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G = C42.13C8order 128 = 27

10th non-split extension by C42 of C8 acting via C8/C4=C2

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

Aliases: C42.13C8, C8.27M4(2), C4.13M5(2), (C2×C4)⋊4C16, (C4×C16)⋊3C2, C42(C4⋊C16), C82(C4⋊C16), C4⋊C1619C2, C4.9(C2×C16), (C4×C8).23C4, C42(C4⋊C16), C82(C22⋊C16), C42(C22⋊C16), C8.98(C4○D4), (C22×C4).16C8, C22.5(C2×C16), C2.3(C22×C16), (C22×C8).32C4, C23.38(C2×C8), (C2×C42).50C4, C2.5(C2×M5(2)), C22⋊C16.11C2, C42(C22⋊C16), C42.341(C2×C4), (C2×C16).66C22, (C4×C8).443C22, (C2×C8).629C23, C4.67(C2×M4(2)), C22.28(C22×C8), C4.76(C42⋊C2), (C22×C8).499C22, C2.4(C42.12C4), (C2×C4×C8).28C2, (C4×C8)(C4⋊C16), (C4×C8)(C22⋊C16), (C2×C4).100(C2×C8), (C2×C8).264(C2×C4), (C2×C4).614(C22×C4), (C22×C4).447(C2×C4), SmallGroup(128,894)

Series: Derived Chief Lower central Upper central Jennings

C1C2 — C42.13C8
C1C2C4C8C2×C8C4×C8C2×C4×C8 — C42.13C8
C1C2 — C42.13C8
C1C4×C8 — C42.13C8
C1C2C2C2C2C4C4C2×C8 — C42.13C8

Generators and relations for C42.13C8
 G = < a,b,c | a4=b4=1, c8=a2, ab=ba, cac-1=a-1b2, bc=cb >

Subgroups: 100 in 80 conjugacy classes, 60 normal (34 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×6], C4 [×2], C22, C22 [×2], C22 [×2], C8 [×4], C8 [×2], C2×C4 [×6], C2×C4 [×4], C2×C4 [×4], 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], C4×C16 [×2], C22⋊C16 [×2], C4⋊C16 [×2], C2×C4×C8, C42.13C8
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C8 [×4], C2×C4 [×6], C23, C16 [×4], C2×C8 [×6], M4(2) [×2], C22×C4, C4○D4 [×2], C2×C16 [×6], M5(2) [×2], C42⋊C2, C22×C8, C2×M4(2), C42.12C4, C22×C16, C2×M5(2), C42.13C8

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

80 conjugacy classes

class 1 2A2B2C2D2E4A···4L4M···4R8A···8P8Q···8X16A···16AF
order1222224···44···48···88···816···16
size1111221···12···21···12···22···2

80 irreducible representations

dim11111111111222
type+++++
imageC1C2C2C2C2C4C4C4C8C8C16M4(2)C4○D4M5(2)
kernelC42.13C8C4×C16C22⋊C16C4⋊C16C2×C4×C8C4×C8C2×C42C22×C8C42C22×C4C2×C4C8C8C4
# reps122214228832448

Matrix representation of C42.13C8 in GL3(𝔽17) generated by

1300
01613
001
,
400
040
004
,
1000
0150
012
G:=sub<GL(3,GF(17))| [13,0,0,0,16,0,0,13,1],[4,0,0,0,4,0,0,0,4],[10,0,0,0,15,1,0,0,2] >;

C42.13C8 in GAP, Magma, Sage, TeX

C_4^2._{13}C_8
% in TeX

G:=Group("C4^2.13C8");
// GroupNames label

G:=SmallGroup(128,894);
// by ID

G=gap.SmallGroup(128,894);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,-2,112,141,58,102,124]);
// 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,b*c=c*b>;
// generators/relations

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