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G = C23.15C42order 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.C4226C2, (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

C1C22 — C23.15C42
C1C2C22C2×C4C22×C4C23×C4C2×C42⋊C2 — C23.15C42
C1C2C22 — C23.15C42
C1C2×C4C23×C4 — C23.15C42
C1C2C2C22×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 [×2], C2 [×6], C4 [×8], C4 [×4], C22 [×3], C22 [×4], C22 [×10], C8 [×8], C2×C4 [×4], C2×C4 [×24], C2×C4 [×8], C23, C23 [×6], C23 [×2], C42 [×4], C22⋊C4 [×4], C4⋊C4 [×4], C2×C8 [×12], M4(2) [×8], M4(2) [×12], C22×C4 [×2], C22×C4 [×16], C24, C2×C42 [×2], C2×C22⋊C4 [×2], C2×C4⋊C4 [×2], C42⋊C2 [×4], C22×C8 [×2], C2×M4(2) [×12], C2×M4(2) [×6], C23×C4, C22.C42 [×4], C2×C42⋊C2, C22×M4(2) [×2], C23.15C42
Quotients: C1, C2 [×7], C4 [×12], C22 [×7], C2×C4 [×18], D4 [×6], Q8 [×2], C23, C42 [×4], C22⋊C4 [×12], C4⋊C4 [×12], C22×C4 [×3], C2×D4 [×3], C2×Q8, C2.C42 [×8], C2×C42, C2×C22⋊C4 [×3], C2×C4⋊C4 [×3], C2×C2.C42, M4(2).8C22 [×2], 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 17)(10 18)(11 19)(12 20)(13 21)(14 22)(15 23)(16 24)
(1 31)(2 28)(3 25)(4 30)(5 27)(6 32)(7 29)(8 26)(9 21)(10 18)(11 23)(12 20)(13 17)(14 22)(15 19)(16 24)
(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 13)(2 10 32 22)(3 23 25 11)(4 16 26 20)(5 21 27 9)(6 14 28 18)(7 19 29 15)(8 12 30 24)

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

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

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

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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