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G = C42.265C23order 128 = 27

126th non-split extension by C42 of C23 acting via C23/C2=C22

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

Aliases: C42.265C23, (C2×C8)⋊26D4, C86D432C2, C89D430C2, C4⋊C850C22, (C4×C8)⋊56C22, C4.132(C4×D4), C8.127(C2×D4), C22≀C2.3C4, C4⋊D4.17C4, C24.80(C2×C4), C22.45(C4×D4), C8⋊C459C22, C22⋊Q8.17C4, C22⋊C844C22, (C2×C8).402C23, (C2×C4).649C24, (C22×C8)⋊51C22, (C4×D4).53C22, C4.195(C22×D4), C82M4(2)⋊31C2, C24.4C432C2, C23.33(C22×C4), C22.D4.3C4, C2.14(Q8○M4(2)), (C2×M4(2))⋊77C22, (C22×M4(2))⋊25C2, C22.176(C23×C4), (C23×C4).526C22, (C22×C4).916C23, C22.19C24.10C2, C42.6C2230C2, C42⋊C2.293C22, C2.47(C2×C4×D4), (C2×C8○D4)⋊20C2, C4⋊C4.114(C2×C4), C4.300(C2×C4○D4), (C2×D4).136(C2×C4), (C2×C4).1412(C2×D4), C22⋊C4.15(C2×C4), (C2×C4).66(C22×C4), (C2×Q8).153(C2×C4), (C22×C8)⋊C228C2, (C2×C4).682(C4○D4), (C22×C4).340(C2×C4), (C2×C4○D4).284C22, SmallGroup(128,1662)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C42.265C23
C1C2C4C2×C4C22×C4C22×C8C22×M4(2) — C42.265C23
C1C22 — C42.265C23
C1C2×C4 — C42.265C23
C1C2C2C2×C4 — C42.265C23

Generators and relations for C42.265C23
 G = < a,b,c,d,e | a4=b4=e2=1, c2=b2, d2=a2b, ab=ba, cac-1=a-1b2, dad-1=eae=ab2, bc=cb, bd=db, be=eb, dcd-1=a2c, ce=ec, ede=b2d >

Subgroups: 380 in 246 conjugacy classes, 140 normal (30 characteristic)
C1, C2, C2 [×2], C2 [×6], C4 [×2], C4 [×2], C4 [×8], C22, C22 [×2], C22 [×18], C8 [×4], C8 [×6], C2×C4 [×2], C2×C4 [×12], C2×C4 [×14], D4 [×10], Q8 [×2], C23, C23 [×4], C23 [×4], C42 [×2], C22⋊C4 [×10], C4⋊C4 [×6], C2×C8 [×2], C2×C8 [×10], C2×C8 [×6], M4(2) [×16], C22×C4 [×2], C22×C4 [×6], C22×C4 [×4], C2×D4, C2×D4 [×6], C2×Q8, C4○D4 [×4], C24, C4×C8 [×2], C8⋊C4 [×2], C22⋊C8 [×8], C4⋊C8 [×4], C42⋊C2, C4×D4 [×4], C22≀C2 [×2], C4⋊D4 [×2], C22⋊Q8 [×2], C22.D4 [×2], C22×C8 [×2], C22×C8 [×2], C2×M4(2) [×2], C2×M4(2) [×6], C2×M4(2) [×4], C8○D4 [×4], C23×C4, C2×C4○D4, C82M4(2), C24.4C4, (C22×C8)⋊C2, C42.6C22, C89D4 [×4], C86D4 [×4], C22.19C24, C22×M4(2), C2×C8○D4, C42.265C23
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], D4 [×4], C23 [×15], C22×C4 [×14], C2×D4 [×6], C4○D4 [×2], C24, C4×D4 [×4], C23×C4, C22×D4, C2×C4○D4, C2×C4×D4, Q8○M4(2) [×2], C42.265C23

Smallest permutation representation of C42.265C23
On 32 points
Generators in S32
(1 16 31 22)(2 13 32 19)(3 10 25 24)(4 15 26 21)(5 12 27 18)(6 9 28 23)(7 14 29 20)(8 11 30 17)
(1 25 5 29)(2 26 6 30)(3 27 7 31)(4 28 8 32)(9 17 13 21)(10 18 14 22)(11 19 15 23)(12 20 16 24)
(1 3 5 7)(2 26 6 30)(4 28 8 32)(9 15 13 11)(10 22 14 18)(12 24 16 20)(17 23 21 19)(25 27 29 31)
(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 31)(2 28)(3 25)(4 30)(5 27)(6 32)(7 29)(8 26)(9 23)(10 20)(11 17)(12 22)(13 19)(14 24)(15 21)(16 18)

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

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

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

44 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I4A4B4C4D4E4F4G···4N8A···8H8I···8T
order12222222224444444···48···88···8
size11112244441111224···42···24···4

44 irreducible representations

dim11111111111111224
type+++++++++++
imageC1C2C2C2C2C2C2C2C2C2C4C4C4C4D4C4○D4Q8○M4(2)
kernelC42.265C23C82M4(2)C24.4C4(C22×C8)⋊C2C42.6C22C89D4C86D4C22.19C24C22×M4(2)C2×C8○D4C22≀C2C4⋊D4C22⋊Q8C22.D4C2×C8C2×C4C2
# reps11111441114444444

Matrix representation of C42.265C23 in GL6(𝔽17)

010000
1600000
00130162
000020
000900
00113154
,
1600000
0160000
0013000
0001300
0000130
0000013
,
1600000
010000
0013000
000400
0000130
0010134
,
0130000
400000
000010
0020916
0013000
0015420
,
100000
010000
001000
000100
0000160
0040016

G:=sub<GL(6,GF(17))| [0,16,0,0,0,0,1,0,0,0,0,0,0,0,13,0,0,1,0,0,0,0,9,13,0,0,16,2,0,15,0,0,2,0,0,4],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,13],[16,0,0,0,0,0,0,1,0,0,0,0,0,0,13,0,0,1,0,0,0,4,0,0,0,0,0,0,13,13,0,0,0,0,0,4],[0,4,0,0,0,0,13,0,0,0,0,0,0,0,0,2,13,15,0,0,0,0,0,4,0,0,1,9,0,2,0,0,0,16,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,4,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16] >;

C42.265C23 in GAP, Magma, Sage, TeX

C_4^2._{265}C_2^3
% in TeX

G:=Group("C4^2.265C2^3");
// GroupNames label

G:=SmallGroup(128,1662);
// by ID

G=gap.SmallGroup(128,1662);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,723,184,2019,124]);
// Polycyclic

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

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