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G = M4(2).24D4order 128 = 27

5th non-split extension by M4(2) of D4 acting via D4/C4=C2

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

Aliases: M4(2).24D4, C4.70(C4×D4), D4⋊C48C4, Q8⋊C48C4, (C2×C8).328D4, C2.16(C8○D8), C426C427C2, C22.166(C4×D4), C2.16(C8.26D4), C4.193(C4⋊D4), C82M4(2)⋊24C2, C4.40(C42⋊C2), C23.206(C4○D4), (C2×C42).300C22, (C22×C8).394C22, C23.24D4.6C2, C22.6(C4.4D4), C22.7C4225C2, (C22×C4).1383C23, C42⋊C2.272C22, C22.15(C422C2), C4.142(C22.D4), (C2×M4(2)).321C22, C2.14(C24.C22), (C2×C8).43(C2×C4), (C2×C4≀C2).10C2, (C2×C8.C4)⋊9C2, C4⋊C4.153(C2×C4), (C2×D4).96(C2×C4), (C2×Q8).81(C2×C4), (C2×C4).1537(C2×D4), (C2×C4).761(C4○D4), (C2×C4).401(C22×C4), (C2×C4○D4).33C22, (C22×C8)⋊C2.15C2, SmallGroup(128,661)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — M4(2).24D4
C1C2C4C2×C4C22×C4C42⋊C2C82M4(2) — M4(2).24D4
C1C2C2×C4 — M4(2).24D4
C1C2×C4C22×C8 — M4(2).24D4
C1C2C2C22×C4 — M4(2).24D4

Generators and relations for M4(2).24D4
 G = < a,b,c,d | a8=b2=d2=1, c4=a4, bab=a5, ac=ca, dad=a-1b, bc=cb, dbd=a4b, dcd=a6c3 >

Subgroups: 212 in 112 conjugacy classes, 48 normal (46 characteristic)
C1, C2 [×3], C2 [×3], C4 [×4], C4 [×5], C22 [×3], C22 [×5], C8 [×7], C2×C4 [×6], C2×C4 [×9], D4 [×4], Q8 [×2], C23, C23, C42 [×3], C22⋊C4, C4⋊C4 [×2], C2×C8 [×4], C2×C8 [×6], M4(2) [×2], M4(2) [×3], C22×C4, C22×C4 [×2], C2×D4, C2×D4, C2×Q8, C4○D4 [×4], C4×C8, C8⋊C4, C22⋊C8 [×2], D4⋊C4 [×2], Q8⋊C4 [×2], C4≀C2 [×2], C8.C4 [×2], C2×C42, C42⋊C2, C22×C8 [×2], C2×M4(2) [×2], C2×C4○D4, C22.7C42, C426C4, C82M4(2), (C22×C8)⋊C2, C23.24D4, C2×C4≀C2, C2×C8.C4, M4(2).24D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, C22×C4, C2×D4 [×2], C4○D4 [×4], C42⋊C2, C4×D4 [×2], C4⋊D4, C22.D4, C4.4D4, C422C2, C24.C22, C8○D8, C8.26D4, M4(2).24D4

Smallest permutation representation of M4(2).24D4
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)
(1 5)(3 7)(10 14)(12 16)(18 22)(20 24)(25 29)(27 31)
(1 12 27 22 5 16 31 18)(2 13 28 23 6 9 32 19)(3 14 29 24 7 10 25 20)(4 15 30 17 8 11 26 21)
(1 32)(2 27)(3 26)(4 29)(5 28)(6 31)(7 30)(8 25)(9 10)(11 12)(13 14)(15 16)(17 18)(19 20)(21 22)(23 24)

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

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

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

38 conjugacy classes

class 1 2A2B2C2D2E2F4A4B4C4D4E4F4G···4N4O8A8B8C8D8E···8N8O8P
order12222224444444···4488888···888
size11112281111224···4822224···488

38 irreducible representations

dim1111111111222224
type++++++++++
imageC1C2C2C2C2C2C2C2C4C4D4D4C4○D4C4○D4C8○D8C8.26D4
kernelM4(2).24D4C22.7C42C426C4C82M4(2)(C22×C8)⋊C2C23.24D4C2×C4≀C2C2×C8.C4D4⋊C4Q8⋊C4C2×C8M4(2)C2×C4C23C2C2
# reps1111111144226282

Matrix representation of M4(2).24D4 in GL4(𝔽17) generated by

1000
0100
00015
0020
,
1000
0100
00160
0001
,
0100
16000
0080
0008
,
1000
01600
00015
0080
G:=sub<GL(4,GF(17))| [1,0,0,0,0,1,0,0,0,0,0,2,0,0,15,0],[1,0,0,0,0,1,0,0,0,0,16,0,0,0,0,1],[0,16,0,0,1,0,0,0,0,0,8,0,0,0,0,8],[1,0,0,0,0,16,0,0,0,0,0,8,0,0,15,0] >;

M4(2).24D4 in GAP, Magma, Sage, TeX

M_4(2)._{24}D_4
% in TeX

G:=Group("M4(2).24D4");
// GroupNames label

G:=SmallGroup(128,661);
// by ID

G=gap.SmallGroup(128,661);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,288,422,58,2804,718,172,124]);
// Polycyclic

G:=Group<a,b,c,d|a^8=b^2=d^2=1,c^4=a^4,b*a*b=a^5,a*c=c*a,d*a*d=a^-1*b,b*c=c*b,d*b*d=a^4*b,d*c*d=a^6*c^3>;
// generators/relations

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