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G = C42.2C8order 128 = 27

2nd non-split extension by C42 of C8 acting via C8/C2=C4

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

Aliases: C42.2C8, C23.25M4(2), (C2×C16)⋊3C4, (C2×C8).1C8, C4.8(C4⋊C8), C4.19(C4×C8), C8.32(C4⋊C4), (C2×C8).26Q8, (C2×C8).182D4, C22.3(C4⋊C8), (C2×C4).84C42, (C2×C42).11C4, (C22×C8).14C4, C2.2(C16⋊C4), C8.49(C22⋊C4), C4.19(C22⋊C8), (C2×M5(2)).6C2, (C2×C4).68M4(2), C22.12(C8⋊C4), C22.22(C22⋊C8), (C22×C8).367C22, C4.25(C2.C42), C2.13(C22.7C42), (C2×C4).73(C2×C8), (C2×C8).237(C2×C4), (C2×C8⋊C4).17C2, (C2×C4).104(C4⋊C4), (C22×C4).466(C2×C4), (C2×C4).346(C22⋊C4), SmallGroup(128,107)

Series: Derived Chief Lower central Upper central Jennings

C1C4 — C42.2C8
C1C2C4C2×C4C2×C8C22×C8C2×C8⋊C4 — C42.2C8
C1C4 — C42.2C8
C1C2×C4 — C42.2C8
C1C2C2C2C2C4C4C22×C8 — C42.2C8

Generators and relations for C42.2C8
 G = < a,b,c | a4=b4=1, c8=b2, ab=ba, cac-1=ab-1, bc=cb >

Subgroups: 104 in 70 conjugacy classes, 44 normal (24 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×4], C4 [×2], C22 [×3], C22 [×2], C8 [×4], C8 [×2], C2×C4 [×6], C2×C4 [×4], C23, C16 [×4], C42 [×2], C42, C2×C8 [×2], C2×C8 [×6], C2×C8 [×2], C22×C4, C22×C4, C8⋊C4 [×2], C2×C16 [×4], M5(2) [×4], C2×C42, C22×C8 [×2], C2×C8⋊C4, C2×M5(2) [×2], C42.2C8
Quotients: C1, C2 [×3], C4 [×6], C22, C8 [×4], C2×C4 [×3], D4 [×3], Q8, C42, C22⋊C4 [×3], C4⋊C4 [×3], C2×C8 [×2], M4(2) [×2], C2.C42, C4×C8, C8⋊C4, C22⋊C8 [×2], C4⋊C8 [×2], C22.7C42, C16⋊C4 [×2], C42.2C8

Smallest permutation representation of C42.2C8
On 32 points
Generators in S32
(2 21 10 29)(3 11)(4 31 12 23)(6 25 14 17)(7 15)(8 19 16 27)(18 26)(22 30)
(1 28 9 20)(2 29 10 21)(3 30 11 22)(4 31 12 23)(5 32 13 24)(6 17 14 25)(7 18 15 26)(8 19 16 27)
(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)

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

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

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

44 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F4G4H4I4J8A···8H8I8J8K8L16A···16P
order12222244444444448···8888816···16
size11112211112244442···244444···4

44 irreducible representations

dim1111111122224
type++++-
imageC1C2C2C4C4C4C8C8D4Q8M4(2)M4(2)C16⋊C4
kernelC42.2C8C2×C8⋊C4C2×M5(2)C2×C16C2×C42C22×C8C42C2×C8C2×C8C2×C8C2×C4C23C2
# reps1128228831224

Matrix representation of C42.2C8 in GL6(𝔽17)

1300000
040000
001000
00141600
0000130
00100164
,
1600000
0160000
0013000
0001300
0000130
0000013
,
020000
900000
000010
009091
00141500
008480

G:=sub<GL(6,GF(17))| [13,0,0,0,0,0,0,4,0,0,0,0,0,0,1,14,0,10,0,0,0,16,0,0,0,0,0,0,13,16,0,0,0,0,0,4],[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],[0,9,0,0,0,0,2,0,0,0,0,0,0,0,0,9,14,8,0,0,0,0,15,4,0,0,1,9,0,8,0,0,0,1,0,0] >;

C42.2C8 in GAP, Magma, Sage, TeX

C_4^2._2C_8
% in TeX

G:=Group("C4^2.2C8");
// GroupNames label

G:=SmallGroup(128,107);
// by ID

G=gap.SmallGroup(128,107);
# by ID

G:=PCGroup([7,-2,2,-2,2,2,-2,-2,56,85,120,352,1018,136,2804,124]);
// Polycyclic

G:=Group<a,b,c|a^4=b^4=1,c^8=b^2,a*b=b*a,c*a*c^-1=a*b^-1,b*c=c*b>;
// generators/relations

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