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G = C23.15C42  order 128 = 27

10th non-split extension by C23 of C42 acting via C42/C22=C22

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

Aliases: C23.15C42, C24.47(C2×C4), (C2×C4).69C42, (C2×C42).19C4, (C22×C4).40Q8, C23.24(C4⋊C4), (C2×M4(2))⋊16C4, (C22×C4).757D4, C22.7(C2×C42), C4○(C22.C42), M4(2).29(C2×C4), C22.C42⋊26C2, (C23×C4).222C22, C23.177(C22×C4), (C22×C4).647C23, C23.116(C22⋊C4), C4.32(C2.C42), (C22×M4(2)).16C2, (C2×M4(2)).300C22, C22.13(C2.C42), C2.3(M4(2).8C22), C4.30(C2×C4⋊C4), C22.15(C2×C4⋊C4), C4.85(C2×C22⋊C4), (C2×C4).113(C2×Q8), (C2×C4).127(C4⋊C4), (C2×C4).1297(C2×D4), (C2×C22⋊C4).20C4, (C22×C4).48(C2×C4), (C2×C4⋊C4).743C22, (C2×C4).175(C22×C4), (C2×C4).115(C22⋊C4), (C2×C42⋊C2).10C2, C22.110(C2×C22⋊C4), C2.19(C2×C2.C42), SmallGroup(128,474)

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

Derived series
C1 — C22 — C23.15C42
Chief series
C1 — C2 — C22 — C2×C4 — C22×C4 — C23×C4 — C2×C42⋊C2 — C23.15C42
Lower central
C1 — C2 — C22 — C23.15C42
Upper central
C1 — C2×C4 — C23×C4 — C23.15C42
Jennings
C1 — C2 — C2 — C22×C4 — C23.15C42

Generators and relations for C23.15C42
 G = < a,b,c,d,e | a2=b2=c2=e4=1, d4=c, ab=ba, eae-1=ac=ca, ad=da, dbd-1=bc=cb, be=eb, cd=dc, ce=ec, ede-1=bcd >

Subgroups: 308 in 194 conjugacy classes, 108 normal (16 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, C23, C23, C23, C42, C22⋊C4, C4⋊C4, C2×C8, M4(2), M4(2), C22×C4, C22×C4, C24, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C42⋊C2, C22×C8, C2×M4(2), C2×M4(2), C23×C4, C22.C42, C2×C42⋊C2, C22×M4(2), C23.15C42
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C42, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×Q8, C2.C42, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C2×C2.C42, M4(2).8C22, C23.15C42

Smallest permutation representation of C23.15C42
►On 32 points
Generators in S32
(1 31)(2 32)(3 25)(4 26)(5 27)(6 28)(7 29)(8 30)(9 21)(10 22)(11 23)(12 24)(13 17)(14 18)(15 19)(16 20)

(1 31)(2 28)(3 25)(4 30)(5 27)(6 32)(7 29)(8 26)(9 17)(10 22)(11 19)(12 24)(13 21)(14 18)(15 23)(16 20)

(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)

(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)

(1 17 31 9)(2 14 32 22)(3 23 25 15)(4 12 26 20)(5 21 27 13)(6 10 28 18)(7 19 29 11)(8 16 30 24)
 
G:=sub<Sym(32)| (1,31)(2,32)(3,25)(4,26)(5,27)(6,28)(7,29)(8,30)(9,21)(10,22)(11,23)(12,24)(13,17)(14,18)(15,19)(16,20), (1,31)(2,28)(3,25)(4,30)(5,27)(6,32)(7,29)(8,26)(9,17)(10,22)(11,19)(12,24)(13,21)(14,18)(15,23)(16,20), (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), (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), (1,17,31,9)(2,14,32,22)(3,23,25,15)(4,12,26,20)(5,21,27,13)(6,10,28,18)(7,19,29,11)(8,16,30,24)>;
 
G:=Group( (1,31)(2,32)(3,25)(4,26)(5,27)(6,28)(7,29)(8,30)(9,21)(10,22)(11,23)(12,24)(13,17)(14,18)(15,19)(16,20), (1,31)(2,28)(3,25)(4,30)(5,27)(6,32)(7,29)(8,26)(9,17)(10,22)(11,19)(12,24)(13,21)(14,18)(15,23)(16,20), (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), (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), (1,17,31,9)(2,14,32,22)(3,23,25,15)(4,12,26,20)(5,21,27,13)(6,10,28,18)(7,19,29,11)(8,16,30,24) );
 
G=PermutationGroup([[(1,31),(2,32),(3,25),(4,26),(5,27),(6,28),(7,29),(8,30),(9,21),(10,22),(11,23),(12,24),(13,17),(14,18),(15,19),(16,20)], [(1,31),(2,28),(3,25),(4,30),(5,27),(6,32),(7,29),(8,26),(9,17),(10,22),(11,19),(12,24),(13,21),(14,18),(15,23),(16,20)], [(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)], [(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)], [(1,17,31,9),(2,14,32,22),(3,23,25,15),(4,12,26,20),(5,21,27,13),(6,10,28,18),(7,19,29,11),(8,16,30,24)]])
 

44 conjugacy classes

class 1 2A2B2C2D···2I4A4B4C4D4E···4J4K···4R8A···8P
order12222···244444···44···48···8
size11112···211112···24···44···4

44 irreducible representations

dim1111111224
type+++++-
imageC1C2C2C2C4C4C4D4Q8M4(2).8C22
kernelC23.15C42C22.C42C2×C42⋊C2C22×M4(2)C2×C42C2×C22⋊C4C2×M4(2)C22×C4C22×C4C2
# reps14124416624

Matrix representation of C23.15C42 ►in GL6(𝔽17)

100000
010000
001000
0001600
000010
0000016
,
1600000
0160000
001000
000100
0000160
0000016
,
100000
010000
0016000
0001600
0000160
0000016
,
0160000
100000
000010
000001
0013000
000400
,
400000
0130000
000100
001000
000001
000010

G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[0,1,0,0,0,0,16,0,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,4,0,0,1,0,0,0,0,0,0,1,0,0],[4,0,0,0,0,0,0,13,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;
 

C23.15C42 in GAP, Magma, Sage, TeX 

C_2^3._{15}C_4^2
 
% in TeX
 
G:=Group("C2^3.15C4^2");
 
// GroupNames label
 
G:=SmallGroup(128,474);
 
// by ID
 
G=gap.SmallGroup(128,474);
 
# by ID
 
G:=PCGroup([7,-2,2,2,-2,2,2,-2,112,141,232,352,2019,1411,172]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^2=b^2=c^2=e^4=1,d^4=c,a*b=b*a,e*a*e^-1=a*c=c*a,a*d=d*a,d*b*d^-1=b*c=c*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=b*c*d>;
 
// generators/relations
 

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