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G = C23⋊M4(2)  order 128 = 27

The semidirect product of C23 and M4(2) acting via M4(2)/C4=C4

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

Aliases: C231M4(2), C23⋊C813C2, (C2×C4)⋊1M4(2), (C2×C42).17C4, C24.23(C2×C4), (C23×C4).17C4, C4.27(C23⋊C4), (C22×D4).20C4, (C22×C4).200D4, C24.4C416C2, C22⋊C8.121C22, C23.40(C22⋊C4), C2.9(C24.4C4), (C23×C4).199C22, C23.166(C22×C4), (C22×C4).429C23, C22.18(C2×M4(2)), C22.M4(2)⋊14C2, C2.7(M4(2).8C22), (C2×C4×D4).3C2, C2.8(C2×C23⋊C4), (C2×C4).1126(C2×D4), (C2×C22⋊C4).18C4, (C22×C4).44(C2×C4), (C2×C4⋊C4).737C22, (C2×C4).350(C22⋊C4), C22.147(C2×C22⋊C4), (C2×C22⋊C4).404C22, SmallGroup(128,197)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C23⋊M4(2)
C1C2C22C2×C4C22×C4C23×C4C2×C4×D4 — C23⋊M4(2)
C1C2C23 — C23⋊M4(2)
C1C22C23×C4 — C23⋊M4(2)
C1C2C22C22×C4 — C23⋊M4(2)

Generators and relations for C23⋊M4(2)
 G = < a,b,c,d,e | a2=b2=c2=d8=e2=1, dad-1=ab=ba, eae=ac=ca, dbd-1=bc=cb, be=eb, cd=dc, ce=ec, ede=d5 >

Subgroups: 348 in 158 conjugacy classes, 50 normal (26 characteristic)
C1, C2 [×3], C2 [×6], C4 [×2], C4 [×7], C22 [×3], C22 [×20], C8 [×4], C2×C4 [×2], C2×C4 [×4], C2×C4 [×19], D4 [×8], C23, C23 [×4], C23 [×8], C42 [×2], C22⋊C4 [×4], C4⋊C4 [×2], C2×C8 [×4], M4(2) [×4], C22×C4 [×5], C22×C4 [×4], C22×C4 [×6], C2×D4 [×6], C24 [×2], C22⋊C8 [×4], C22⋊C8 [×2], C2×C42, C2×C22⋊C4 [×2], C2×C4⋊C4, C4×D4 [×4], C2×M4(2) [×2], C23×C4 [×2], C22×D4, C23⋊C8 [×2], C22.M4(2) [×2], C24.4C4 [×2], C2×C4×D4, C23⋊M4(2)
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, C22⋊C4 [×4], M4(2) [×4], C22×C4, C2×D4 [×2], C23⋊C4 [×2], C2×C22⋊C4, C2×M4(2) [×2], C24.4C4, C2×C23⋊C4, M4(2).8C22, C23⋊M4(2)

Smallest permutation representation of C23⋊M4(2)
On 32 points
Generators in S32
(1 5)(2 6)(3 29)(4 30)(7 25)(8 26)(9 19)(10 20)(11 15)(12 16)(13 23)(14 24)(17 21)(18 22)(27 31)(28 32)
(2 32)(4 26)(6 28)(8 30)(10 24)(12 18)(14 20)(16 22)
(1 31)(2 32)(3 25)(4 26)(5 27)(6 28)(7 29)(8 30)(9 23)(10 24)(11 17)(12 18)(13 19)(14 20)(15 21)(16 22)
(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)
(1 13)(2 10)(3 15)(4 12)(5 9)(6 14)(7 11)(8 16)(17 29)(18 26)(19 31)(20 28)(21 25)(22 30)(23 27)(24 32)

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

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

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

32 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I4A···4H4I···4N8A···8H
order12222222224···44···48···8
size11112244442···24···48···8

32 irreducible representations

dim11111111122244
type+++++++
imageC1C2C2C2C2C4C4C4C4D4M4(2)M4(2)C23⋊C4M4(2).8C22
kernelC23⋊M4(2)C23⋊C8C22.M4(2)C24.4C4C2×C4×D4C2×C42C2×C22⋊C4C23×C4C22×D4C22×C4C2×C4C23C4C2
# reps12221222244422

Matrix representation of C23⋊M4(2) in GL6(𝔽17)

1620000
010000
001000
0001600
0001110
0080016
,
1600000
0160000
001000
000100
00126160
00810016
,
100000
010000
0016000
0001600
0000160
0000016
,
1120000
1160000
009702
0051120
00107612
00107108
,
1620000
010000
0001600
0016000
00116016
00116160

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,2,1,0,0,0,0,0,0,1,0,0,8,0,0,0,16,11,0,0,0,0,0,1,0,0,0,0,0,0,16],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,12,8,0,0,0,1,6,10,0,0,0,0,16,0,0,0,0,0,0,16],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[1,1,0,0,0,0,12,16,0,0,0,0,0,0,9,5,10,10,0,0,7,11,7,7,0,0,0,2,6,10,0,0,2,0,12,8],[16,0,0,0,0,0,2,1,0,0,0,0,0,0,0,16,1,1,0,0,16,0,16,16,0,0,0,0,0,16,0,0,0,0,16,0] >;

C23⋊M4(2) in GAP, Magma, Sage, TeX

C_2^3\rtimes M_4(2)
% in TeX

G:=Group("C2^3:M4(2)");
// GroupNames label

G:=SmallGroup(128,197);
// by ID

G=gap.SmallGroup(128,197);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,1430,352,1123,851,172]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^8=e^2=1,d*a*d^-1=a*b=b*a,e*a*e=a*c=c*a,d*b*d^-1=b*c=c*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^5>;
// generators/relations

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