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G = C42.6C8order 128 = 27

3rd non-split extension by C42 of C8 acting via C8/C4=C2

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

Aliases: C42.6C8, C4.8M5(2), C8.23M4(2), C22.6M5(2), C4⋊C1615C2, (C4×C8).39C4, C165C48C2, C8.99(C4○D4), C22⋊C16.9C2, (C2×C42).51C4, (C22×C4).12C8, C23.39(C2×C8), (C22×C8).48C4, C2.8(C2×M5(2)), (C2×C8).630C23, C42.297(C2×C4), (C4×C8).374C22, (C2×C16).53C22, C4.68(C2×M4(2)), C22.50(C22×C8), C4.77(C42⋊C2), (C22×C8).500C22, C2.11(C42.12C4), (C2×C4×C8).67C2, (C2×C4).65(C2×C8), (C2×C8).253(C2×C4), (C22×C4).448(C2×C4), (C2×C4).615(C22×C4), SmallGroup(128,895)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C42.6C8
C1C2C4C8C2×C8C4×C8C2×C4×C8 — C42.6C8
C1C22 — C42.6C8
C1C2×C8 — C42.6C8
C1C2C2C2C2C4C4C2×C8 — C42.6C8

Generators and relations for C42.6C8
 G = < a,b,c | a4=b4=1, c8=a2, ab=ba, cac-1=a-1b2, cbc-1=a2b >

Subgroups: 100 in 76 conjugacy classes, 52 normal (34 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×2], C4 [×4], C22, C22 [×2], C22 [×2], C8 [×4], C8 [×2], C2×C4 [×6], C2×C4 [×6], C23, C16 [×4], C42 [×4], C2×C8 [×4], C2×C8 [×4], C22×C4 [×3], C4×C8 [×4], C2×C16 [×4], C2×C42, C22×C8 [×2], C165C4 [×2], C22⋊C16 [×2], C4⋊C16 [×2], C2×C4×C8, C42.6C8
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C8 [×4], C2×C4 [×6], C23, C2×C8 [×6], M4(2) [×2], C22×C4, C4○D4 [×2], M5(2) [×4], C42⋊C2, C22×C8, C2×M4(2), C42.12C4, C2×M5(2) [×2], C42.6C8

Smallest permutation representation of C42.6C8
On 64 points
Generators in S64
(1 5 9 13)(2 19 10 27)(3 7 11 15)(4 21 12 29)(6 23 14 31)(8 25 16 17)(18 22 26 30)(20 24 28 32)(33 60 41 52)(34 38 42 46)(35 62 43 54)(36 40 44 48)(37 64 45 56)(39 50 47 58)(49 53 57 61)(51 55 59 63)
(1 51 22 36)(2 60 23 45)(3 53 24 38)(4 62 25 47)(5 55 26 40)(6 64 27 33)(7 57 28 42)(8 50 29 35)(9 59 30 44)(10 52 31 37)(11 61 32 46)(12 54 17 39)(13 63 18 48)(14 56 19 41)(15 49 20 34)(16 58 21 43)
(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)

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

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

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

56 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E···4N8A···8H8I···8T16A···16P
order12222244444···48···88···816···16
size11112211112···21···12···24···4

56 irreducible representations

dim11111111112222
type+++++
imageC1C2C2C2C2C4C4C4C8C8M4(2)C4○D4M5(2)M5(2)
kernelC42.6C8C165C4C22⋊C16C4⋊C16C2×C4×C8C4×C8C2×C42C22×C8C42C22×C4C8C8C4C22
# reps12221422884488

Matrix representation of C42.6C8 in GL4(𝔽17) generated by

4000
11300
0040
0004
,
16000
4100
0004
0040
,
91300
8800
0026
001115
G:=sub<GL(4,GF(17))| [4,1,0,0,0,13,0,0,0,0,4,0,0,0,0,4],[16,4,0,0,0,1,0,0,0,0,0,4,0,0,4,0],[9,8,0,0,13,8,0,0,0,0,2,11,0,0,6,15] >;

C42.6C8 in GAP, Magma, Sage, TeX

C_4^2._6C_8
% in TeX

G:=Group("C4^2.6C8");
// GroupNames label

G:=SmallGroup(128,895);
// by ID

G=gap.SmallGroup(128,895);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,-2,112,141,1430,58,102,124]);
// Polycyclic

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

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