12 56)(13 22)(14 50)(15 24)(16 52)(17 43)(19 45)(21 47)(23 41)(25 38)(26 64)(27 40)(28 58)(29 34)(30 60)(31 36)(32 62)(42 51)(44 53)(46 55)(48 49)

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

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

44 conjugacy classes

class 1 2A···2G2H2I2J2K4A···4H4I···4P8A···8P
order12···222224···44···48···8
size11···122222···24···44···4

44 irreducible representations

dim111111222444
type++++-++-
imageC1C2C2C4C4C8D4Q8M4(2)C23⋊C4C4.D4C4.10D4
kernelC23.19C42C2×C2.C42C2×C22⋊C8C22⋊C8C23×C4C22×C4C22×C4C22×C4C23C22C22C22
# reps1128416314211

Matrix representation of C23.19C42 ►in GL8(𝔽17)

160000000
016000000
001600000
000160000
00001000
00000100
000000160
000058016
,
10000000
01000000
00100000
00010000
000016000
000001600
000000160
000000016
,
10000000
01000000
001600000
000160000
000016000
000001600
000000160
000000016
,
315000000
514000000
00230000
003150000
00000010
0000613101
0000161500
00001504
,
130000000
54000000
000130000
00400000
00006600
0000141100
0000210156
000015082

G:=sub<GL(8,GF(17))| [16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,5,0,0,0,0,0,1,0,8,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16],[3,5,0,0,0,0,0,0,15,14,0,0,0,0,0,0,0,0,2,3,0,0,0,0,0,0,3,15,0,0,0,0,0,0,0,0,0,6,16,1,0,0,0,0,0,13,15,5,0,0,0,0,1,10,0,0,0,0,0,0,0,1,0,4],[13,5,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,0,6,14,2,15,0,0,0,0,6,11,10,0,0,0,0,0,0,0,15,8,0,0,0,0,0,0,6,2] >;
 

C23.19C42 in GAP, Magma, Sage, TeX 

C_2^3._{19}C_4^2
 
% in TeX
 
G:=Group("C2^3.19C4^2");
 
// GroupNames label
 
G:=SmallGroup(128,12);
 
// by ID
 
G=gap.SmallGroup(128,12);
 
# by ID
 
G:=PCGroup([7,-2,2,-2,2,2,-2,2,56,85,120,758,570,248]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^2=b^2=c^2=e^4=1,d^4=c,d*a*d^-1=a*b=b*a,a*c=c*a,a*e=e*a,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*d>;
 
// generators/relations
 

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