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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), (C2xC4):4C16, (C4xC16):3C2, C4o2(C4:C16), C8o2(C4:C16), C4:C16:19C2, C4.9(C2xC16), (C4xC8).23C4, C42o(C4:C16), C8o2(C22:C16), C4o2(C22:C16), C8.98(C4oD4), (C22xC4).16C8, C22.5(C2xC16), C2.3(C22xC16), (C22xC8).32C4, C23.38(C2xC8), (C2xC42).50C4, C2.5(C2xM5(2)), C22:C16.11C2, C42o(C22:C16), C42.341(C2xC4), (C2xC16).66C22, (C4xC8).443C22, (C2xC8).629C23, C4.67(C2xM4(2)), C22.28(C22xC8), C4.76(C42:C2), (C22xC8).499C22, C2.4(C42.12C4), (C2xC4xC8).28C2, (C4xC8)o(C4:C16), (C4xC8)o(C22:C16), (C2xC4).100(C2xC8), (C2xC8).264(C2xC4), (C2xC4).614(C22xC4), (C22xC4).447(C2xC4), SmallGroup(128,894)

Series: Derived Chief Lower central Upper central Jennings

C1C2 — C42.13C8
C1C2C4C8C2xC8C4xC8C2xC4xC8 — C42.13C8
C1C2 — C42.13C8
C1C4xC8 — C42.13C8
C1C2C2C2C2C4C4C2xC8 — 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, C2, C4, C4, C4, C22, C22, C22, C8, C8, C2xC4, C2xC4, C2xC4, C23, C16, C42, C2xC8, C2xC8, C22xC4, C4xC8, C2xC16, C2xC42, C22xC8, C4xC16, C22:C16, C4:C16, C2xC4xC8, C42.13C8
Quotients: C1, C2, C4, C22, C8, C2xC4, C23, C16, C2xC8, M4(2), C22xC4, C4oD4, C2xC16, M5(2), C42:C2, C22xC8, C2xM4(2), C42.12C4, C22xC16, C2xM5(2), C42.13C8

Smallest permutation representation of C42.13C8
On 64 points
Generators in S64
(1 5 9 13)(2 61 10 53)(3 7 11 15)(4 63 12 55)(6 49 14 57)(8 51 16 59)(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 48 64 22)(2 33 49 23)(3 34 50 24)(4 35 51 25)(5 36 52 26)(6 37 53 27)(7 38 54 28)(8 39 55 29)(9 40 56 30)(10 41 57 31)(11 42 58 32)(12 43 59 17)(13 44 60 18)(14 45 61 19)(15 46 62 20)(16 47 63 21)
(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)(2,61,10,53)(3,7,11,15)(4,63,12,55)(6,49,14,57)(8,51,16,59)(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), (1,48,64,22)(2,33,49,23)(3,34,50,24)(4,35,51,25)(5,36,52,26)(6,37,53,27)(7,38,54,28)(8,39,55,29)(9,40,56,30)(10,41,57,31)(11,42,58,32)(12,43,59,17)(13,44,60,18)(14,45,61,19)(15,46,62,20)(16,47,63,21), (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)(2,61,10,53)(3,7,11,15)(4,63,12,55)(6,49,14,57)(8,51,16,59)(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), (1,48,64,22)(2,33,49,23)(3,34,50,24)(4,35,51,25)(5,36,52,26)(6,37,53,27)(7,38,54,28)(8,39,55,29)(9,40,56,30)(10,41,57,31)(11,42,58,32)(12,43,59,17)(13,44,60,18)(14,45,61,19)(15,46,62,20)(16,47,63,21), (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),(2,61,10,53),(3,7,11,15),(4,63,12,55),(6,49,14,57),(8,51,16,59),(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)], [(1,48,64,22),(2,33,49,23),(3,34,50,24),(4,35,51,25),(5,36,52,26),(6,37,53,27),(7,38,54,28),(8,39,55,29),(9,40,56,30),(10,41,57,31),(11,42,58,32),(12,43,59,17),(13,44,60,18),(14,45,61,19),(15,46,62,20),(16,47,63,21)], [(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)C4oD4M5(2)
kernelC42.13C8C4xC16C22:C16C4:C16C2xC4xC8C4xC8C2xC42C22xC8C42C22xC4C2xC4C8C8C4
# 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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