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G = M5(2).19C22order 128 = 27

6th non-split extension by M5(2) of C22 acting via C22/C2=C2

p-group, metabelian, nilpotent (class 3), monomial

Aliases: M5(2).19C22, (C2×D4).6C8, (C2×Q8).6C8, C8.124(C2×D4), (C2×C8).390D4, C8(C23.C8), C23.C88C2, C23.4(C2×C8), (C2×M5(2))⋊9C2, (C22×C8).20C4, C8.63(C22⋊C4), C4.15(C22⋊C8), (C2×C8).385C23, (C2×C4).24M4(2), C4.50(C2×M4(2)), (C2×M4(2)).30C4, C22.5(C22⋊C8), C22.13(C22×C8), (C22×C8).416C22, (C2×M4(2)).329C22, (C2×C4).5(C2×C8), (C2×C8).250(C2×C4), (C2×C4○D4).18C4, (C2×C8○D4).17C2, C2.25(C2×C22⋊C8), C4.116(C2×C22⋊C4), (C22×C4).77(C2×C4), (C2×C4).559(C22×C4), (C2×C4).267(C22⋊C4), SmallGroup(128,847)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — M5(2).19C22
C1C2C4C8C2×C8C22×C8C2×C8○D4 — M5(2).19C22
C1C2C22 — M5(2).19C22
C1C8C22×C8 — M5(2).19C22
C1C2C2C2C2C4C4C2×C8 — M5(2).19C22

Generators and relations for M5(2).19C22
 G = < a,b,c,d | a16=b2=c2=d2=1, bab=a9, cac=ab, ad=da, bc=cb, bd=db, dcd=a8c >

Subgroups: 172 in 110 conjugacy classes, 58 normal (20 characteristic)
C1, C2, C2 [×5], C4 [×2], C4 [×2], C4 [×2], C22, C22 [×2], C22 [×5], C8 [×2], C8 [×2], C8 [×2], C2×C4 [×2], C2×C4 [×6], C2×C4 [×4], D4 [×6], Q8 [×2], C23, C23 [×2], C16 [×4], C2×C8 [×2], C2×C8 [×6], C2×C8 [×4], M4(2) [×6], C22×C4, C22×C4 [×2], C2×D4, C2×D4 [×2], C2×Q8, C4○D4 [×4], C2×C16 [×2], M5(2) [×4], M5(2) [×2], C22×C8, C22×C8 [×2], C2×M4(2), C2×M4(2) [×2], C8○D4 [×4], C2×C4○D4, C23.C8 [×4], C2×M5(2) [×2], C2×C8○D4, M5(2).19C22
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C8 [×4], C2×C4 [×6], D4 [×4], C23, C22⋊C4 [×4], C2×C8 [×6], M4(2) [×2], C22×C4, C2×D4 [×2], C22⋊C8 [×4], C2×C22⋊C4, C22×C8, C2×M4(2), C2×C22⋊C8, M5(2).19C22

Smallest permutation representation of M5(2).19C22
On 32 points
Generators in S32
(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)
(2 10)(4 12)(6 14)(8 16)(18 26)(20 28)(22 30)(24 32)
(2 10)(3 11)(6 14)(7 15)(17 25)(20 28)(21 29)(24 32)
(1 21)(2 22)(3 23)(4 24)(5 25)(6 26)(7 27)(8 28)(9 29)(10 30)(11 31)(12 32)(13 17)(14 18)(15 19)(16 20)

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

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

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

44 conjugacy classes

class 1 2A2B2C2D2E2F4A4B4C4D4E4F4G8A8B8C8D8E···8J8K8L8M8N16A···16P
order1222222444444488888···8888816···16
size1122244112224411112···244444···4

44 irreducible representations

dim111111111224
type+++++
imageC1C2C2C2C4C4C4C8C8D4M4(2)M5(2).19C22
kernelM5(2).19C22C23.C8C2×M5(2)C2×C8○D4C22×C8C2×M4(2)C2×C4○D4C2×D4C2×Q8C2×C8C2×C4C1
# reps1421422124444

Matrix representation of M5(2).19C22 in GL4(𝔽17) generated by

60150
10001
116110
10030
,
1000
0100
60160
140016
,
16000
0100
00160
3001
,
01600
16000
51409
111020
G:=sub<GL(4,GF(17))| [6,10,1,10,0,0,16,0,15,0,11,3,0,1,0,0],[1,0,6,14,0,1,0,0,0,0,16,0,0,0,0,16],[16,0,0,3,0,1,0,0,0,0,16,0,0,0,0,1],[0,16,5,11,16,0,14,10,0,0,0,2,0,0,9,0] >;

M5(2).19C22 in GAP, Magma, Sage, TeX

M_5(2)._{19}C_2^2
% in TeX

G:=Group("M5(2).19C2^2");
// GroupNames label

G:=SmallGroup(128,847);
// by ID

G=gap.SmallGroup(128,847);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,-2,112,141,723,2019,1411,102,124]);
// Polycyclic

G:=Group<a,b,c,d|a^16=b^2=c^2=d^2=1,b*a*b=a^9,c*a*c=a*b,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=a^8*c>;
// generators/relations

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